This book is DUE on last date stamped below SOUTHERN BRANCH, OF C7VLIF0RNIA, HISTORY OF THE THEORY OF NUMBERS VOLUME I DIVISIBILITY AND PRIMALITY By Leonard Eugene Dickson Professor of Mathematics in the University of Chicago Published by the Carnegie Institution of Washington Washington, 1919 / 1 i 5 'J CARNEGIE INSTITUTION OF WASHINGTON Publication No. 256, Vol. I PRESS OF GIBSON BROTHERS WASHINGTON. D. C. 'fl i ^ r^ , _ &!£:: steering* '^vi-^ r7' Mathenrstical 2^H-j Sciences :D/^ifii PREFACE. libraiy The efforts of Cantor and his collaborators show that a chronological history of mathematics down to the nineteenth century can be written in four large volumes. To cover the last century with the same elaborateness, it has been estimated that about fifteen volumes would be required, so extensive is the mathematical literature of that period. But to retain the chronological order and hence devote a large volume to a period of at most seven years would defeat some of the chief purposes of a history, besides making it very inconvenient to find all of the material on a particular topic. In any event there is certainly need of histories which treat of particular branches of mathematics up to the present time. The theory of numbers is especially entitled to a separate history on account of the great interest which has been taken in it continuously through the centuries from the time of Pythagoras, an interest shared on the one extreme by nearly every noted mathematician and on the other extreme ^ by numerous amateurs attracted by no other part of mathematics. This v history aims to give an adequate account of the entire literature of the \ theory of numbers. The first volume presents in twenty chapters the material relating to divisibility and primality. The concepts, results, and Jl authors cited are so numerous that it seems appropriate to present here an introduction which gives for certain chapters an account in untechnical language of the main results in their historical setting, and for the remaining • chapters the few remarks sufficient to clearly characterize the nature of their v^, contents. J' ' ' Perfect numbers have engaged the attention of arithmeticians of every *»>• century of the Christian era. It was while investigating them that Fermat discovered the theorem which bears his name and which forms the basis of a large part of the theory of numbers. A_perfect number is one, like 6 = 1+2+3, which equals the sum of its divisors other than itself . Euclid ,. proved that 2^~'^{2^ — \) is a perfect numbeflf 2^ — 1 is a prime. For p = 2, 3, 5, 7, the values 3, 7, 31, 127 of 2''-l are primes, so that 6, 28, 496, 8128 are perfect numbers, as noted by Nicomachus (about A. D. 100). A manu- script dated 1456 correctly gave 33550336 as the fifth perfect number; it cor- * ! responds to the value 13 of p. Very many early writers believed that 2^ — 1 I is a prime for every odd value of p. But in 1536 Regius noted that 2^-1 = 511 = 7-73, 211-1=2047 = 23-89 are not primes and gave the above fifth perfect number. Cataldi, who founded at Bologna the most ancient known academy of mathematics, IV PREFACE. noted in 1603 that 2'' — 1 is composite if p is composite and verified that it is a prime for p = 13, 17, and 19; but he erred in stating that it is also a prime for p = 23, 29, and 37. In fact, Fermat noted in 1640 that 2'-'-l has the factor 47, and 2^'-l the factor 223, while Euler observed in 1732 that 2'' — 1 has the factor 1103. Of historical importance is the statement made by Mersenne in 1644 that the first eleven perfect numbers are given by 2P-i(2P_i) for p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257; but he erred at least in including 67 and excluding 61, 89, and 107. That 2" — 1 is com- posite was proved by Lucas in 1876, while its actual factors were found by Cole in 1903. The primality of 2^^ — 1, a number of 19 digits, was estab- lished by Pervusin in 1883, Seelhoff in 1886, and Hudelot m 1887. Both Powers and Fauquembergue proved in 1911-14 that 2^^ — 1 and 2^°^ — 1 are primes. The primality of 2'^ — 1 and 2™ — 1 had been estabhshed by Euler and Lucas respectively. Thus 2^— 1 is known to be a prime, and hence lead to a perfect number, for the twelve values 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107 and 127 of p. Since 2^' — 1 is known (pp. 15-31) to be composite for 32 primes p ^257, only the eleven values p = 137, 139, 149, 157, 167, 193, 199, 227, 229, 241, 257 now remain in doubt. Descartes stated in 1638 that he could prove that every even perfect number is of Euclid's type and that every odd perfect number must be of the form ps^, where p is a prime. Euler's proofs (p. 19) were published after his death. Xd. immediate proof of the former fact was given by Dickson (p. 30). According to Sylvester (pp. 26-27), there exists no odd perfect number with fewer than six distinct prime factors, and none with fewer than eight if not divisible by 3. But the question of the existence of odd perfect numbers remains unanswered. A multiply perfect number, like 120 and 672, is one the sum of whose divisors equals a multiple of the number. They were actively investigated during the years 1631-1647 by IMersenne, Fermat, St. Croix, Frenicle, and Descartes. Many new examples hav^e been found recently by American writers. Two numbers are called amicable if each equals the sum of the aliquot divisors of the other, where an aliquot divisor of a number means a divisor other than the number itself. The pair 220 and 284 was known to the Pythagoreans. In the ninth century, the Arab Thabit ben Korrah noted that 2"/!« and 2"s are amicable numbers if /j=3-2''-l, t = 2>'2'^^-l and s = 9.22"-! _i are all primes, and n> 1. This result leads to amicable numbers for n = 2 (giving the above pair), n = 4 and n = 7, but for no further value ^ 200 of n. The chief investigation of amicable numbers is that by Euler who listed (pp. 45, 46) 62 pairs. At the age of 16, Paganini announced in 1866 the remarkable new pair 1184 and 1210. A few new pairs of very large numbers have been found by Legendre, Seelhoff, and Dickson. PREFACE. V Interesting amicable triples and amicable numbers of higher order have been recently found by Dickson and Poulet (p. 50). Although it had been employed in the study of perfect and amicable numbers, the explicit expression for the sum a{n) of all the divisors of n is reserved for Chapter II, in which is presented the history of Fermat's two problems to solve (T(x^)=y^ and <t{x^) =y^ and John Wallis's problem to find solutions other than a; = 4 and y = 5 oi (T{x^)=o-{y^). Fermat stated in 1640 that he had a proof of the fact, now known as Fermat's theorem, that, if p is any prime and x is any integer not divisible by p, then x^~^ — l is divisible by p. This is one of the fundamental theo- rems of the theory of numbers. The case x = 2 was known to the Chinese as early as 500 B. C. The first published proof was given by Euler in 1736. Of first importance is the generalization from the case of a prime p to any integer n, published by Euler in 1760: if (/)(n) denotes the munber of positive integers not exceeding n and relatively prime to n, then x*^"^ — 1 is divisible . by n for every integer x relatively prime to n. Another elegant theorem states that, if p is a prime, l+jl-2-3. . . .{p — l)\ is divisible by p; it was first pubUshed by Waring in 1770, who ascribed it to Sir John Wilson. This theorem was stated at an earlier date in a manuscript by Leibniz, who with Newton discovered the calculus. But Lagrange was the first one to publish (in 1773) a proof of Wilson's theorem and to observe that its converse is true. In 1801 Gauss stated and suggested methods to prove the generali- zation of Wilson's theorem: if P denotes the product of the positive integers less than A and prime to A, then P+1 is divisible by A if A =4, p"" or 2p"*, where p is an odd prime, while P — 1 is divisible by A if A is not of one of these three forms. A very large number of proofs of the preceding theorems are given in the first part of Chapter III. Various generalizations are then presented (pp. 84-91). For instance, if iV = p/' . . . p/*, where Pi, ..., p« are distinct primes, a^-(a^/P'+ . . . +a^/PO + (a^/P'P'+ ...)-••• . +(-l)''a^/^---P'' is divisible by N, a fact due to Gauss for the case in which a is a prime. Many cases have been found in which o"~^ — 1 is divisible by n for a composite number n. But Lucas proved the following converse of Fermat's theorem : if a^ — 1 is divisible by n when x = n — l, but not when x is a divisor |<n — 1 of 71 — 1, then w is a prime. Any integral symmetric function of degree d of 1, 2, . . ., p — 1 with I integral coefficients is divisible by the prime p if c^ is not a multiple of p — 1. A generalization to the case of a divisor p" is due to Meyer (p. 101) . Nielsen proved in 1893 that, if p is an odd prime and if k is odd and l<fc<p — 1, the sum of the products of 1, 2, . . ., p — 1 taken A; at a time is divisible by p^. Taking fc = p — 2, we see that if p is a prime > 3 the numerator of the fraction A VI PREFACE. equal to 1 + 1/2+1/3+ . . . +l/(p — 1) is divisible by p'^, a result first proved by Wolstenholme in 1862. Sylvester stated in 1866 that the sum of all products of n distinct numbers chosen from 1, 2, . . . , w is divisible by each prime > n + 1 which is contained in any term of the set m — n + 1 , . . . , w, m + 1 . There are various theorems analogous to these. In Chapter IV are given properties of the quotient {uP~^ — l)/p, which plays an important role in recent investigations on Fermat's last theorem (the impossibiUty of x'^-\-y^ = z^ if p>2), the history of which will be treated in the final chapter of Volume II. Some of the present papers relate to (w*^"^ — 1)/«, where n is not necessarily a prime. TMiile Euler's ^-function was defined above in order to state his general- ization of Fermat's theorem, its numerous properties and generalizations are reserved for the long Chapter V. In 1801 Gauss gave the result that 4>{d^ + . . . -\-<i>{dk) = n, if di, . . . , d^ are the divisors of n; this was generalized by Laguerre in 1872, H. G. Cantor in 1880, Busche in 1888, Zsigmondy in 1893, Vahlen in 1895, Elliott in 1901, and Hammond in 1916. In 1808 Legendre proved a simple formula for the number of integers ^ n which are divisible by no one of any given set of primes. The asymptotic value of (f>{l)-\- . . . +0(G) for G large was discussed by Dirichlet in 1849, Mertens in 1874, Perott in 1881, Sylvester in 1883 and 1897, Cesaro in 1883 and 1886-8, Berger in 1891, and Kronecker in 1901. The solution of 4>{x)=g was treated by Cayley in 1857, Mmin in 1897, Pichler in 1900, Carmichael in 1907-9, Ranum in 1908, and Cunningham in 1915. H. J. S. Smith proved in 1875 that the m-rowed determinant, ha\ing as the element in the ith row and ji\i column any function fib) of the greatest common divisor 5 of i and j, equals the product of F{\), F{2),. . ., F(m), where F(m)=/(m)-2/g)+2/(^J-...., m = py In particular, F{m)=<t>{m) if f{8)=8. In several papers (pp. 128-130) Cesaro considered analogous determinants. The fact that 30 is the largest number such that all smaller numbers relatively prime to it are primes was first proved by Schatunowsky in 1893. A. Thacker in 1850 evaluated the sum 4>k{n) of the kth. powers of the integers ^n which are prime to n. His formula has been expressed m^ various symbolic forms by Ces^o and generalized by Glaisher and Nielsen./ Crelle had noted in 1845 that <piin) = |n0( n). In 1869 Schemmel considered the number of sets of n consecutive integers each < m and prime to m. In connection with linear congruence groups, Jordan evaluated the number of different sets of k positive integers ^?i whose greatest common divisor is prime to n. This generalization of Euler's (^-function has properties as simple as the latter function and occurs in many papers under a variety of notations. It in turn has been generalized (pp. 151-4). PREFACE. VII The properties of the set of all irreducible fractions, arranged in order of magnitude, whose numerators are ^ m and denominators are ^ n (called a Farey series if m = n), have been discussed by many writers and applied to the approximation of numbers, to binary quadratic forms, to the composi- tion of linear fractional substitutions, and to geometry (pp. 155-8). Some of the properties of periodic decimal fractions are already familiar tq the reader in view of his study of arithmetic and the chapter of alge- bra dealing with the sum to infinity of a geometric progression. For the generalization to periodic fractions to any base h, not necessarily 10, the length of the period of the periodic fraction for 1/d, where d is prime to h, is the least positive exponent e such that h^ — \ is divisible by d. Hence this Chapter VI, which reports upon more than 160 papers, is closely related to the following chapter and furnishes a concrete introduction to it. The subject of exponents and primitive roots is one of the important topics of the theory of numbers. To present the definitions in the customary, compact language, we shall need the notion of congruence. If the differ- ence of two integers a and 6 is divisible by m, they are called congruent modulo m and we write a=& (mod m). For example, 8=2 (mod 6). If n'= 1 (mod m), but n*^ 1 (mod m) for 0<s<e, we say that n belongs to the exponent e modulo m. For example, 2 and 3 belong to the exponent 4 modulo 5, while 4 belongs to the exponent 2. In view of Euler's generaliza- tion of Fermat's theorem, stated above, e never exceeds 0(m). If n belongs to this maximum exponent ^(n) modulo m, n is called a primitive root of m. For example, 2 and 3 are primitive roots of 5, while 1 and 4 are not. Lam- bert stated in 1769 that there exists a primitive root of any prime p, and Euler gave a defective proof in 1773. In 1785 Legendre proved that there are exactly 4>{e) numbers belonging modulo p to any exponent e which divides p — 1. In 1801 Gauss proved that there exist primitive roots of m if and only if m = 2, 4, p* or 2p*, where p is an odd prime. In particular, for a primitive root a of a prime modulus p and any integer N not divisible by p, there is an exponent ind N, called the index of N by Gauss, such that N=a''"^^ (mod p). Indices play a role similar to logarithms, but we re- quire two companion tables for each modulus p. The extension to a power of prime modulus is immediate. For a general modulus, systems of indices were employed by Dirichlet in 1837 and 1863 and by Kronecker in 1870. Jacobi's Canon Arithmeticus of 1839 gives companion tables of indices for each prime and power of a prime < 1000. Cunningham's Binary Canon of 1900 gives the residues of the successive powers of 2 when divided by each prime or power of a prime < 1000 and companion tables showing the powers of 2 whose residues are 1, 2, 3, . . .. In 1846 Arndt proved that, if ^ is a primitive root of the odd prime p, g belongs to the exponent p"~"^(p — 1) modulo p'* if and only ii G = g^~^ — 1 is divisible by p^, but not by p^'^\ where VIII PREFACE. X<n; taking X= 1, we see that, if G is not divisible by p^, g' is a primitive root of p^ and of all higher powers of p. This Chapter VII presents many more theorems on exponents, primitive roots, and binomial congruences, and cites various lists of primitive roots of primes < 10000. Lagrange proved easily that a congruence of degree n has at most n roots if the modulus is a prime. Lebesgue found the number of sets of solutions of 01^1"*+ • • • -\-akXk"=a (mod p), when p is a prime such that p — 1 is divisible by m. Konig (p. 226) employed a cyclic determinant and its minors to find the exact number of real roots of any congruence in one unknown; Gegen- bauer (p. 228) and Rados (p. 233) gave generalizations to congruences in several unknowns. Galois's introduction of imaginary roots of congruences has not only led to an important extension of the theory of numbers, but has given rise to wide generalizations of theorems which had been obtained in subjects like linear congruence groups by applying the ordinary theory of numbers. Instead of the residues of integers modulo p, let us consider the residues of polynomials in a variable x with integral coefficients with respect to two moduH, one being a prime p and the other a polynomial f{x) of degree n which is irreducible modulo p. The residues are the p" polynomials in x of degree n — 1 whose coefficients are chosen from the set 0, 1, . . . , p — 1 . These residues form a Galois field within which can be performed addition, sub- traction, multiplication, and division (except by zero) . As a generahzation of Fermat's theorem, Galois proved that the power p" — 1 of any residue except zero is congruent to unity with respect to our pair of moduli p and f{x). He avoided our second modulus f{x) by introducing an undefined imaginary root i of f{x) = 0 (mod p) and considering the residues modulo p of polynomials in i; but the above use of the two moduH affords the only logical basis of the theory. In view of the fullness of the reports in the text (pp. 233-252) of the papers on this subject, further comments here are unnecessary. The final topics of this long Chapter VIII are cubic congru- ences and miscellaneous results on congruences and possess little general interest. In Chapter IX are given Legendre's expression for the exponent of the highest power of a prime p which divides the factorial 1-2. . .m, and the generalization to the product of any integers in arithmetical progression; many theorems on the divisibility of one product of factorials by another product and on the residues of multinomial coefficients ; various determina- tions of the sign in 1-2... (p — l)/2==tl (mod p); and miscellaneous congruences involving factorials. In the extensive Chapter X are given many theorems and formulas concerning the sum of the kth. powers of all the divisors of n, or of its even or odd divisors, or of its divisors which are exact sth powers, or of those divisors PREFACE. IX whose complementary divisors are even or odd or are exact sth powers, and the excess of the sum of the A;th powers of the divisors of the form 4m +1 of a number over the sum of the A;th powers of the divisors of the form 4m -f 3, as well as more technical sums of divisors defined on pages 297, 301-2, 305, 307-8, 314-5 and 318. For the important case k = 0, such a sum becomes the number of the divisors in question. There are theorems on the number of sets of positive integral solutions of UiU^. . .Uk = n or of x''y^ = n. Also Glaisher's cancellation theorems on the actual divisors of numbers (pp. ^' 310-11, 320-21). Scattered through the chapter are approximation and asymptotic formulas involving some of the above functions. In Chapter XI occur Dirichlet's theorem on the number of cases in the division of n by 1, 2, . . . , p in turn in which the ratio of the remainder to the divisor is less than a given proper fraction, and the generalizations on pp. 330-1; theorems on the number of integers ^n which are divisible by no ,( exact sth power > 1 ; theorems on the greatest divisor which is odd or has / specified properties; many theorems on greatest coromon divisor and least I common multiple ; and various theorems on mean values and probability. ' The casting out of nines or of multiples of 11 or 7 to check arithmetical computations is of early origin. This topic and the related one of testing the divisibility of one number by another have given rise to the numerous elementary papers cited in Chapter XII. The frequent need of the factors of numbers and the excessive labor required for their direct determination have combined to inspire the construction of factor tables of continually increasing limit. The usual method is essentially that given by Eratosthenes in the third century B. C. A special method is used by Lebon (pp. 355-6). Attention is called to Lehmer's Factor Table for the First Ten Millions and his List of Prime Numbers from 1 to 10,006,721, published in 1909 and 1914 by the Carnegie Institution of Washington. Since these tables were constructed anew with the greatest care and all variations from the chief former tables were taken account of, they are certainly the most accurate tables extant. Absolute accuracy is here more essential than in ordinary tables of continuous func- tions. Besides giving the history of factor tables and lists of primes, this Chapter XIII cites papers which enumerate the primes in various intervals, prime pairs (as 11, 13), primes of the form 4n+l, and papers listing primes written to be base 2 or large primes. Chapter XIV cites the papers on factoring a number by expressing it as a difference of two squares, or as a sum of two squares in two ways, or by use of binary quadratic forms, the final digits, continued fractions. Pell equa- tions, various small moduli, or miscellaneous methods. Fermat expressed his belief that Fn = 2^"+l is a prime for every value of n. While this is true if n = 1, 2, 3, 4, it fails forn = 5 as noted by Euler. Later, X PREFACE. Gauss proved that a regular polygon of m sides can be constructed by ruler and compasses if m is a product of a power of 2 and distinct odd primes each of the form Fn, and stated correctly that the construction is impossible if m is not such a product. In view of the papers cited in Chapter XV, F„ is composite if n = 5, 6, 7, 8, 9, 11, 12, 18, 23, 36, 38 and 73, while nothing is known for other values >4 of n. No conoment will be made on the next chapter which treats of the factors of numbers of the form o"±6" and of certain trinomials. In Chapter XVII are treated questions on the divisors of terms of a recurring series and in particular of Lucas' functions a — o where a and h are roots oi x'^ — Px-\-Q = Q, P and Q being relatively prime integers. By use of these functions, Lucas obtained an extension of Euler's generaUzation of Fermat's theorem, which requires the correction noted by Carmichael (p. 406), as well as various tests for primality, some of which have been emploj^ed in investigations on perfect numbers. Many papers on the algebraic theory of recurring series are cited at the end of the chapter. Euchd gave a simple and elegant proof that the number of primes is infi- nite. For the generalization that every arithmetical progression n, n+m, n-\-2m,. . ., in which n and m are relatively prime, contains an infinitude of primes, Legendre offered an insufficient proof, while Dirichlet gave his classic proof by means of infinite series and the classes of binary quadratic forms, and extended the theorem to complex integers. Mertens and others obtained simpler proofs. For various special arithmetical progressions, the theorem has been proved in elementary ways by many writers. Dirichlet also obtained the theorems that, if a, 26, and c have no common factor, ax'^+2hxy-\-cy^ represents an infinitude of primes, while an infinitude of these primes are representable by any given linear form Mx+N with M and N relatively prime, pro\^ded a, h, c, M, N are such that the quadratic and linear forms can represent the same number. No complete proof has been found for Goldbach's conjecture in 1742 that every even integer is a sum of two primes. One of various analogous un- proved conjectures is that every even integer is the difference of two consec- utive primes in an infinitude of ways (in particular, there exists an infinitude of pairs of primes differing by 2). No comment will be made on the further topics of this Chapter XVIII: polynomials representing numerous primes, primes in arithmetical progression, tests for primality, number of primes between assigned limits, Bertrand's postulate of the existence of at least one prime between x and 2x — 2 for x>3, miscellaneous results on primes, diatomic series, and asymptotic distribution of primes. PREFACE. XI If F(m)=2/(d), summed for all the divisors d of m, we can express /(m) in terms of F by an inversion formula given in Chapter XIX along with generalizations and related formulas. Bougaief called F{m) the numerical integral of /(m). The final Chapter XX gives many elementary results involving the digits of numbers mainly when written to the base 10. Since the history of each main topic is given separately, it has been possible without causing confusion to include reports on minor papers and isolated problems for the sake of completeness. In the cases of books and journals not usually accessible, the reports are quite full with some indication of the proofs. In other cases, proofs are given only when necessary to differentiate the paper from others deriving the same result. The references were selected mainly from the Subject Index of the Royal Society of London Catalogue of Scientific Papers, volume 1, 1908 (with which also the proof-sheets were checked), and the supplementary annual volumes forming the International Catalogue of Scientific Literature, Jahrbuch iiber die Fortschritte der Mathematik, Revue semestrielle des publications math^matiques, Poggendorff's Handworterbuch, Kliigel's Mathematische Worterbuch, Wolffing's Mathematischer Biicherschatz (a list of mathemat- ical books and pamphlets of the nineteenth century), historical journals, such as Bulletino di bibliografia e di storia delle scienze matematiche e fisiche, Bolletino . . . . , BibUotheca Mathematica, Abhandlungen zur Geschichte der mathematischen Wissenschaften, various histories and encyclopedias, including the Enclyclop^die des sciences mathematiques. Further, the author made a direct examination at the stacks of books and old journals in the libraries of Chicago, California, and Cambridge Universities, and Trinity College, Cambridge, and the excellent John Crerar Library at Chi- cago. He made use of G. A. Plimpton's remarkable collection, in New York, of rare books and manuscripts. In 1912 the author made an extended investigation in the libraries of the British Museum, Kensington Museum, Royal Society, Cambridge Philosophical Society, Bibliotheque Nationale, Universite de Paris, St. Genevieve, I'lnstitut de France, Uni- versity of Gottingen, and the Konigliche Bibhothek of Berlin (where there is a separate index of the material on the theory of numbers). Many books have since been borrowed from various libraries; the Ladies' and other Diaries were loaned by R. C. Archibald. At the end of the volume is a separate index of authors for each of the twenty chapters, which will facilitate the tracing of the relation of a paper to kindred papers and hence will be of special service in the case of papers inaccessible to the reader. The concluding volume will have a combined index of authors from which will be omitted minor citations found in the chapter indices. XII PREFACE. The subject index contains a list of symbols; while [x] usually denotes the greatest integer ^x, occasionally such square brackets are used to inclose an addition to a quotation. The symbol * before an author's name signifies that his paper was not available for report. The symbol f before a date signifies date of death. Initials are given only in the first of several immediately successive citations of an author. Although those volumes of Euler's Opera Omnia which contain his Com- mentationes Arithmeticae CoUectse have been printed, they are not yet available; a table showing the pages of the Opera and the corresponding pages in the present volume of this history will be given in the concluding volume. The author is under great obligations to the following experts in the theory of numbers for numerous improvements resulting from their reading the initial page proofs of this volume: R. D. Carmichael, L. Chanzy, A. Cunningham, E. B. Escott, A. Gerardin, A. J. Kempner, D. N. Lehmer, E. Maillet, L. S. Shively, and H. J. Woodall; also the benefit of D. E. Smith's accurate and extensive acquaintance with early books and writers was for- tunately secured ; and the author's special thanl<:s are due to Carmichael and Kempner, who read the final page proofs with the same critical attention as the initial page proofs and pointed out various errors and obscurities. To these eleven men who gave so generously of their time to perfect this volume, and especially to the last two, is due the gratitude of every devotee of number theory who may derive benefit or pleasure from this history. In return, such readers are requested to further increase the usefulness of this work by sending corrections, notices of omissions, and abstracts of papers marked not available for report, for insertion in the concluding volume. Finally, this laborious project would doubtless have been abandoned soon after its inception seven years ago had not President Woodward approved it so spontaneously, urged its completion with the greatest thoroughness, and given continued encouragement. L. E. Dickson. November, 1918. ^ TABLE OF CONTENTS. Chapter. page. I. Perfect, multiply perfect, and amicable numbers 3 \ 11. Formulas for the number and simi of divisors, problems of Fermat and Wallis 51 III. Fermat's and Wilson's theorems, generalizations and converses; symmetric functions of 1, 2, . . . , p— 1, modulo p 59 IV. Residue of (wp~^ — l)/p modulo p 105 V. Euler's (^function, generalizations; Farey series 113 VI. Periodic decimal fractions; periodic fractions; factors of 10" =•=!... . 159 VII. Primitive roots, exponents, indices, binomial congruences 181 VIII. Higher congruences 223 IX. Divisibility of factorials and multinomial coefficients 263 X. Sum and number of divisors 279 XI. Miscellaneous theorems on divisibility, greatest common divisor, least common multiple 327 XII. Criteria for divisibility by a given number 337 XIII. Factor tables, lists of primes 347 v^IV. Methods of factoring 357 • XV. Fermat numbers F„ = 22"+l 375 XVI. Factors of a"±6« 381 XVII. Recurring series; Lucas' Un, Vn 393 ^VIII. Theory of prime numbers 413 XIX. Inversion of functions; Mobius' function ix{n); numerical integrals and derivatives 441 XX. Properties of the digits of numbers 453 Author index 467 Subject index 484 1 (. CHAPTER I. PERFECT. MULTIPLY PERFECT. AND AMICABLE NUMBERS. Perfect, Abundant, and Deficient Numbers. By the aliquot parts or divisors of a number are meant the divisors, including unity, which are less than the number. A number, like 6 = 1 -h 2+3, which equals the sum of its aliquot divisors is called perfect (voll- kommen, vollstandig) . If the sum of the aliquot divisors is less than the number, as is the case with 8, the number is called deficient (diminute, defective, unvollkommen, unvollstandig, mangelhaft). If the sum of the aliquot divisors exceeds the number, as is the case with 12, the number is called abundant (superfluos, plus quam-perfectus, redundantem, exc^dant, iibervollstandig, iiberflussig, iiberschiessende) . Euclid^ proved that, if p = 1+2+2^+ • • • +2" is a prime, 2"p is a perfect number. He showed that 2"p is divisible by 1, 2, . . . , 2", p, 2p, . . . , 2'*~^p, but by no further number less than itself. By the usual theorem on geometrical progressions, he showed that the sum of these divisors is 2"^. The early Hebrews^" considered 6 to be a perfect number. Philo Judeus^'' (first century A. D.) regarded 6 as the most productive of all numbers, being the first perfect number. Nicomachus^ (about A. D. 100) separated the even numbers (book I, chaps. 14, 15) into abundant (citing 12, 24), deficient (citing 8, 14), and perfect, and dwelled on the ethical import of the three types. The perfect (I, 16) are between excess and deficiency, as consonant sound between acuter and graver sounds. Perfect numbers will be found few and arranged with fitting order; 6, 28, 496, 8128 are the only perfect numbers in the respective intervals between 1, 10, 100, 1000, 10000, and they have the property of ending alternately in 6 and 8. He stated that Euclid's rule gives all the perfect numbers without exception. Theon of Smyrna^ (about A. D. 130) distinguished between perfect (citing 6, 28), abundant (citing 12) and deficient (citing 8) numbers. ^Elementa, liber IX, prop. 36. Opera, 2, Leipzig, 1884, 408. ^"S. Rubin, "Sod Hasfiroth" (secrets of numbers), Wien, 1873, 59; citation of the Bible, Kings, II, 13, 19. **Treatise on the account of the creation of the world as given by Moses, C. D. Young's transl. of Philo's works, London, 1854, vol. 1, p. 3. 'Nicomachi Gerasini arithmeticse Ubri duo. Nunc primdm typis excusi, in lucem eduntur. Parisiis, 1538. In officina Christian! WecheU. (Greek.) Theologumena arithmeticae. Accedit Nicomachi Gerasini institutio arithmetica ad fidem codicum Monacensium emendata. Ed., Fridericus Astius. Lipsiae, 1817. (Greek.) Nicomachi Geraseni Pythagorei introductionis arithmeticae libri ii. Recensvit Ricardus Hoche. Lipsiae, 1866. (Greek.) 'Theonis Smymaei philosophi Platonici expositio rerum mathematicarum ad legendum Platonem utiHum. Ed., Ed. Hiller, Leipzig, 1878, p. 45. Theonis Smymaei Platonici, Latin by Ismaele BuUialdo. Paris, 1644, chap. 32, pp. 70-72. 3 4 History of the Theory of Numbers. [Chap. I lamblichus* (about 283-330) repeated in effect the remarks by Nico- machus on perfect, abundant, and deficient numbers, but made erroneous additions. He stated that there is one and but one perfect number in the successive intervals between 1, 10, 100,..., 100000, etc., to infinity. "Examples of a perfect number are 6, and 28, and 496, and 8128, and the like numbers, alternately ending in 6 and 8." He remarked that the Pythag- oreans called the perfect number 6 marriage, and also health and beauty (on account of the integrity of its parts and the agreement existing in it). Aurelius Augustinus^ (354-430) remarked that, 6 being the first perfect number, God effected the creation in 6 daj's rather than at once, since the perfection of the work is signified by the number 6. The sum of the aUquot parts of 9 falls short of it; likewise for 10. But the sum of the aliquot parts of 12 exceeds it. Anicius Manhus Severinus Boethius^ (about 481-524), in a Latin exposi- tion of the arithmetic of Nicomachus, stated that perfect numbers are rare, easily counted, and generated in a very regular order, while abundant (superfluos) and deficient (diminutos) numbers are found to an unlimited extent and not in regular order. The perfect numbers below 10000 are 6, 28, 496, 8128. And these numbers alwaj^s end alternately in 6 and 8. Munyos^ stated that Boethius added to EucUd's idea of perfect number that of deficient (diminute) and abundant (redundantem) numbers. Isidorus of Seville^ (570-636) distinguished even and odd numbers, perfect and abundant numbers, linear, flachen and Korper Zahlen (primes, products of two, products of three factors). Alcuin^ (735-804) , of York and Tours, explained the occurrence of the number 6 in the creation of the universe on the ground that 6 is a perfect number. The second origin of the human race arose from the deficient number 8; indeed, in Noah's ark there were 8 souls from which sprung the entire human race, showing that the second origin was more imperfect than the first, which was made according to the number 6. ^lamblichus Chalcidensis ex Coele-Syria in Nicomachi Geraseni arithmeticam introduc- tionem, et de Fato. Accedit Joachimi Camerarii explicatio in duos libros Nicomachi. Ed., Samuel Tennulius. Amhemiae, 1668, pp. 43-47. (Greek text and Latin translation in parallel columns.) lamblichi in Nicomachi arithmeticam introductionem Uber ad fidem codicis Florentini. Ed., H. Pistelli. Lipsiae, 1894. (Greek.) *De Civitate Dei, hber XI, cap. XXX, ed., B. Dombart, Lipsiae, 1877, 1, p. 504. The reference by Frizzo"' i' to lib. II, cap. 39. "Arithmetica boetij, Augsburg, 1488; Cologne, 1489; Leipzig, 1490; Venice, 1491-2, 1499; Paris, [1496, 1501], 1503, etc.; lib. 1, cap. 20. "De generatione numeri perfecti." Opera Boetii, Venice, 1491-2, etc.; ed., Friedlein, Leipzig, 1867. "Institvtiones arithmeticae ad percipiendam astrologiam et mathematicas facultates neces- sariae. Auctore Hieronymo Mimj'os, Valentiae, 1566, f. 5, verso. *Incipit epistola Isidori iunioris hispalensis . . . Finit Uber etymologiarum . . . [Augsburg, 1472]; Venice, 1483, etc. In this book of etymologies, arithmetic is treated very briefly in Book 3, beginning f . 15. •Bibhotheca Rerum Germanicarum, tomus sextus: Monumenta Alcuiniana, Berlin, 1873, epistolae 259, pp. 818-821. Cf. Migne, Patrologiae, vol. 100, 1851, p. 665; Hankel, Geschichte Math., p. 311. Chap. I] PERFECT, MULTIPLY PERFECT, AND AmICABLE NUMBERS. 5 Thabit ben Korrah,^° in a manuscript composed the last half of the ninth century, attributed to Pythagoras and his school the employment of perfect and amicable numbers in illustration of their philosophy. Let s = 1+2+ ... +2". Then (prop. 5), 2'*s is a perfect number if s is a prime; 2"p is abundant if p is a prime <s, deficient if p is a prime >s, and the excess or deficiency of the sum of all the divisors over the number equals the difference of s and p. Let (prop. 6) p' and p" be distinct primes >2; the sum of the divisors <N oi N = p'p"2" is a = (2"+i-l)(l+p'+p") + (2"-l)py. Hence N is abundant or deficient according as a-iV=(2"+^-l)(l+p'+p")-py>0or <0. Hrotsvitha,^^ a nun in Saxony, in the second half of the tenth century, mentioned the perfect numbers 6, 28, 496, 8128. Abraham Ibn Ezra^^" (tll67), in his commentary to the Pentateuch, Ex. 3, 15, stated that there is only one perfect number between any two successive powers of 10. Rabbi Josef b. Jehuda Ankin^^'', at the end of the twelfth century, recom- mended the study of perfect numbers in the program of education laid out in his book "Healing of Souls." Jordanus Nemorarius^^ (tl236) stated (in Book VII, props. 55, 56) that every multiple of a perfect or abundant number is abundant, and every divisor of a perfect number is deficient. He attempted to prove (VII, 57) the erroneous statement that all abundant numbers are even. Leonardo Pisano, or Fibonacci, cited in his Liber Abbaci^^ of 1202, revised about 1228, the perfect numbers 1 2^(2^-1) =6, i 2^(2^-1) =28, | 2^(2^-1) =496, excluding the exponent 4 since 2^ — 1 is not prime. He stated that by pro- ceeding so, you can find an infinitude of perfect numbers. i^Manuscript 952, 2, Suppl. Arabe, Bibliotheque imperiale, Paris. Textual transl., except of the proofs which are given in modem algebraic notation as foot-notes [as numbers were represented by line, in the manuscript], by Franz Woepcke, Journal Asiatique, (4), 20, 1852, 420-9. "See Ch. Magnin, Theatre de Hrotsvitha, Paris, 1845. ""Mikrooth Gedoloth, Warsaw, 1874 ("Large Bible" in Hebrew). Samuel Ben Sdadias Ibn Motot; a Spaniard, wrote in 1370 a commentary on Ibn Ezra's commentary, Perush ai Perush Ibn Ezra, Venice, 1554, p. 19, noting the perfect numbers 6, 28, 496, 8128, and citing EucUd's rule. Steinschneider, in his book on Ibn Ezra, Abh. Geschichte Math. Wiss., 1880, p. 92, stated that Ibn Ezra gave a rule for finding all perfect numbers. As this rule is not given in the Mikrooth Gedoloth of 1874, Mr. Ginsburg of Columbia University infers the existence of a fuller version of Ibn Ezra's commentary. "^Quoted by Giideman, Das Jiidische Unterrichtswesen wahrend der Spanish Arabischen Periode, Wien, 1873. *^In hoc opere contenta. Arithmetica decern libris demonstrata .... Epitome i libros arithmeticos diui Seuerini Boetij . . . , Paris, 1496, 1503, etc. It contains Jordanus' "Elementa arithmetica decern libris, demonstrationibus Jacobi Fabri Stapulensis," and "Jacobi Fabri Stapulenais epitome in duos Hbros arithmeticos diui Seuerini Boetij." i^Il Liber Abbaci di Leonardo Pisano. Roma, 1857, p. 283 (Scritti, vol. 1). 6 History of the Theory of Numbers. [Chap. I In the manuscripts^ Codex lat. Monac. 14908, a part dated 1456 and a part 1461, the first four perfect numbers are given (J. 33') as usual and the fifth perfect number is stated correctly to be 33550336. Nicolas Chuquet^^ defined perfect, deficient, and abundant numbers, indicated a proof of EucHd's rule and stated incorrectly that perfect num- bers end alternately in 6 and 8. Luca Paciuolo, de Borgo San Sepolcro,^^ gave (f. 6) Euclid's rule, saying one must find by experiment whether or not the factor 1+2+4+. . . is prime, stated (f. 7) that the perfect numbers end alternately in 6 and 8, as 6, 28, 496, etc., to mfinity. In the fifth article (ff. 7, 8), he illustrated the finding of the aliquot divisors of a perfect number by taking the case of the fourteenth perfect number 9007199187632128. He gave its half, then the half of the quotient, etc., until after 26 divisions by 2, the odd number 134217727, marked " Indi^dsibilis " [prime]. Dividing the initial number by these quotients, he obtained further factors [1,2,..., 2'^, but written at length]. The proposed number is said to be evidently perfect, since it is the sum of these factors [but he has not employed all the factors, since the above odd number equals 2'-'^ — 1 and has the factor 2^ — 1 = 7] . Although Paciuolo did not list the perfect numbers between 8128 and 90 . . .8, the fact that he called the latter the fourteenth perfect number imphes the error expressly committed bj^ Bo^illus.^" Thomas Bradwardin^" (1290-1349) stated that there is only one perfect number (6) between 1 and 10, one (28) up to 100, 496 up to 1000, 8128 up to 10000, from which these numbers, taken in order, end alternately in 6 and 8. He then gave EucUd's rule. Faber Stapulensis^^ or Jacques Lefevre (born at Etaples 1455, tl537) stated that all perfect numbers end alternately in 6 and 8, and that Euclid's rule gives all perfect numbers. Georgius Valla^^ gave the first four perfect numbers and observed that "The manuscript is briefly described by Gerhardt, Monatsber. Berlin Ak., 1870, 141-2. See Catalogus codicum latinorum bibliothecae regiae Monacensis, Tomi II, pars II, codices nuna. 11001-15028 complectus, Munich, 1876, p. 250. An extract of ff. 32-34 on perfect numbers was published by MaximiUan Curtze, BibUotheca Mathematica, (2), 9, 1895, 39-42. "Triparty en la science des nombres, manuscript No. 1436, Fonds Fran^ais, BibliothSque Nationale de Paris, written at Lyons. 1484. Published by Aristide Marre, Bull. Bibl. Storia Sc. Mat. et Fis.. 13 (1880), 593-659, 693-814; 14 (1881), 417-460. See Part 1, Ch. Ill, 3, 619-621, manuscript, ff. 20-21. "Summa de Arithmetica geometria proportioni et proportionalita. [Suma . . , Venice, 1494.] Toscolano, 1523 (two editions substantially the same). "Arithmetica thome brauardini. Tractatus perutilis. In arithmetica speculativa a magistro thoma Brauardini ex libris eucUdis boecij & ahorum qua optimne excerptus. Parisiis, 1495, 7th unnumbered page. Arithmetica Speculativa Thome Brauardini nuper mendis Plusculis tersa et diligenter Impressa, Parisiis [1502], 6th and 7th unnumbered pages. Also undated edition [1510], 3d page. "Epitome (iii) of the arithmetic of Boethius in Faber's edition of Jordanus," 1496, etc. Also in Introductio Jacobi fabri Stapulesis in Arithmecam diui Seuerini Boetij pariter Jordani, Paris, 1503, 1507. Also in Stapulensis, Jacobi Fabri, Arithmetica Boethi epitome, Basileae, 1553, 40. "De expetendis et fvgiendis rebvs opvs, Aldus, 1501. Liber I ( = Arithmeticae I), Cap. 12. Chap. I] PERFECT, MULTIPLY PERFECT, AND AmICABLE NuMBERS. 7 "these happen to end in 6 or 8. . .and these terminal numbers will always be found alternately." Carolus Bovillus^" or Charles de Bouvelles (1470-1553) stated that every perfect number is even, but his proof applies only to those of Euclid's type. He corrected the statement of Jordanus^^ that every abundant number is even, by citing 45045 [ = 5-9-7-ll-13] and its multiples. He stated that 2" — 1 is a prime if n is odd, expUcitly citing 511 [ = 7-73] as a prime. He listed as perfect numbers 2"~^(2'* — 1), n ranging over all the odd numbers ^ 39 [Cataldi^ later indicated that 8 of these are not perfect]. He repeated the error that all perfect numbers end alternately in 6 and 8. He stated (f. 175, No. 25) that if the sum of the digits of a perfect number >6 be divided by 9, the remainder is unity [proved for perfect numbers of Euclid's type by Cataldi,^^ p. 43]. He noted (f. 178) that any divisor of a perfect number is deficient, any multiple abundant. He stated (No. 29) that one or both of 6n=i=l are primes and (No. 30) conversely any prime is of the form 6n=t 1 [Cataldi,^ p. 45, corrects the first statement and proves the second]. He stated (f. 174) that every perfect number is triangular, being 2" (2'' — l)/2. Martinus^^ gave the first four perfect numbers and remarked that they end alternately in 6 and 8. Gasper Lax'^^ stated that the perfect numbers end alternately in 6 and 8. V. Rodulphus Spoletanus^^ was cited by Cataldi,'*^ with the implication of errors on perfect numbers. [Copy not seen.] Girardus Ruffus^^ stated that every perfect number is even, that most odd numbers are deficient, that, contrary to Jordanus,^^ the odd number 45045 is abundant, and that for n odd 2^* — 1 always leads to a perfect num- ber, citing 7, 31, 127, 511, 2047, 8191 as primes [the fourth and fifth are composite]. Feliciano^^ stated that all perfect numbers end alternately in 6 and 8. Regius^^ defined a perfect number to be an even number equal to the sum of its aliquot divisors, indicated that 511 and 2047 are composite, gave correctly 33550336 as the fifth perfect number, but said the perfect numbers ^''Caroli Bouilli Samarobrini Liber De Perfectis Numeris (dated 1509 at end), one (ff. 172-180) of 13 tracts in his work, Que hoc volumine continetur: Liber de intellectu, . . . De Numeris Perfectis, . . . , dated on last page, 1510, Paris, ex ofEcina Henrici Stephani. Biography in G. Maupin, Opinions et Curiosit^s touchant la Math., Paris, 1, 1901, 186-94. "Ars Arithmetica loannis Martini, Silicei: in theoricen & praxim. 1513, 1514. Arithmetica loannis Martini, Scilicei, Paris, 1519. "Arithmetica speculatiua magistri Gasparis Lax. Paris, 1515, Liber VII, No. 87 (end). *3De proportione proportionvm dispvtatio, Rome, 1515. "Divi Severini Boetii Arithmetica, dvobvs discreta hbris, Paris, 1521; ff. 40-44 of the commen- tary by G. Ruffus. "Libro di Arithmetica & Geometria speculatiua & praticale: Composto per maestro Fran- cesco FeUciano da Lazisio Veronese Intitulato Scala Grimaldelli: Nouamente stampato. Venice, 1526 (p. 3), 1527, 1536 (p. 4), 1545, 1550, 1560, 1570, 1669, Padoua, 1629, Verona, 1563, 1602. *Vtrivsqve Arithmetices, epitome ex uariis authoribus concinnata per Hvdalrichum Regium. Strasburg, 1536. Lib. I, Cap. VI: De Perfecto. Hvdalrichvs Regius, Vtrivsque. . . ex variis . . . , Friburgi, 1550 [and 1543], Cap. VI, fol. 17-18. 8 History of the Theory of Numbers. [Chap. I end alternately in 6 and 8. A multiple of an abundant or perfect number is abundant, a divisor of a perfect number is deficient. Cardan^^ (1501-1576) stated that perfect numbers were to be formed by Euclid's rule and always end with 6 or 8; and that there is one between any two successive powers of ten. De la Roche-^ stated in effect that 2""^ (2" — 1) is perfect for every odd n, citing in particular 130816 and 2096128, given by n = 9, n = ll. This erroneous law led him to believe that the successive perfect numbers end alternately in 6 and 8. Noviomagus-^ or Neomagus or Jan Bronckhorst (1494-1570) gave Euclid's rule correctly and stated that among the first 10 numbers, 6 alone is perfect, . . . , among the first 10000 numbers, 6, 28,496, 8128 alone are perfect, etc., etc. [implying falsely that there is one and but one perfect number with any prescribed number of digits]. In Lib. II, Cap. IV, is given the sieve (or crib) of Eratosthenes, with a separate column for the multiples of 3, a separate one for the multiples of 5, etc. WilUchius^'^ (tl552) listed the first four perfect numbers and stated that to these are to be added a very few others, whose nature is that they end either in 6 or 8. Michael StifeP^ (1487-1567) stated that all perfect numbers except 6 are multiples of 4, while 4(8-1), 16(32-1), 64(128-1), 256(512-1), etc., to infinity, are perfect [error, Kraft^°]. He later^- repeated the latter error, listing as perfect 2X3, 4X7, 16X31, 64X127, 256X511, 1024X2047, "& so fort an ohn end." Every perfect munber is triangular. Peletier^^ (1517-1582) stated (1549, V left; 1554, p. 20) that the perfect numbers end in 6 or 8, that there is a single perfect number between any two successive powers of 10, and (1549, C III left; 1554, pp. 270-1) that 4(8-1), 16(32-1), 64(128-1), 256(511),. . .are perfect. The first two statements were also given later by Peletier.^ ^'Hieronimi C. Cardani Medici Mediolanensis, Practica Arithmetice, & Mensurandi singu- laris. Milan, 1537, 1539; Xiirnberg, 1541, 1542, Cap. 42, de proprietatibus numerorum mirificis. Opera IV, Lyon, 1663. -*Larismetique & Geometrie de maistre Estienne de la Roche diet Ville Franche, Nouuelle- ment Imprimee & des fautea corrigee, Lyon, 1538, fol. 2, verso. Ed. 1, 1520. '"De Nvmeris libri dvo .... authore loanne Nouiomago, Paris, 1539, Lib. II, Cap. III. Reprinted, Cologne, 1544; Deventer, 1551. Edition by G. Frizzo, Verona, 1901, p. 132. '°Iodoci Vvillichii Reselliani, Arithmeticae libri tres, Argentorati, 1540, p. 37. '^Arithmetica Integra, Norimbergae, 1544, ff. 10, 11. "Die Cosa Cbristoffs Rudolffs Die schonen Exempeln der Coss Durch Michael Stifel Gebessert vnd sehr gemehrt, Konigsperg in Preussen, 1553, Anbang Cap. I, f. 10 verso, f. 11 (f. 27 v.), and 1571. "L'Arithmetiqve de lacqves Peletier dv Mans, departie en quatre Liures, Poitiers, 1549, 1550, 1553. . . . , ff. 77 v, 78 r. Reviie e augmentee par 1' Auteur, Lion, 1554 Troisieme edition, reucue et augmentee, par lean de Tovmes, 1607. "Arithmeticae Practicae methodvs facilis, per Gemmam Frisivm, Medicvm, ac Mathematicum conscripta .... In eandem loannis Steinii & lacobi Peletarii Annotationes. Antver- piae, 1581, p. 10. Chap. I] PERFECT, MULTIPLY PERFECT, AND AmICABLE NUMBERS. 9 Postello^^ stated erroneously that 130816 [ = 256-511] is perfect. Lodoico Baeza^^ stated that Euchd's rule gives all perfect numbers. Pierre Forcadel" (tl574) gave 130816 as the fifth perfect number, implying incorrectly that 511 is a prime. Tartaglia^^ (1506-1559) gave an erroneous [Kraft ^^J list of the first twenty perfect numbers, viz., the expanded forms of 2"~^(2'* — 1), for n = 2 and the successive odd numbers as far as n = 39. He stated that the sums 1+2+4, 1+2+4+8, .. .are alternately prime and composite; and that the perfect numbers end alternately in 6 and 8. The third ''notable prop- erty" mentioned is that any perfect number except 6 yields the remainder 1 when divided by 9. Robert Recorde^^ (about 1510-1558) stated that all the perfect numbers under 6-10^ are 6, 28, 496, 8128, 130816, 2096128, 33550336, 536854528 [the fifth, sixth, eighth of these are not perfect]. Petrus Ramus^° (1515-1572) stated that in no interval between succes- sive powers of 10 can you find more than one perfect number, while in many intervals you will find none. At the end of Book I (p. 29) of his Arith- meticae libri tres, Paris, 1555, Ramus had stated that 6, 28, 496, 8128 are the only perfect numbers less than lOOpOO. Franciscus Maurolycus*^ (1494-1575) gave an argument to show that every perfect number is hexagonal and hence triangular. Peter Bungus^^ (fieoi) gave (1584, pars altera, p. 68) a table of 20 numbers stated erroneously to be the perfect numbers with 24 or fewer digits [the same numbers had been given by Tartaglia^^]. In the editions of 1591, etc., p. 468, the table is extended to include a perfect number of 25 digits, one of 26, one of 27, and one of 28. He stated (1584, pp. 70-71 ; 1591, pp. 471-2) that all perfect numbers end alternately in 6 and 28; employing Euclid's formula, he observed that the product of a power of 2 ending in 4 by a number ending in 7 itself ends in 28, while the product of one ending in 6 by one ending in 1 ends in 6. He verified (1585, pars ^"Theoricae Arithmetices Compendium h Guilielmo Postello, Lutetiae, 1552, a syllabus on one large sheet of arithmetic definitions. "Nvmerandi Doctrina, Lvtetiae, 1555, fol. 27-28. ''L'Arithmeticqve de P. Forcadel de Beziers, Paris, 1556-7. Livre I (1556), fol. 12 verso. 3*La seconda Parte del General Trattato di Nvmeri, et Misvre di Nicolo Tartagha, Vinegia, 1556, f. 146 verso. L' Arithmetiqve de Nicolas Tartagha Brescian .... Recueillie, & traduite d'ltalien en FranQois, par Gvillavme Gosselin de Caen, .... Paris, 1578, f. 98 verso, f. 99. '®The Whetstone of witte, whiche is the seconde parte of Arithmetike, London, 1557, eighth unnumbered page. ^''Petri Rami Scholarum Mathematicarum, Libri unus et triginta, k Lazaro Schonero recog- niti & emendati, Francofvrti, 1599, Libr. IV (Arith.), p. 127, and Basel, 1578. "Arithmeticorvm hbri dvo, Venetiis, 1575, p. 10; 1580. Published with separate paging, at end of Opuscula mathematica. *^Mysticae nvmerorvm significationis liber in dvas divisvs partes, R. D. Petro Bongo Canonico Bergomate avctore. Bergomi. Pars prior, 1583, 1585. Pars altera, 1584. Petri Bungi Bergomatis Numerorum mysteria, Bergomi, 1591, 1599, 1614, Lutetiae Parisio- rum, 1618, all four with the same text and paging. Classical and biblical citations on numbers (400 pages on 1, 2, . . , 12). On the 1618 edition, see Font^s, M6m. Acad. So. Toulouse, (9), 5, 1893, 371-380. 10 History of the Theory of Numbers. [Chap, i prior, p. 238; 1591, p. 343) for the first seven numbers of his table [two being imperfect, however] that the sum of the digits of a perfect number exceeds by unity a multiple of 9. Every perfect number is triangular (1591, p. 270). Every multiple of a perfect number is abundant, every divisor deficient (1591, p. 464). Unicornus^^ (1523-1610) cited Bungus and repeated his error that 2"- 1 (2^ — 1) is always perfect for n odd and that all perfect numbers end alternately in 6 and 8. Cataldi"^ (1548-1626) noted in his Preface that Paciuolo's^^ fourteenth per- fect number 90. . .8 is in fact abundant since it arose from 1+2+4+ • ■ • +2^^ = 134217727, which is divisible by 7,whereas Paciuolo said it was prime. Citing the error of the latter, Bovillus,^° and others, that all perfect num- bers end alternately in 6 and 8, Cataldi observed (p. 42) that the fifth per- fect number is 33550336 and the sixth is 8589869056, from 8191 =2'^- 1 and 131071=2^^ — 1, respectively, proved to be primes (pp. 12-17) by actually trying as possible divisor every prime less than their respective square roots. He gave (pp. 17-22) the corresponding work showing 2^^ — 1 to be prime. He stated (p. 11) that 2'*-! is a prime forn = 2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 37, remarking that the prime n = ll does not yield a perfect number since (p. 5) 2^^ — 1=2047 = 23*89, while it is composite if n is composite. He proved (p. 8) that the perfect numbers given by Euclid's rule end in 6 or 8. He gave (pp. 28-40, 48) a table of all divisors of all even and odd numbers ^ 800, and a table of primes < 750. Georgius Henischiib^^ (1549-1618) stated that the perfect numbers end alternately in 6 and 8, and that one occurs between any two successive powers of 10. He applied Euclid's formula without restricting the factor 2"—! to primes. Johan Rudolff von Graff enried"*^ stated that all perfect numbers are given by Euclid's rule, which he applied without restricting 2" — 1 to primes, expressly citing 256X511 as the fifth perfect number. Every perfect number is triangular. Bachet de Mezirac''^ (1581-1638) gave (f. 102) a lengthy proof of Euclid's theorem that 2'*p is perfect if p = l+2+ . . . +2^* is a prime, but "De I'arithmetica vniversale del Sig. loseppo Vnicorno, Venetia, 1598, f. 57. "Trattato de nvmeri perfetti di Pierto Antonio Cataldo, Bologna, 16C3. According to the Preface, this work was composed in 1588. Cataldi founded at Bologna the Academia Erigende, the most ancient known academy of mathematics; his interest in perfect numbers from early youth is shown by the end of the first of his "due lettioni fatte nell' Academia di Perugia" (G. Libri, Hist. Sc. Math, en ItaUe, 2d ed., vol. 4, Halle, 1865, p. 91). G. Wertheim, BibHotheca Math., (3), 3, 1902, 76-83, gave a summary of the Trattato. "Arithmetica Perfecta et Demonstrata, Georgii Henischiib, Augsburg [1605], 1609, pp. 63-64. "Arithmeticae Logistica Popularis Librii IIII. Jn welchen der Algorithmus in gantzen Zahlen u. Fracturen . . . . , Bern, 1618, 1619, pp. 236-7. *^Elementorum arithmeticorum libri XIII auctori D . . . , a Latin manuscript in the Biblio- thfeque de I'lnstitut de France. On the inside of the front cover is a comment on the sale of the manuscript by the son of Bachet to DaUbert, treasurer of France. A general account of the contents of the manuscript was given by Henry, Bull. Bibl. Storia Sc. Mat. e Fis., 12, 1879, pp. 619-641. The present detailed account of Book 4, on perfect numbers, was taken from the manuscript. Chap. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 11 (f. 103, verso) is abundant if p is composite. Every multiple of a perfect or abundant number is abundant, every divisor of a perfect number is deficient (ff. 104 verso, 105). The product of two primes, other than 2X3, is deficient (f. 105 verso). The odd number 945 is abundant, the sum of its ahquot divisors being 975 (f. 107). Commenting (f. Ill verso, f. 112) on the statement of Boethius^ and Cardan^^ that the perfect numbers end alternately in 6 and 8, he stated that the fourth is 8128 and the fifth is 2096128 [an error], the fifth not being 130816 = 256X511, since 511 =7X73. Jean Leurechon^^ (about 1591-1670) stated that there are only ten perfect numbers between 1 and 10^^, listed them (noting the admirable property that they end alternately in 6 and 8) and gave the twentieth per- fect number. [They are the same as in Tartaglia's^^ list.] Lantz^^ stated that the perfect numbers are 2(4-1), 4(8-1), 16(32-1), 64(128-1), 256(512-1), 1024(2048-1), etc. Hugo Sempilius^° or Semple (Scotland, 1594-Madrid, 1654) stated that there are only seven perfect numbers up to 40,000,000; they end alternately in 6 and 8. Casper Ens^^ stated that there are only seven perfect numbers <4-10'', viz., 6, 28, 496, 8128, 130816, 1996128 [for 2096128], 33550336, and that they end alternately in 6 and 8. Daniel Schwenter^^ (1585-1636) made the same error as Casper Ens.^^ Erycius Puteanus^^ quoted from Martiano Capella, lib. VII, De Nuptiis Philologiae, to the effect that the perfect number 6 is attributed to Venus; for it is made by the union of the two sexes, that is, from triad, which is male since it is odd, and from diad, which is feminine since it is even. Puteanus said that the perfect numbers in order are 6, 28, 496, 8128, 130816, 2096128, 33550336, and gave all their divisors [implying that 511, 2047, 8191 are primes], and stated that these seven and all the remaining end alternately in 6 and 8. Between any two successive powers of 10 is one perfect number. That they are all triangular adds perfection to the perfect. Joannes Broscius^"^ or Brocki remarked that there is no perfect number between 10000 and 10000000, contrary to Stifel,^^ Bungus,^^ Sempilius.^" Puteanus,^^ and the author of Selectarum Propositionum Mathematicarum, quas propugnavit, Mussiponti, Anno 1622, Maximilianus Willibaldus, Baro ^^Recreations math^matiques, Pont-^-Mousson, 1624; London, 1633, 1653, 1674 (these three EngUsh editions by Wm. Oughtred), p. 92. The authorship is often attributed to Leurechon's pupil Henry Van Etten, whose name is signed to the dedicatory epistle. Cf. Poggendorff, Handworterbuch, 1863, 2, p. 250 (under C. Mydorge); Bibliotheque des 6crivains de la compagnie de Jesus, par A. de Backer, 2, 1872, 731; Biograpliie Generale, 31, 1872, 10. "Institutionum Arithmeticarum hbri quatuor h loanne Lantz, Coloniae Agrippinae, 1630, p. 54. "De Mathematicis Disciphnis hbri Duodecim, Antverpiae, 1635, Lib. 2, Cap. 3, N. 10, p. 46. There is (pp. 263-5) an index of writers on geometry and one for arithmetic. "Thaumaturgus Math., Munich, 1636, p. 101; Coloniae. 1636, 1651; Venice, 1706. "Dehciae Physico-Mathematicae oder Mathemat: vnd Philosophische Erquickstunden, part I (574 pp.), Numberg, 1636, p. 108. "De Bissexto Liber: nova temporis facula qua intercalandi arcana .... Lovanii, 1637; 1640, pp. 103-7. Reproduced by J. G. Graevius, Thesaurus Antiquitatum Romanarum (12 vols., 1694-9), Lugduni Batavorum, vol. 8. 12 History of the Theory of Numbers. [Chap, i in Waldpurg. WTiile they considered 511X256 and 2047X1024 as perfect, 511 has the factor 7, and (as pointed out to him by Stanislaus Pudlowski) 2047 has the factor 23. Broscius stated that 2^-1 has the factor 3 5 7 11 13 17 19 23 29 31 if n is a multiple of 2 4 3 10 12 8 18 11 28 5. The contents of the second dissertation are given below under the date 1652. Ren^ Descartes,^^ in a letter to Mersenne, November 15, 1638, thought he could prove that every even perfect number is of Euclid's type, and that every odd perfect number must have the form ps^, where p is a prime. He saw no reason why an odd perfect number may not exist. For p = 22021, s = 3'7-ll-13, ps^ would be perfect if p were prime [but p = 61-19^]. In a letter to Frenicle, January 9, 1639, Oeuvres, 2, p. 476, he expressed his belief that an odd perfect number could be found by replacing 7, 11, 13 in s by other values. Fermat^^ stated that he possessed a method of solving all questions relating to aliquot parts. Citing this remark, Frenicle^' challenged Fermat to find a perfect number of 20 or 21 digits. Fermat^^ replied that there is none with 20 or 21 digits, contrary to the opinion of those who believe that there is a perfect number between any two consecutive powers of 10. Fermat,^^ in a letter to Mersenne, June (?), 1640, stated three proposi- tions which he had proved not without considerable trouble and which he called the basis of the discovery of perfect numbers: if n is composite, 2" — 1 is composite; if n is a prime, 2" — 2 is divisible by 2n, and 2" — 1 is divisible by no prime other than those of the form 2kn-\-l [cf . Euler^']. For example, 2"-l = 23-89, 2^^-l has the factor 223. Also 2"^^-! has the factor 47, Oeuvres, 2, p. 210, lett-er to Frenicle, October 18, 1640. Mersenne^° (1588-1648) stated that, of the 28 numbers* exhibited by "De numeris perfectis disceptatio qua ostenditur a decern millibus ad centies centena millia, nullum esse perfectum numenim atque ideo ab unitate usque ad centies centena millia quatuor tantum perfectos numerari, Amsterdam, 1638. Reproduced as the first (pp. 115-120) of two dissertations on perfect numbers, they forming pp. 111-174 of Apologia pro Aristotele & Evchde, contra Petrvm Ramvm, & aUos. Addititiae sunt Dvae Discep- tationes de Nvmeris Perfectis. Authore loanne Broscio, Dantiaci, 1652 (with a some- what different title, Amsterdam, 1699). "Oeuvres de Descartes, II, Paris, 1898, p. 429. s«Oeuvres de Fermat, 2, Paris, 1894, p. 176; letter to Mersenne, Dec. 26, 1638. *'Oeuvres de Fermat, 2, p. 185; letter to Mersenne, March, 1640. osQeuvres, 2, p. 194; letter to Mersenne, May (?), 1640. "Oeuvres de Fermat, 2, pp. 198-9; Varia Opera Math. d. Petri de Fermat, Tolosae, 1679, p. 177; Precis des Oeuvres math, de P. Fermat et de 1' Arithmdtique de Diophante, par E. Brassinne, M6m. Ac. Imp. Sc. Toulouse, (4), 3, 1853, 149-150. •°F. Marini Mersenni minimi Cogitata Physico Mathematica, Parisiis, 1644. Praefatio Generahs, No. 19. C. Henry (Bull. Bibl. Storia Sc. Mat. e Fis., 12, 1879, 524-6) beheved that these remarks were taken from letters from Fermat and Frenicle, and that Mersenne had no proof. A similar opinion was expressed by W. W. Rouse Ball, Messenger Math., 21, 1892, 39 (121). On documents relating to Mersenne see Tinterm^diaire des math., 2, 1895, 6; 8, 1901, 105; 9, 1902, 101, 297; 10, 1903, 184. Cf. Lucas."* *Only 24 were given by Bungus. While his table has 28 lines, one for each number of digits, there are no entry of numbers of 5, 11, 17, 23 digits. Chap. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 13 Bungus,^^ chap. 28, as perfect numbers, 20 are imperfect and only 8 are perfect : 6, 28, 496, 8128, 23550336 [for 33. . .], 8589869056, 137,438691328, 2305843008139952128, which occur at the lines marked 1, 2, 3, 4, 8, 10, 12 and 29 [for 19] of Bungus' table [indicating the number of digits]. Perfect numbers are so rare that only eleven are known, that is, three different from those of Bungus; norf is there any perfect number other than those eight, unless you should surpass the exponent 62 in 1+2+2^+ .. . The ninth perfect number is the power with the exponent 68 less 1; the tenth, the power 128 less 1 ; the eleventh, the power 258 less l,i.e., the power 257, decreased by unity, multiplied by the power 256. [The first 11 perfect numbers are thus said to be 2"-'(2"-l) for n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257, in error as to n = 61, 67, 89, 107 at least.] He who would find 11 others will know that all analysis up to the present will have been exceeded, and will remember in the meantime that there is no perfect number from the power 17000 to 32000, and no interval of powers can be assigned so great but that it can be given without perfect numbers. For example, if the exponent be 1050000, there is no larger exponent n up to 2090000 for which 2" — 1 is a prime. One of the greatest difficulties in mathematics is to exhibit a prescribed number of perfect numbers; and to tell if a given number of 15 or 20 digits is prime or not, all time would not suffice for the test, what- ever use is made of what is already known. Mersenne" stated that 2^ — 1 is a prime if p is a prime which exceeds by 3, or by a smaller number, a power of 2 with an even exponent. Thus 2^-1 is a prime since 7 = 2^3; again, since 67 = 3+2*^, 2^^ + 1 = 1... 7 [for 2®^ — 1] is a prime and leads to a perfect number [error corrected by Cole^^^]. Understand this only of primes 2^ — 1. Wherefore this property does not belong to the prime 5, but to 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, and all such. Numbers expressible as the sum or difference of two squares in several ways are composite, as 65 = 1+64 = 16+49. As he speaks of Frenicle's knowledge of numbers, at least part of his results are doubtless due to the latter. In 1652, J. Broscius (Apologia,^^ p. 121) observed that while perfect numbers were deduced by Euclid from geometrical progressions, they may be derived from arithmetical progressions: 6 = 1+2+3, 28 = 1+2+3+4+5+6+7, 496 = 1+2+3+ ... +31. fNeque enim vllus est alius perfectus ab illis octo, nisi superes exponentem numerum 62, progressionis duplae ab 1 incipientis. Nonus enim perfectus est potestas exponentis 68, minus 1. Decimus, potestas exponentis 128, minus 1. Vndecimus denique, potestas 258, minus 1, hoc est potestas 257, unitate decurtata, multiplicata per potestatem 256. *T. Marini Mersenni Novarvm Observationvm Physico-Mathematicarum^ Tomvs III, Parisiis, 1647, Cap. 21, p. 182. The Reflectiones Physico-Math. begin with p. 63; Cap. 21 is quoted in Oeuvres de Fennat, 4, 1912, pp. 67-8. 14 History of the Theory of Numbers. [Chap, i He stated that while perfect numbers end with 6 or 28, the proof by Bungus*' does not show that they end alternately with 6 and 28, since Bungus included imperfect as well as perfect numbers. The numbers 130816 and 2096128, cited as perfect by Puteanus,^^ are abundant. After giving a table of the expanded form of 2" forn = 0, 1, . . . , 100, Broscius (p. 130, seq.) gave a table of the prime divisors of 2" — 1 (n = 1, . . . , 100), but showing no prime factor when n is any one of the primes, other than 11 and 23, less than 100. For n = ll, the factors are 23, 89; for n = 23, the factor 47 is given. Thus omitting unity, there remain only 23 numbers out of the first hundred which can possibly generate perfect numbers. Contrary to Car- dan,^^ but in accord with Bungus,^^ there is (p. 135) no perfect number between 10* and 10\ Of Bungus' 24 numbers, only 10 are perfect (pp. 135-140): those with 1, 2, 3, 4, 8, 10, 12, 18, 19, 22 digits, and given by 2'-i(2'*-l) for n = 2, 3, 5, 7, 13, 17, 19, 29, 31, 37, respectively. The pri- maUty of the last three was taken on the authority of unnamed predecessors. There are only 21 abundant numbers between 10 and 100, and all of them are even; the only odd abundant number <1000 is 945, the sum of whose aliquot di\isors is 975 (p. 146). The statement by Lucas, Th^orie des nombres, 1, Paris, 1891, p. 380, Ex. 5, that 3^-5-79 [deficient] is the smallest abundant number is probably a misprint for 945 = 3^-5-7. This error is repeated in Encyclopedic Sc. Math., I, 3, Fas. 1, p. 56. Johann Jacob Heinlin^- (1588-1660) stated that the only perfect num- bers <4-10' are 6, 28, 496, 8128, 130816, 2096128, 33550336, and that all perfect numbers end alternately in 6 and 8. Andrea Tacquet^^ (Antwerp, 1612-1660) stated (p. 86) that Euclid's rule gives all perfect numbers. Referring to the 11 numbers given as perfect by Mersenne,^^ Tacquet said that the reason why not more have been found so far is the greatness of the numbers 2^ — 1 and the vast labor of testing their primaUty. Frenicle^ stated in 1657 that EucUd's formula gives all the even perfect numbers, and that the odd perfect numbers, if such exist, are of the form p/c^, where p is a prime of the form 4n+l [cf. Euler^^]. Frans van Schooten^^ (the younger, 1615-1660) proposed to Fermat that he prove or disprove the existence of perfect numbers not of Euclid's type. Joh. A. Leuneschlos^^ remarked that the infinite multitude of numbers contains only ten perfect numbers; he who will find ten others will know '*Joh. Jacobi Heinlini, Synopsis Math, praecipuas totius math .... Tubmgae, 1653. Synopsis Math. Universalis, ed. Ill, Tubingae, 1679, p. 6. English translation of last by Venterus Mandey, London, 1709, p. 5. "Arithmeticae Theoria et Praxis, Lovanii, 1656 and 1682 (same paging), [1664, 1704]. Hia opera math., Antwerpiae, 1669, does not contain the Arithmetic. "Correspondence of Chr. Huygens, No. 389; Oeuvres de Fermat, 3, Paris, 1896, p. 567. "Oeuvres de Huygens, II, Correspondence, No. 378, letter from Schooten to J. Wallis, Mar. 18, 1658. Oeuvres de Fermat, 3, Paris, 1896, p. 558. "Mille de Quantitate Paradoxa Sive Admiranda, Heildelbergae, 1658, p. 11, XLVI, XLVII. Chap. I] PERFECT, MULTIPLY PERFECT, AND AmICABLE NUMBERS. 15 that he has surpassed all analysis up to the present. Goldbach" called Euler's attention to these remarks and stated that they were probably taken from Mersenne, the true sense not being followed. Wm. Leybourn^^ hsted as the first ten perfect numbers and the twentieth those which occur in the table of Bungus.^^ "The number 6 hath an emi- nent Property, for his parts are equal to himself." Samuel Tennulius, in his notes (pp. 130-1) on lamblicus,^ 1668, stated that the perfect numbers end alternately in 6 and 8, and included 130816 = 256-511 and 2096128 = 1024-2047 among the perfect numbers. Tassius®^ stated that all perfect numbers end in 6 or 8. Any multiple of a perfect or abundant number is abundant, any divisor of a perfect number is deficient. He gave as the first eight known perfect numbers the first eight listed by Mersenne.^" Joh. Wilh. Pauli^° (Philiatrus) noted that if 2" — 1 is a prime, n is, but not conversely. For n = 2, 3, 5, 7, 13, 17, 19, 2"-l is a prime; but 2^^-l is divisible by 23, 2^^ — 1 by 47, and 2*^ — 1 by 83, the three divisors being 2n+l. G. W. Leibniz'^^ quoted in 1679 the facts stated by Pauli and set himself the problem to find the basis of these facts. Returning about five years later to the subject of perfect numbers, Leibniz implied incorrectly that 2^^ — 1 is a prime if and only if p is. Jean Prestet^^ (tl690) stated that the fifth, . . . , ninth perfect numbers are 23550336 [for 33. . .], 8589869056, 137438691328, 238584300813952128 [for 2305. . .39952128], 2''^-2^'\ [Hence 2'*-^(2'*-l) for 7i = 13, 17, 19, 31, 257. The numerical errors were noted by E. Lucas,i24 p 7g4 j Jacques Ozanam^^ (1640-1717) stated that there is an infinitude of perfect numbers and that all are given by Euclid's rule, which is to be applied only when the odd factor is a prime. Charles de Neuveglise^^ proved that the products 3-4, . . ., 8-9 of two consecutive numbers are abundant. All multiples of 6 or an abundant number are abundant. "Correspondence Math. Phy8.,ed.,Fus8, 1, 1843; letters to Euler, Oct. 7, 1752 (p. 584), Nov. 18 (p. 593). '^Arithmetical Recreations; or Enchriridion of Arithmetical Questions both Delightful and Profitable, London, 1667, p. 143. "Arithmeticae Empiricae Compendium, Johannis Adolfi Tassii. Ex recensione Henrici Siveri, Hamburgi, 1673, pp. 13, 14. ^"De nvmiero perfecto, Leipzig, 1678, Magister-disputation. "Manuscript in the Hannover Library. Cf. D. Mahnke, Bibhotheca Math., (3), 13, 1912-3, 53-4, 260. "Nouveaux elemens des Mathematiques, ou Principes generaux de toutes les sciences, Paris, 1689, I, 154-5. "Recreations mathematiques et physiques, Paris and Amsterdam, 2 vols., 1696, I, 14, 15. "Traits methodique et abreg6 de toutes les mathematiques, Trevoux, 1700, tome 2 (L'arith- m^tique ou Science des nombres), 241-8. 16 History of the Theory of Numbers. [Chap, i John Harris,"^ D. D., F. R. S., stated that there are but ten perfect numbers between unity and one million of millions. John Hiir^ stated that there are only nine perfect numbers up to a hundred thousand million. He gave (pp. 147-9) a table of values of 2" forn = l,. . ., 144. Christian Wolf" (1679-1754) discussed perfect numbers of the form y"x [where x, y are primes]. The sum of its aliquot parts is l+y+ . . . +i/"+a:+?/x+ . . . +i/""'x, which must equal y'^x. Thus x = {l-\-y+ . . . +2/")M d = y^-l-y- . . . -7/""^ He stated* that x is an integer only when d = l, and that this requires y = 2, x = l +2+ . . . +2". Then if this x is a prime, 2"x is a perfect number. This is said to be the case forn = 8 and n = 10, since 2^ — 1 = 51 1 and 2^' — 1 = 2047 are primes, errors pointed out bj^ Euler.^^ A. G. Kastner^^ was not satisfied with the argument leading to the conclusion y = 2. Jacques Ozanam^^ listed as perfect numbers 2(4-1), 4(8-1), 16(32-1), 64(128-1), 256(512-1), 1024(2048-1),. . . without expUcit mention of the condition that the final factor shall be prime, and stated that perfect numbers are rare, only ten being known, and all end in 6 and 8 alternately. [Criticisms by Montucla,®^ Gruson.-^°°] Johann Georg Liebnecht^° said there were scarcely 5 or 6 perfect num- bers up to 4.10"; they always end alternately in 6 and 8. Alexander ]Malcolm^^ observed that it is not yet proved that there is no perfect number not in Euclid's set. He stated that, if pA is a perfect number, where p is a prime, and if M<p and M is not a factor of A, then MM is an abundant number [probably a misprint for MA, as the condi- tions are satisfied when p = 7, .4=4, M = 5, and MA =20 is abundant, while Af^ = 25 is deficient]. Christian Wolf^- made the same error as Casper Ens.^^ ^'Lexicon Technicum, or an Universal English Dictionarj' of Arts and Sciences, vol. I, London, 1704; ed. 5, vol. 2, London, 1736. "Arithmetik, London, ed. 2, 1716, p. 3. ^'Elementa Matheseos Universae, Halae Magdeburgicae, vol. I, 1730 and 1742, pp. 383-^, of the five volume editions [first printed 1713-41]; vol. I, 1717, 315-6, of the two volume edition. Quoted, with other errors, Ladies' Diary, 1733, Q. 166; Leybourn's ed., 1, 1817, 218; Button's ed., 2, 1775, 10; Diarian Repository, by Soc. Math., 1774, 289. *"Jam ut X sit numerus integer, nee in casu speciali, si y per numerum explicetur, numerus partium aliquotarum diversus sit a numero earundem in formula general!; necesse est ut d = l." "Math. Anfangsgrlinde, I, 2 (Fortsetzung der Rechnenkunst, ed. 2, 1801, 546-8). "Recreations math., new ed. of 4 vols., 1723, 1724, 1735, etc., I, 29-30. '"Grund-Satze der gesammten Math. Wiss. u. Lehren, Giessen u. Franckfurt, 1724, p. 21. *iA new system of arithmetik, theoretical & practical, London, 1730, p. 394. "Mathematisches Lexicon, I, 1734 (under Vollkommen Zahl). Chap. Il PERFECT, MULTIPLY PERFECT, AND AmICABLE NUMBERS. 17 Leonard Euler^^ (1707-1783) noted that 2'' — 1 may be composite for n a prime; for instance, 2^' — 1 = 23-89, contrary to Wolf.'^^ If n = 4m — 1 and 8m — 1 are primes, 2" — 1 has the factor 8m — 1, so that 2^-1 is com- posite for n = ll, 23, 83, 131, 179, 191, 239, etc. [Proof by Lucas.^^sj Furthermore, 2^^-l has the factor 223, 2^^-l the factor 431, 22^-1 the factor 1103, 2"^ — 1 the factor 439, etc. ''However, I venture to assert that aside from the cases noted, every prime less than 50, and indeed than 100, makes 2"~^(2" — 1) a perfect number, whence the eleven values 1, 2, 3, 5, 7, 13, 17, 19, 31, 41, 47 of n yield perfect numbers. I derived these results from the elegant theorem, of whose truth I am certain, although I have no proof: aJ^ — V is divisible by the prime n+1, if neither a nor h is." [For later proofs by Euler, see Chapter III on Fermat's theorem.] Euler's errors as to n = 41 and 47 were corrected by Winsheim,®^ Euler^^ himself, and Plana.i^o Michael Gottlieb Hansch^ stated that 2^*— 1 is a prime if n is any of the twenty- two primes ^79 [error, Winsheim,^^ Kraft^^]. George Wolfgang Kraft^^ corrected Stifel's^^ error that 511-256 is per- fect and the error of Ozanam (Elementis algebrae, p. 290) that the sum of all the divisors of 2*" is a prime, by noting that the sum forn = 2 is 511 = 7-73 ; and n6ted that false perfect numbers were listed by Ozanam.'^^ Kraft presented (pp. 9-11) an incomplete proof, communicated to him by Tobias Maier [cf. Fontana^^^], that every perfect number is of Euclid's type. Let 1, m, n, . . .,p, A,. . .be the aliquot parts of any perfect number pA, where p and A are the middle factors [as 4 and 7 jn 28]. Then q r n m Solving for A, he stated that the denominator must be unity, whence 'p = 2q/D, D = q — l—q/r — q/n — q/m. Again, D = l, whence g = 2r/D', D' = r — l—r/n—r/m. From I>' = 1, r = 2n/I)", D" = n — l—n/m. From D" = l, n = 2m/(m — 1), m — 1 = 1, m = 2, n = 4, r = 8, etc. Thus the aliquot parts up to the middle must be the successive powers of 2, and A must be a prime, since otherwise there would be new divisors. For p = 2"~\ we get A =2" — 1. Kraft observed that if we drop from Tartaglia's^^ list of 20 numbers those shown to be imperfect by Euler's^^ results, we have left only eight perfect numbers 2"-^(2"-l) for n^39, viz., those for n = 2, 3, 5, 7, 13, 17, 19, 31. For these, other than the first, as well as for the false ones of Tartaglia, if we add the digits, then add the digits of that sum, etc., we finally get unity (p. 14) [proof by WantzeP^^]. All perfect numbers end in 6 or 28. *3Comm. Acad. Petropol., 6, 1738, ad annos 1732-3, p. 103. Commentationes Arithmeticae Collectae, I, Petropoli, 1849, p. 2. ^Epistola ad mathematicos de theoria arithmetices nouis a se inuentis aucta, Vindobonae [Vienna], 1739. "De numeris perfectis, Comm. Acad. Petrop., 7, 1740, ad annos 1734-5, 7-14. 18 History of the Theory of Numbers. [Chap, i Johann Christoph Heilbronner^^ stated that the perfect numbers up to 4-10' are 6, 28, 496, 8128, 130816,2096128. "The fathers of the early church and many wTiters always held this number 6 in high esteem. God com- pleted the creation in 6 days and since all things created by Him came out perfect, he wished the work of creation completed according to the number 6 as being a perfect number." L. Euler" deduced from Fermat's theorem, which he here proved by use of the binomial theorem, the result* that, if m is a prime, 2"* — 1, when composite, has no prime factors other than those of the form wn+l. J. Landen^^ noted that 196 is the least number 4a;'*, where x is prime, the sum of whose ahquot parts exceeds the number by 7. L. Euler^^ gave a table of the prime factors of 2" — 1 for n^37. C. N. de Winsheim^° noted that 2^'^ — 1 has the factor 2351, and stated that 2" — 1 is a prime for n = 2, 3, 5, 7, 13, 17, 19, 31, composite for the remaining n<48, but was doubtful as to n = 41, thus reducing the Hst of perfect numbers given by Euler^ by one or perhaps two. He suspected that n = 41 leads to an imperfect number since it was excluded by the acute Mersenne,^° who gave instead 2^^(2^" — 1) as the ninth perfect number. He remarked that the basis of Mersenne's assertion is doubtless to be found in the stupendous genius of Mersenne which perhaps recognized more truths than he could demonstrate. He discussed the error of Hansch^ that 2" — ! is a prime if n is a prime ^ 79. G. W. Kraft^^ considered perfect numbers AP, where P is a prime [not dividing A]. Thus a{P-\-l)=2AP, where a is the sum of all the divisors of A. Hence a/ {2 A— a) equals the prime P. Let 2A— a = l, a property holding for A =2"". Then P = 2"'+^ — 1 and the resulting numbers are of Euchd's type. L. Euler,^- in a letter to Goldbach, October 28, 1752, stated that he knew only seven perfect numbers, viz., 2p~^(2^ — 1) for p = 2, 3, 5, 7, 13, 17, 19, and was uncertain whether 2^^ — 1 is prime or not (a factor is necessarily of the form 64n+l and none are <2000). ^^Historia matheseos universae. Accedit recensio elementorum compendiorum et openim math, atque historia arithmetices ad nostra tempora, Lipsiae, 1742, 755-6. There is a 63-page Ust of arithmetics of the 16th century. «^^ovi Comm. Ac. Petrop., 1, 1747-8, 20; Comm. .\rith., I, 56, §39. *We may simpUfy the proof by using the fact that 2 belongs to an e.xponent e modulo p (p a prime) such that e divides p — 1. For, if p is a factor of 2'"— 1, m is a multiple of e, whence e equals the prime m. Thus p — 1 =n7«. If we take m>2, we see that n is even since p is odd and conclude with Fermat^' that, if m is an odd prime, 2"»— 1 is divisible by no primes other than those of the form 2km + l. "•Ladies' Diary, 1748, Question 305. The Diarian Repository, Collection of all the mathe- matical questions from the Ladies' Diary, 1704-1760, by a society of mathematicians, London, 1774, 509. Button's The Diarian Miscellany (from Ladies' Diarj-, 1704-1773), London, 1775, vol. 2, 271. Leyboiu-n's Math. Quest, proposed in Ladies' D., 2, 1817, 9-10. "Opuscula varii argumenti, Berlin, 2, 1750, 25; Comm. Arith., 1, 1849, 104. •"Novi Comm. Ac. Petrop., 2, 1751, ad annum 1749, mem., 68-99. "/bid., mem., 112-3. •^Corresp. Math. Phys. (ed., Fuss), I, 1843, 590, 597-8. Chap. I] PERFECT, MULTIPLY PERFECT, AND AmICABLE NUMBERS. 19 G. W. Kraft^^ stated (p. 114) that Euler had communicated to him pri- vately in 1741 the fact that 2*^-1 is divisible by 2351. He stated (p. 121) that if 2^ — 1 is composite {p being prime), it has a factor of the form 2q'^p + l, where g is a prime [including unity], using as illustrations the factorizations noted by Euler. ^^ Of the numbers 2" — 1, n a prime ^71, stated to be prime by Hansch,^^ six are composite, while the cases 53, . . . , 71 are in doubt (p. 115). A. Saverien^^ repeated the remarks by Ens^^ without reference. L. Euler^^ stated in a letter to Bernoulli that he had verified that 2^^ — 1 is a prime by examining the primes up to 46339 which are contained in the possible forms 248n+l and 248n+63 of divisors. L. Euler^^ gave a prime factor of 2"=»= 1 for various values of n, but no new cases 2^—1 with n a prime. L. Euler,^^ in a posthumous paper, proved that every even perfect number is of Euclid's type. Let o = 2"6 be perfect, where b is odd. Let B denote the sum of the divisors of b. The sum {2'*^^ — l)B of the divisors of a must equal 2a. Thus 6/5 = (2"+^-l)/2"+\ a fraction in its lowest terms. Hence 6 = (2^+^ — 1 )c. If c = l, 6 = 2"'*'^ — 1 must be a prime since the sum of its divisors is 5 = 2""^^ whence Euclid's formula. If c>l, the sum B of the divisors of b is not less than 6+2""''^ — 1+c+l; hence ^^2"+nc+l) 2"+' 6= h '^2"+^-l' contrary to the earlier equation. The proof given in another posthumous paper by Euler^^ is not complete. L. Euler^^ proved that any odd perfect number must be of the form y.4x+ip2^ where r is a prime of the form 4nH-l [Frenicle®^]. Express it as a product ABC. . . of powers of distinct primes. Denote by a, b, c, . . .the sums of the divisors oi A, B,C,. . ., respectively. Then abc . . . = 2 ABC .... Thus one of the numbers a, b, . . . , say a, is the double of an odd number, and the remaining ones are odd. Thus B, C,. . . are even powers of primes, while A =r*^"^^ In particular, no odd perfect number has the form 4n+3. Amplifications of this proof have been given by Lionnet,^^^ Stern, ^^'^ Syl- vester, ^^^ Lucas. ^" See also Liouville^° in Chapter X. Montucla^^ remarked that Euclid's rule does not give as many perfect numbers as believed by various writers; the one often cited [Paciuolo^®] as the fourteenth perfect number is imperfect; the rule by Ozanam^^ is false since 511 and 2047 are not primes. "Novi Comm. Ac. Petrop., 3, 1753, ad annos 1750-1. "Dictionnaire universel de math, et physique, two vols., Paris, 1753, vol. 2, p. 216. »^Nouv. Mim. Acad. BerUn, ann6e 1772, hist., 1774, p. 35; Euler, Comm. Arith., 1, 1849, 584. "Opusc. anal., 1, 1773, 242; Comm. Arith., 2, p. 8. "De numeris amicabihbus, Comm. Arith., 2, 1849, 630; Opera postuma, 1, 1862, 88. '^Tractatus de numerorum doctrina, Comm. Arith., 2, 514; Opera postuma, 1, 14-15. "Recreations math, et physiques par Ozanam, nouvelle 6d. par M., Paris, 1, 1778, 1790, p. 33. Engl, transl. by C. Hutton, London, 1803, p. 35. / 20 History of the Theory of Numbers. [Chap. I Johann Philipp Griison^^'' made the same criticism of Ozanam"^ and noted that, if 2"x is perfect and x is an odd prime, 1+2+ . . .+2'' = 2'*x-a:-2x-. . .-2'*-^x = x. M. Fontana^"^ noted that the theorem that all perfect numbers are triangular is due to Maurolycus^^ and not to T. Maier (cf. Kraft^^). Thomas Taylor^"- stated that only eight perfect numbers have been found so far [the 8 listed are those of Mersenne^^j. J. Struve^^^ considered abundant numbers which are products ahc of three distinct primes in ascending order; thus ob+o+M-l 2 ^ .- — ; ; >C, >C + 1. ah-a-h-1 ' i_i_l_jL a h ab The case a^3 is easily excluded, also a = 2, 6^5 [except 2-5'7]. For a = 2, 6 = 3, c any prime > 3, 6c is abundant. Next, abed is abundant if ^"^' >d+i. a6c — (a6+ac+6c+a+6+c+l)' For a = 2, 6 = 3, c = 5 or 7, and for a = 2, 6 = 5, c = 7, abed is abundant for any prime d [>c]. Of the numbers ^ 1000, 52 are abundant. J. Westerberg^*^ gave the factors of 2"='=1 for n = l,..., 32, and of 10''±l,n = l,..., 15. O. Terquem^°^ Usted 2*^-1 and 2*^-1 as primes. L. WantzeP"® proved the remark of Kraft*^ that if A^i be the sum of the digits of a perfect number N>6 [of Euclid's type], and N2 the sum of the digits of A^i, etc., a certain iV, is unity. Since iV=l(mod 9), each Ni=l (mod 9), while the NiS decrease. V. A. Lebesgue^°^ stated that he had a proof that there is no odd perfect number with fewer than four distinct prime factors. For an even perfect number 2"?/ V . . . , y'^" ■ ■ • +prrY = (1 +y+ ■ ■ ■ +y') d +^+ • • • +^') ■ • • » ""Enthiillte Zaubereyen und Geheimnisse der Arithmetik, erster Theil, Berlin, 1796, p. 85, and Zusatz (end of Theil I). »<»Memorie dell' Istituto Nazionale Ital., mat., 2, pt. 1, 1808, 285-6. '°*The elements of a new arithmetical notation and of a new arithmetic of infinites, with an appendix .... of perfect, amicable and other numbers no less remarkable than novel, London, 1823, 131. ^''Ueber die so gennannten numeri abundantes oder die Ueberfluss mit sich fiihrenden Zahlen, besonders im ersten Tausend unsrer Zahlen, Altona, 1827, 20 pp. *'**De factoribus numerorum compositorum dignoscendis, Disquisitio Acad. CaroUna, Lundae, 1838. In the volume, Meditationum Math publice defendent C. J. D. Hill, Pt. II, 1831. i^Nouv. Ann. Math., 3, 1844. 219 (cf. 553). ^<*Ibid., p. 337. i"76id., 552-3. Chap. I] PERFECT, MULTIPLY Perfect, and Amicable Numbers. 21 the impossibility of which is evident when the exponents j3, 7, . . . are other than 1, 0, 0, . . ., a case giving Euclid's solution [cf. Desboves^^']. C. G. Reuschle^^* gave in his table C the exponent to which 2 belongs modulo p, for each prime p<5000. Thus 2" — 1 has the factor 1399 for n = 233, the factor 2687 forn = 79, and 3391 for n = 113 [as stated exphcitly by Le Lasseur^^^'^^^]. ^iso 23514513 for n = 47, 1433 for w = 179, and 1913 for n = 239. In the addition (p. 22) to Table A, he gave the prime factors of 2^* — 1 for various n's to 156, 37 being the least n for which the decomposition is not given completely, while 41 is the least n for which no factor is known. For 34 errata in Table C, see Cunningham^^° of Ch. VII. F. Landry^^^ gave a new proof that 2^^ — 1 is a prime. Jean Plana^^° gave (p. 130) the factorization into two primes: 2*^-1 = 13367X 164511353. His statement (p. 141) that 2^^ — 1 has no factor < 50033 was corrected by Landry^^^ (quoted by Lucas, "^ p. 280) and Gerardin."' Giov. Nocco^" showed that an odd perfect number has at least three distinct prime factors. For, if a"*6'* is perfect, 2a- = V-T^' 6" = ^ -^, 0—1 a— 1 whence ^ ^ Q^""^^ _(a-l)b"+l 2(5-1)" 2(6 -l)a"»" 6"+^-l ' a+fe(a6"+26"-'+2)=2+6(26"+2a6"-^). But the minunum values of a, h are 3, 5. Thus 6(a— 2)>2a — 2, a6''-26" = 6"-^-6(a-2)>6'*-^(2a-2), a6'»+26"-'>26"+2a6''-\ contrary to the earlier equation. In attempting to prove that every even perfect number 2'"6Vd' ... is of EucUd's type, he stated without proof that 2-+16V. . . =(2"'+^-l)J5C. . ., B= \ /, C = - -,.. . 0 — 1 c — 1 require that 2"*+^ = B, 6" = 2^"+^ - 1 , d' = C, . . . (the first two of which results yield Euclid's formula). F. Landry^^^ stated (p. 8) that he possessed the complete decomposi- tion of 2"±l(n^64) except for 2^^±1, 2«Hl, and gave (pp. 10-11) the factors of 2^^-l and of 2"+l for n = 65, 66, 69, 75, 90, 105. "^Mathematische Abhandlung, enthaltend neue Zahlentheoretische Tabellen sammt einer dieselben betreflfenden Correspondenz mit dem verewigten C. G. J. Jacobi. Prog., Stutt- gart, 1856, 61 pp. Described by Kummer, Jour, fur Math., 53, 1857, 379. ^°'Proc6des nouveaux pour demontrer que le nonabre 2147483647 est premier. Paris, 1859. Reprinted in SpLinx-Oedipe, Nancy, 1909, 6-9. ""Mem. Reale Ac. Sc. Torino, (2), 20, 1863, dated Nov. 20, 1859. '"Alcune teorie su'numeri pari, impari, e perfetti, Lecce, 1863. "^Aux math^maticiens de toutes les parties du monde: communicatidn but la decomposition des nombres en leurs facteurs simples, Paris, 1867, 12 pp. 22 History of the Theory of Numbers. [Chap. I F. Landry"^ soon published his table. It includes the entries (quoted byLucas^2o.i22). 2«-l=431-9719-2099863, 2*^-1=23514513.13264529, 253-1 = 6361.69431-20394401, 2^^-1 = 179951.3203431780337, the least factors of the first two of which had been given by Euler.^' •' This table was republished by Lucas^-^ (p. 239), who stated that only three entries remain in doubt: 2^^ — 1, (2^^ + l)/3, 2^*+l, each being conjectured a prime by Landry. The second was believed to be prime by Kraitchik.^^^" Landr>''s factors of 2"+l, for 28^n^64 were quoted elsewhere."^'' Jules Carvallo^^* announced that he had a proof that there exists no odd perfect number. Without indication of proof, he stated that an odd per- fect number must be a square and that the ratio of the sum of the divisors of an odd square to itself cannot be 2. The first statement was abandoned in his published erroneous proof, ^^^ while the second follows at once from the fact that, when p is an odd prime, the sum of the 2n+l divisors, each odd, of p^" is odd. E. Lucas^^^ stated that long calculations of his indicated that 2°^ — 1 and 2^^ - 1 are composite [cf . Cole,^" Powers^^^]. See Lucas-° of Ch. XVII. E. Lucas^^^ stated that 2^^ — 1 and 2^^^ — 1 are primes. E. Catalan^^^ remarked that, if we admit the last statement, and note that 2^ — 1, 2^ — 1, 2^ — 1 are primes, we may state empirically that, up to a certain limit, if 2" — 1 is a prime p, then 2^ — 1 is a prime g, 2^ — 1 is a prime, etc. [cf. Catalan^^^]. G. de Longchamps^^'^ suggested that the composition of 2"±1 might be obtained by continued multiplications, made by simple displacements from right to left, of the primes written to the base 2. E. Lucas^^^ verified once only that 2^^^ — 1, a number of 39 digits, is a prime. The method will be given in Ch. XVII, where are given various results relating indirectly to perfect numbers. He stated (p. 162) that he had the plan of a mechanism which will permit one to decide almost instan- taneously whether the assertions of Mersenne and Plana that 2" — 1 is a prime for n = 53, 67, 127, 257 are correct. The inclusion of n = 53 is an error of citation. He tabulated prime factors of 2" — 1 for n^40. E. Lucas^^^ gave a table of primes with 12 to 16 digits occurring as a factor in 2"-l for n = 49, 59, 65, 69, 87, and in 2''+l forn = 43, 47, 49, 53, 69, 72, 75, 86, 94, 98, 99, 135, and several even values of n>100. The '"Decomposition des nombres 2"=!= 1 en leurs facteurs premiers de n = 1 ^ n = 64, moins quatre, Paris, 1869, 8 pp. "3<»Sphinx-0edipe, 1911, 70, 95. "'^L'interm^diaire des math., 9, 1902, 186. '"Comptes Rendus Paris, 81, 1875, 73-75. "'Sur la th^orie des nombres premiers, Turin, 1876, p. 11; TWorie des nombres, 1891, 376. "«Nouv. Corresp. Math., 2, 1876, 96. i^Comptes Rendus Paris, 85, 1877, 950-2. "sBull. Bibl. Storia Sc. Mat. e Fis.; 10, 1877, 152 (278-287). Lucas"- " of Ch. XVII. "'Atti R. Ac. Sc. Torino, 13, 1877-8, 279. Chap. I] PERFECT, MULTIPLY PERFECT, AND AmICABLE NUMBERS. 23 verification of the primality was made by H. Le Lasseur. To the latter is attributed (p. 283) the factorization of 2'*-! for n = 73, 79, 113. These had been given without reference by Lucas. ^^° E. Lucas^^^ proposed as a problem the proof that if 8q+7 is a prime, 24a+3_i is not. E. Lucas^^^ stated as new the assertion of Euler^^ that if 4m — 1 and 8m — 1 are primes, the latter divides A = 2*"*~^ — 1. E. Lucas^^^ proved the related fact that if 8m — 1 is a prime, it divides A. For, by Fermat's theorem, it divides 2^"*"^ — 1 and hence divides A or 2^~^H-1. That the prime 8m — 1 divides A and not the latter, follows from Euler's criterion that 2^^"^^''^ — 1 is divisible by the prime p if 2 is a quad- ratic residue of p, which is the case if p = 8m='=l. No reference was made to Euler, who gave the first seven primes 4m — 1 for which 8m — 1 is a prime. Lucas gave the new cases 251, 359, 419, 431, 443, 491. Lucas^^^ elsewhere stated that the theorem results from the law of reciprocity for quadratic residues, again without citing Euler. Later, Lucas^^^ again expressly claimed the theorem as his own discovery. T. Pepin^^^ noted that if p is a prime and q = 2^ — 1 is a quadratic non- residue of a prime 4n + 1 = a^ + 6^, then qisa, prime if and only if (a — hi) / (o + hi) is a quadratic non-residue of q. A. Desboves^^^ amplified the proof by Lebesgue^^^ that every even per- fect number is of Euclid's type by noting that the fractional expression in Lebesgue's equation must be an integer which divides y^z'^ . . . and hence is a term of the expansion of the second member. Hence this expansion produces only the two terms in the left member, so that (j8+l)(7+l) . . . = 2. Thus one of the exponents, say /3, is unity and the others are zero. The same proof has been given by Lucas^^^ (pp. 234-5) and Th^orie des Nombres, 1891, p. 375. Desboves (p. 490, exs. 31-33) stated that no odd perfect number is divisible by only 2 or 3 distinct primes, and that in an odd perfect number which is divisible by just n distinct primes the least prime is less than 2". F. J. E. Lionnet^^* amplified Euler's^^ proof about odd perfect numbers. F. Landry^^^ stated that 2^^=*= 1 are the only cases in doubt in his table."' Moret-Blanc^^° gave another proof that 2^^ — 1 is a prime. ""Assoc. franQ. avanc. sc, 6, 1877, 165. »2iNouv. Corresp. Math., 3, 1877, 433. i"Mess. Math., 7, 1877-8, 186. AJso, Lucas."" i«Amer. Jour. Math., 1, 1878, 236. i^^BuU. Bibl. Storia Sc. Mat. e Fis., 11, 1878, 792. The results of this paper will be cited in Ch. XVI. ^Recreations math., ed. 2, 1891, 1, p. 236. "^Comptes Rendus Paris, 86, 1878, 307-310. "'Questions d'algebre 616mentau-e, ed. 2, Paris, 1878, 487-8. '"Nouv. Ann. Math., (2), 18, 1879, 306. "sBull. Bibl. Storia Sc. Mat., 13, 1880, 470, letter to C. Henry. "«Nouv. Ann. Math., (2), 20, 1881, 263. Quoted, with Lucas' proof, Sphinx-Oedipe, 4, 1909, 9-12. 24 History of the Theory of Numbers. [Chap, i H. LeLasseur found after^^^ 1878 and apparently just before^^^ 1882 that 2'*-l has the prime factor 11447 if n = 97, 15193 if n = 211, 18121 if 71 = 151, 18287 if n = 223, and that there is no divisor <30000 of 2"-l for the 24 prime values of n, n^257, which remain in doubt, viz. [cf. Lucas^^^], 61, 67, 71, 89, 101, 103, 107, 109, 127, 137, 139, 149, 157, 163, 167, 173, 181, 193, 197, 199, 227, 229, 241, 257. J. Carvallo^^^ attempted again^^* to prove the non-existence of odd perfect numbers y^'z^ . . .u\ where y,. . .u are distinct odd primes. He began by noting that one and only one of the exponents n, . . ., r is odd [Euler^^]. Let y<z< . . .<u, and call their number jjl. From the definition of a perfect number, y — l "' u — 1 ' y — l'u — l The fractions in this inequality form a decreasing series. Hence fe)'>^. ^<A' -~i>^' H^r Thus w(2 — A;)<2. By a petitio principii (the division by 2— A:, not known to be positive), it was concluded (p, 10) that ^<2i:^' ^^<2, y>2i/(M-i)_i- [This error, repeated on p. 15, was noted by P. Mansion. ^^] For a given n, there is at most one prime between the two limits (of difference < 2) for y. A superior limit is found for 2 as a function of y. An incomplete computation is made to show that, if /x>8, z <y-\-l. It is shown (p. 7) that an odd perfect number has a prime factor greater than the prime factor w entering to an odd power, since w+l divides the sum of the divisors. In a table (p. 30) of the first ten perfect numbers, 2^^ — 1 and 2*^ — 1 are entered as primes [contrary to Euler^^ and Plana^^°]. E. Catalan^^^ stated that 2^ — 1 is a prime if p is a prime of the form 2^ — 1. If correct this would imply that 2^^^ — 1 is a prime [cf. Catalan^^^]. E. Lucas^^^ repeated the remark of LeLasseur^^^ on the 24 prime values of n^257 for which the composition of 2^ — 1 is in doubt. According to a "iSince these four values of n are included in the list by Lucas^** of the 28 values of n ^ 257 for which the composition of 2"—! is unknown. Cf. Lucas^^^ p. 236. "2Lucas, Recreations math., 1, 1882, 241; 2, 1883, 230. Later, Lucas^^s credited LeLasseur with these four cases as well as n = 73 [Eulers^] and n = 79, 113, 233 [cf. Reuschlei"]. The last four cases were given by Lucas"*, while the last three do not occur in the table (Lucasi24^ pp 7gg_9) by LeLasseur of the proper divisors of 2"— 1 for each odd n, n<79, and for a few larger composite n's. The last three were given also by Lucas"^ (p. 236) without reference. '"Th^orie des nombres parfaits, par M. Jules Carvallo, Paris, 1883, 32 pp. "<Mathesis, 6, 1886, 147. '"Melanges Math., Bruxelles, 1, 1885, 376. '"Mathesis, 6, 1886, 146. Chap. IJ PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 25 communication from Pellet, 2" — 1 is divisible by 6n+l if n and 6n+l are primes such that 6n+l =4L^+27M^ [provided* n= 1 (mod 4), i. e., L is odd]. M. A. Stern^" amplified Euler's^^ proof concerning odd perfect numbers. E. Lucas^^^ repeated the statement [Desboves^^^] that an odd perfect number must contain at least four distinct primes. G. Valentin^^^ gave a table, computed in 1872, showing factors of 2"— 1 for n = 79, 113, 233, etc., but not the new cases of LeLasseur.^^^ The primality of iV = 2®^ — 1, a number of 19 digits, considered composite by Mersenne and prime by Landry, was established by J. Pervusin"° and P. Seelhoff ^^^ independently. The latter claimed to verify that there is no factor <N^'^ of the form 8?i+7, abbreviating the work by use of various numbers of which iV is a quadratic residue; thus iV is a prime or the product of two primes. Since iV = 2(2^°)^ — 1, 2 is a quadratic residue of any prime factor of N, so that the factor is 8n=i= 1. It was verified that 3^= 1 (mod N), where i8 = (iV — 1)/9. If N=fF, where F is the prime factor 8n-|-l, then 3^^1(mod F) and, by Fermat's theorem, 3^~^=l(mod F). It is stated without proof that one of the exponents /S and F — 1 divides the other. Cole^^^ regarded the proof as unsatisfactory. Seelhoff proved that a perfect number of the form pV is of Euclid's type if p and r are primes and p<r. The condition is r''+\2-p)-2r''{l-p)-p If p > 2, the denominator is negative. Hence p = 2 and ^'=2K^' 2'+' = r+;j-j, p = l, r=2-+'-l. His statements (p. 177) about the factors of 2" — 1, n = 37, 47, 53, 59, were corrected by him {ibid., p. 320) to accord with Landry.^^^ P. Seelhoff ^^^ obtained the known factors of these 2" — 1 and proved that 2^^ — 1 is a prime, by use of his method of quadratic residues. H. Novarese^^^ proved that every perfect number of Euclid's type ends in 6 or 28, and that each one > 6 is of the form 9A;+1. Jules Hudelot"^ verified in 54 hours that 2^^ — 1 is a prime by use of the test by Lucas, Recreations math., 2, 1883, 233. ♦Correction by Kraitchik, Sphinx-Oedipe, 6, 1911, 73; Pellet, 7, 1912, 15. "'Mathesis, 6, 1886, p. 248. "8/6td., p. 250. ""Archiv Math. Phys., (2), 4, 1886, 100-3. ""Bull. Acad. Sc. St. Petersb., (3), 31, 1887, p. 532; Melanges math. astr. ac. St. P^tersb., 6, 1881-8, 553; communicated Nov. 1883. "iZeitschr. Math. Phys., 31, 1886, 174-8. »"Archiv Math. Phys., (2), 2, 1885, 327; 5, 1887, 221-3 (misprint forn = 41). *«Jomal de sciencias math, e astr., 8, 1887, 11-14. [Servais"*.] "♦Mathesis, 7, 1887, 46. Sphinx-Oedipe, 1909, 16. 26 History of the Theory of Numbers. [Chap, i CI. Servais"^ republished the proofs by Novarese^"*^ and proved that a'"6'* is not perfect if a and h are odd primes. For, by the equations [Nocco"^] a-+i_l = 6"(a-l), b''+^-l=2a"'(6-l), we obtain, by subtraction, Thus2a'">6". Since a^3, a'"+^^3a"'>a'"+6'*>a+6-l. He next proved that, if an odd perfect number is divisible by only three distinct primes a, h, c, two of them are 3 and 5, since [as by Carvallo^^^] 04)04)04)<l Taking a = 3, 6 = 5, we have c<16, whence c = 7, 11, or 13. He quoted from a letter from Catalan that the sum of the reciprocals of the divisors of a perfect number equals 2. E. Cesaro^*^ proved that in an odd perfect jiumber containing n distinct prime factors, the least prime factor is ^n\/2. CI. Servais^^^ showed that it does not exceed n since, if a<h<c< . . . , b a+1 c a-\-2 6 — 1 a c — \ a+1 ah a a+1 a+2 a-\-n — \ 2i^. . . ^ . . . . > a—lb—1 ' a— r a a+1 a+n— 2 whence 2(a — l)<a+n — 1, a<n+l. If I is the (m — l)th prime factor and s is the 772th, and if a b a-l'b-l"l-l then ^L<2, s+1 s+n—m _. ^L{n—m)-\-2 >2, s< s — l s ' ' ' s-\-n—m-\-l ' 2—L J. J. Sylvester ^'^^ reproduced Euler's^^ proof that every even perfect number is of EucUd's type. From the fact that ■|.|-<2, he concluded that there is no odd perfect number a'"6'*. For the case of three prime factors he obtained the result of Servais^'*^ in the same manner. He proved that no odd perfect number is divisible by 105 and stated that there is none with fewer than six distinct prime factors. Sylvester^^^ and Servais^^*^ gave complete proofs that there exists no odd perfect number with only three distinct prime factors. i«Mathesis, 7, 1887, 228-230. »«/6id., 245-6. "'Mathesis, 8, 1888, 92-3. "8Nature, 37, Dec. 15, 1887, 152 (minor correction, p. 179); Coll. Math. Papers, 4, 1912, 588. "'Comptes Rendus Paris, 106, 1888, 403-5 (correction, p. 641); reproduced with notes by P. Mansion, Mathesis, 8, 1888, 57-61. Sylvester's Coll. Math. Papers, 4, 1912, 604, 615. ""Mathesis, 8, 1888, 135. Chap. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 27 Sylvester ^^^ proved there is no odd perfect number not divisible by 3 with fewer than eight distinct prime factors. Sylvester^^^ proved there is no odd perfect number with four distinct prime factors. Sylvester^^^ spoke of the question of the non-existence of odd perfect numbers as a "problem of the ages comparable in difficulty to that which previously to the labors of Hermite and Lindemann environed the subject of the quadrature of the circle." He gave a theorem useful for the investi- gation of this question: For r an integer other than 1 or —1, the sum l+r+r^+ . . . -\-r^~^ contains at least as many distinct prime factors as p contains divisors >1, with a possible reduction by one in the number of prime factors when r= — 2, p even, and when r = 2, p divisible by 6. E. Catalan ^^^ proved that if an odd perfect number is not divisible by 3, 5, or 7, it has at least 26 distinct prime factors and thus has at least 45 digits. In fact, the usual inequality gives lTl3-- I ^2 ^^^^"3 5 7 11 • I <2 3 5 7<^-^^^^- By Legendre's table IX, Theorie des nombres, ed. 2, 1808; ed. 3, 1830, of the values of P{w) up to w; = 1229, we see that I ^ 127. But 127 is the 27th prime >7. R. W. D. Christie^^^ erroneously considered 2^^ — 1 and 2^^ — 1 as primes. E. Lucas^^® proved that every even perfect number, aside from 6 and 496, ends with 16, 28, 36, 56, or 76; any one except 28 is of the form 7k='= 1 ; any one except 6 has the remainder 1,2, 3, or 8 when divided by 13, etc. E. Lucas^" reproduced his^^^ proofs and the proof by Euler,^^ and gave (p. 375) a list of known factorizations of 2" — 1. Genaille^^^ stated that his machine "piano arithm^tique " gives a prac- tical means of applying in a few hours the test by Lucas {ibid., 5, 1876, 61) for the primality of 2" — 1 . J. Fitz-Patrick and G. ChevreP^^ stated that 2^8(229-1) is perfect. E. Fauquembergue^®^ found that 2®'' — 1 is composite by a process not yielding its factors [cf. Mersenne,^" Lucas,^^^ Cole^'^^]. A. Cunningham^^^ called 2^ — 1 a Lucassian if p is a prime of the form 4A;+3 such that also 2p+l is a prime, stating that Lucas^^^ had proved that 2'' — 1 has the factor 2p+l. Cunningham listed all such primes p<2500 i"Comptes Rendus Paris, 106, 1888, 448-450; Coll. M. Papers, IV, 609-610. ^mid., 522-6; Coll. M. Papers, IV, 611-4. ""Nature, 37, 1888, 417-8; Coll. M. Papers, IV, 625-9. ""Mathesis, 8, 1888, 112-3. M6m. soc. sc. Ukge, (2), 15, 1888, 205-7 (Melanges math., III). »*Math. Quest. Educat. Times, 48, 1888, p. xxxvi, 183; 49, p. 85. "«Mathe8is, 10, 1890, 74-76. "'Theorie des nombres, 1891, 424-5. "'Assoc, frang. avanc. sc, 20, I, 1891, 159. "'Exercices d'Arith., Paris, 1893, 363. ""L'interm^diaire des math., 1, 1894, 148; 1915, 105, for representations by u*+67t;*. ""British Assoc. Reports, 1894, 563. 28 History of the Theory of Numbers. [Chap, i and considered it probable that primes of the forms 2''±1, 2'^S (if not yielding Lucassians) generally yield prime values of 2^ — 1, and that no other primes will. All known and conjectured primes 2^ — 1, with p prime, fall under this rule. In a letter to Tannery/^- Lucas stated that Mersenne^°'^^ implied that a necessary and sufficient condition that 2^—1 be a prime is that p be a prime of one of the forms 2^"+l, 2^''±3, 2^"+^ — 1. Tannery expressed his behef that the theorem was empirical and due to Frenicle, rather than to Fermat, and noted that the sufficient condition would be false if 2^' — 1 is composite [as is the case, Fauquembergue^^°]. Goulard and Tannery^^^ made minor remarks on the subject of the last two papers. A. Cunningham^^ found that 2^^' - 1 has the factor 7487. This contra- dicts LeLasseur's^^- statement on di visors < 30000 of Mersenne's numbers. A. Cunningham^^^ found 13 new cases (317, 337, 547, 937, . . .) in which 2^—1 is composite, and stated that for the 22 outstanding primes 5^257 [above list^^- except 61, 197] 2^ — 1 has no divisor < 50,000 (error as to q = lSl, see Woodall^^). The factors obtained in the mentioned 13 cases were found after much labor by the indirect method of Bickmore,^^^ who gave the factors 1913 and 5737 of 2-^^-1. A. Cunningham^" gave a factor of 2^-1 for g = 397, 1801, 1367, 5011 and for five larger primes q. C. Bourlet"^ proved that the sum of the reciprocals of all the divisors di of a perfect number n equals 2 [Catalan ^^^], by noting that n/di ranges with di over the divisors of n, so that 2n = 'Zn/di. The same proof occurs in II Pitagora, Palermo, 16, 1909-10, 6-7. M. Stuyvaert^^^ remarked that an odd perfect number, if it exists, is a sum of two squares since it is of the form pk^, where p is a prime 4n+l [Frenicle,^ Euler^sj T. Pepin^"° proved that an odd perfect number relatively prime to 3-7, 3-5 or 3-5-7 contains at least 11, 14 or 19 distinct prime factors, respectively, and can not have the form 6/cH-5. F. J. Studnicka^^^ called Ep = 2''-\2''-l) an Euclidean number if 2^-1 is a prime. The product of all the divisors <Ep of Ep is E/~'^. When Ep is written in the diadic system (base 2), it has 2p — l digits, the first p of which are unity and the last p — 1 are zero. "^L'interm^diaire des math., 2, 1895, 317. i«/6Mi., 3, 1896, 115, 188, 281. ^"Nature, 51, 1894-5, 533; Proc. Lond. Math. Soc, 26, 1895, 261; Math. Quest. Educat. Times, 5, 1904, 108, last footnote. i«British Assoc. Reports, 1895, 614. i«On the numerical factors of a"-l, Messenger Math., 25, 1895-6, 1-44; 26, 1896-7, 1-38. French transl. by Fitz-Patrick, Sphinx-Oedipe, 1912, 129-144. 155-160. "Troc. London Math. Soc, 27, 1895-6, 111. "8Nouv. Ann. Math., (3), 15, 1896, 299. "•Mathesis, (2), 6, 1896, 132. ''"Memou-e Accad. Pont. Nuovi Lincei, 13, 1897, 345-420. "'Sitzungsber. Bohm. Gesell., Prag, 1899, math, nat., No. 30. Chap. I] PERFECT, MULTIPLY PERFECT, AND AmICABLE NUMBERS. 29 Mario Lazzarini^" attempted to prove that there is no odd perfect num- ber a'^b^c'^, but made the error of thinking that a is relatively prime to 6^+. . .+6+1. He attempted to show that p = 2" — 1 is a prime if and only if p divides iV = 3*^+1, where fc = 2"~^ — 1 [false for a = 2, since p = S, N = 4:]. He restricted his argument to the case a odd, whence p = 1 (mod 3) . Then, if p is a prime, —3 is a quadratic residue of p, so that (— 3)^^~^^^^=1 (mod p), whence p divides N. Conversely, when this congruence holds, he concluded falsely that z^=—3 (mod p) has two and only two roots, so that p is expressible in a single way as a sum of a square and the triple of a square and hence is prime. To show the error, let p = ab, where a = 23, 6 = 3851 are primes; then g-l 6-1 (-3)11 + 1= _2a6,(-3) 2 =-l(mod5),(-3) ^ =(-3)^^-^^^= -1 (mod a), whence (—3)^^"^^^^=! (mod p). Cipolla remarked (p. 288) that we may deduce from a result of Lucas^^° that p is a prime if it divides N without dividing 3*+l for any divisor 8 of p=2"~^ — 1. F. N. Cole^'^ found that 2^"^ - 1 is the product of the two primes 193707721 , 761838257287. In the footnote to p. 136, he criticized the proof by Seel- hoff"^ of the primality of A^ = 2^^ — 1 and stated he had verified that N is prime by an actual computation of a series of primes of which iV is a quadratic residue. R. D. Carmichael^^^ proved that any even perfect number Tp2\ . .p/" is of Euclid's type. Write d for 2'*+^ — 1. Then, as usual, d pf d \ p/ If n>2. Pi is less than d, being an aliquot divisor of it, so that 1 + 1/p, exceeds the left member of the inequality. Hence n = 2, p2 = d. A. Cunningham^^^ gave the residues of A; =2^"*, 2*, etc., modulo 2^ — 1 for primes g^lOl. A. Turcaninov^"^ (Turtschaninov) proved that an odd perfect number has at least four distinct prime factors and exceeds 2000000. A. Gerardin^^^ noted the error by Plana. ^^° A. Gerardin^^^ stated the empirical laws: If n is a prime of the form 24a^+ll and if 2" — 1 is composite, the least factor is of the form 24?/ +23 "^Periodico di mat. insegn. sec, 18, 1903, 203; criticized by C. Ciamberlini, p. 283, and by M. Cipolla, p. 285. i"Bull. Amer. Math. Soc, 10, 1903-4, 134-7. French transl., Sphinx-Oedipe, 1910, 122-4. Cf. Fauquembergue.^^" i^^Annals of Math., (2), 8, 1906-7, 149. I'sproc. London Math. Soc, (2), 5, 1907, 259 [250]. i'6 Vest, opytn. fiziki (Spaczinskis Bote), Odessa, 1908, No. 461 (pp. 106-113), No. 463 (162-3), No. 465-6 (213-9), No. 470 (314-8). In Russian. Cf. Bourlet.is* '"L'interm^diaire des math., 15, 1908, 230-1. '"Sphinx-Oedipe, Nancy, 3, 1908-9, 113-123; Assoc, frang. avanc sc, 1909, 145-156. In Wis- kundig Tijdschrift, 10, 1913, 61, he added that in the remaining three cases <257, n = 107, 167, 227, the least divisor (necessarily >1 mUlion) is respectively 5136 y+2783, 8016 y+335, 10896 J/+5903. 30 History of the Theory of Numbers. [Chap, i {e. g., n = ll, 59, 83, 131, 179, 251). If n is a prime 24a:+23 and 2'*-l is composite, the least factor is of the form 48?/ +47 (e. g., n = 47, ?/ = 48, factor 2351; n = 23, 71, 191, 239). Gerardin^^^ gave tables of the possible, but (unverified, factors of 2" — 1, n<257. A. Cunningham^^o gave the factor 150287 of 2^^^-\. A. Cunningham^^^ found the factor 228479 of 2^^-l. T. M. Putnam^^^ proved that not all of the r distinct prime factors of a perfect number exceed 1 +r/loge2 and hence do not all equal or exceed 1 +3r/2. L. E. Dickson^^^ gave an immediate proof that every even perfect num- ber is of Euclid's type. Let 2"g be perfect, where q is odd and n > 0. Then (2"+^ — l)s = 2'^'^^q, where s is the sum of all the divisors of q. Thus s = q-\-d, where d = q/{2''^^ — l). Hence d is an integral divisor of q, so that q and d are the only divisors of q. Hence d = \ and 5 is a prime. H. J. WoodalP^^ obtained the factor 43441 of 2^^^-l. R. E. Powers^^^ verified that 2^^ — 1 is a prime by use of Lucas' test on the series 4, 14, 194, .... H. Tarry^^^ made an incomplete examination. E. Fauquembergue^^^ proved that 2^^ — 1 is a prime by writing the residues of that series to base 2. A. Cunningham^^^ noted that 2^ — 1 is composite for three primes of 8 digits. On the proof-sheets of this history, he noted that the first two should be g = 67108493, p = 134216987; 5 = 67108913, p = 134217827. A G^rardin^^^'' observed that 2'^''+^-\=F^-2(?, F = 2"+^±l = 2m+l, G = 2"±l, G2 = m2+(m+l)2-(2")2. H. Tarry^^^^ verified for the known composite numbers 2^—1, where p is a prime, that, if a is the least factor, 2" — 1 is composite. A. Gerardin added empirically that, if p is any number and a any di- visor of 2^ — 1 , a = 8m =t 1 not being of the form 2" — 1 then 2" — 1 is composite. A. Cunningham^^^ noted that, if g is a prime, M^ = 2^-\ = T^-2{quY={qtf-2U\ If Mq is a prime it can be expressed in the forms A^-]-?>B^ — G'^-\-QH'^, and in one or the other of the pairs of forms f^au^ {ci = '^, 14, 21, 42). He discussed M^ to the base 2. >'»Sphinx-Oedipe, 3, 1908-9, 118-120, 161-5, 177-182; 4, 1909, 1-5, 158, 168; 1910, 149, 166. ""Proc. London Math. Soc, (2), 6, 1908, p. xxii. "iL'intermddiaire des math., 16, 1909, 252; Sphinx-Oedipe, 4, 1909, 4e Trimestre. 36-7. "2Amer. Math. Monthly, 17, 1910, 167. ^^Ibid., 18, 1911, 109. '"Bull. Amer. Math. Soc, 16, 1910-11, 540 (July, 1911). Proc. London Math. Soc, (2), 9, 1911, p. xvi. Mem. and Proc. Manchester Literary and Phil. Soc, 56, 1911-12, No. 1, 5 pp. Sphinx-Oedipe, 1911, 92. Verification by J. Hammond, Math. Quest. Solutions, 2, 1916, 30-2. i«Bull. Amer. Math. Soc, 18, 1911-12, 162 (report of meeting Oct., 1911). Amer. Math. Monthly, 18, 1911, 195. Sphinx-Oedipe, Feb., 1912, 17-20. "«Sphinx-Oedipe, Dec, 1911, p. 192; 1912, 15. (Proc. London Math. Soc, (2), 10, 1912, Records of Meetings, 1911-12, p. ii.) ^"Ibid., 1912, 20-22. "^Messenger Math., 41, 1911, 4. "saBuU. Soc. Philomatiquesde Paris, (10), 3, 1911, 221. isseSphinx-Oedipe, 6, 1911, 174 186, 192. ""Math. Quest. Educ Times, (2), 19, 1911, 81-2; 20, 1911, 90-1, 105-6; 21, 1912, 58-9, 73. Chap.i] Perfect, Multiply Perfect, and Amicable Numbers. 31 A. Cunningham^^o found the factor 730753 of 2^^^-l. V. Ramesam^^i verified that the quotient of 2"^-l by the factor 228479 [Cunningham^^^] is the product of the primes 48544121 and 212885833. A. Aubry^^^ stated erroneously that ''Mersenne affirmed that 2" — 1 is a prime, for n^257, only for n = l, 2, 3, 4, 8, 10, 12, 29, 61, 67, 127, 257 (which has now been almost proved) ; this proposition seems to be due to Frenicle.^'" What Mersenne^" actually stated was that the first 8 perfect numbers occur at the lines marked 1, 2, 3, 4, 8, etc., in the table by Bungus. A. Cunningham^^^'' noted that M113, M151, M251 have the further factors 23279-65993, 55871, 54217, respectively. Cf. Reuschle^o^ Lucas^^s A. Gerardin^^-^ noted that there is no divisor < 1000000 of the composite Mersenne numbers not already factored. Let d denote the least divisor of 2«- 1, g a prime ^257. li q = 60z^+43, then d=47 (mod 96), except for the cases given by Euler's^^ theorem (verified for 43, 163, 223). If 5 = 40w+33, d=7 (mod 24), verified for 73, 113, 233. If 5 = 30m+l, d=l (mod 24), verified for 31, 61, 151, 181, 211. E. Fauquembergue^^^'' proved that 2^°^ — 1 is composite by means of Lucas' test with 4, 14, 194,. . ., written to base 2 (Ch. XVII). L. E. Dickson^^^ called a non-deficient number primitive if it is not a multiple of a smaller non-deficient number, and proved that there is only a finite number of primitive non-deficient numbers having a given number of distinct odd prime factors and a given number of factors 2. As a corollary, there is not an infinitude of odd perfect numbers with any given number of distinct prime factors. There is no odd abundant number with fewer than three distinct prime factors; the primitive ones with three are 3^5-7, 32527, 325.72, 3^5211, 3^13, 3*5^13, 3*52132, 3^5^132. There is given a list of the numerous primitive odd abundant numbers with four distinct prime factors and lists of even non-deficient numbers of certain types. In particular, all primitive non-deficient numbers < 15000 are determined (23 odd and 78 even). In view of these lists, there is no odd perfect number with four or fewer distinct prime factors (cf . Sylvester^*^"^^^) . A. Cunningham^^* gave a summary of the known results on the composi- tion of the 56 Mersenne numbers Mq = 2^ — 1, q a prime ^257. Of these, 12 have been proved prime: M^, 5 = 1, 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 127; while 29 of them have been proved composite. Thus only 15 remain in ""British Assoc. Reports, 1912, 406-7. Sphinx-Oedipe, 7, 1912, 38 (1910, 170, that 730753 is a possible factor). Cf. Cunningham"*. i"Nature, 89, 1912, p. 87; Sphinx-Oedipe, 1912, 38. Jour, of Indian Math. Soc, Madras, 4 1912, 56. "K)euvres de Fermat, 4, 1912, 250, note to p. 67. "2" Mem. and Proc. Manchester Lit. and Phil. Soc, 56, 1911-2, No. 1. i«* Sphinx-Oedipe, 7, 1912, num^ro special, 15-16. "'•^ Ibid., Nov., 1913, 176. i«Amer. Jour. Math., 35, 1913, 413-26. "*Proc. Fifth International Congress, I, Cambridge, 1913, 384-6. Proc. London Math. Soc, (2), 11, 1913, Record of Meeting, Apr. 11, 1912, xxiv. British Assoc. Reports, 1911, 321. Math. Quest. Educat. Times, (2), 23, 1913, 76. 32 History of the Theory of Numbers. [Chap. I doubt: M„ q = 101, 103, 107, 109, 137, 139, 149, 157, 167, 193, 199, 227, 229, 241, 257. The last has no factor under one million, as verified by R. E. Powers. ^^^^ No one of the other 14 has a factor under one milhon, as verified t^dce with the collaboration of A. G^rardin. Up to the present three errors have been found in Mersenne's assertion; Mqj has been proved composite (Lucas,^^° Cole^^^), while Mqi and Mgg have been proved prime (Pervusin,"° Seelhoff,^*^ Cole,^"^ Powers^^^). It is here announced that M^^ has the factor 730753, found with the collaboration of A. Gerardin. J. ]McDonnell^^^ commented on a test by Lucas in 1878 for the primality of 2"-l. L. E. Dickson^^® gave a table of the even abundant numbers <6232. R. Niewiadomski^^^ noted that 2^^^ — 1 has the factor 4567 and gave known factors of 2'* — 1. He gave the formula 2^'"+^-l = (2^'"+2"*-l)^+(22"'-2'"-l)^ + l. G. Ricalde^^^ gave relations between the primes p, q and least solutions of 22«+i_i = pg, al-2h^ = p, c'-2dr = q. R. E. Powers^^^ proved that 2^°" — 1 is a prime by means of Lucas'^^ test in Ch. XVII. E. Fauquembergue^'''' proved that 2^ — 1 is prime for p = 107 and 127, composite for p = 101, 103, 109. T. E. Mason-°^ described a mechanical de\'ice for applying Lucas'"^ method for testing the primality of 2^^'*'^ — 1. R. E. Powers^°^ proved that 2^°^ — 1 and 2^°^ — 1 are composite by means of Lucas' tests with 3, 7, 47, . . .and 4, 14, 194. . . (Ch. XVII), respectively. A. Gerardin^°^ gave a history of perfect numbers and noted that 2^—1 can be factored if we find t such that m = 2pt-\-\ is a prime not dividing 8 = 1+2^+22^+ . . . +2^2'-^^^ since 2-p'-1= (2^-1)8 (mod m). Or we may seek to express 2^—1 in two ways in the form x^—2y'. On tables of exponents to which 2 belongs, see Ch. VII, Cunningham and Woodam'^ Kraitchik.^-^ Additional Papers of a Merely Expository Character. E. Catalan, Mathesis, (1), 6, 1886, 100-1, 178. W. W. Rouse Ball, Messenger Math., 21, 1891-2, 34-40, 121. - Pontes (on Bovnius^"), Mem. Ac. Sc. Toulouse, (9), 6, 1894, 155-67. J. Bezdicek, Casopis Mat. a Fys., Prag, 25, 1896, 221-9. Hultsch (on lamblichus), Nachr. Kgl. Sachs. Gesell., 1895-6. H. Schubert, Math. Mussestunden, I, Leipzig, 1900, 100-5. M. Nasso, Revue de math. (Peano), 7, 1900-1, 52-53. i»*'Sphinx-Oedipe, 1913, 49-50. i«*London Math. Soc, Records of Meeting, Dec, 1912, v-vi. "«Quart. Jour. Math., 44, 1913, 274-7. '"L'interm^diaire des math., 20, 1913, 78, 167. "s/btd., 7-8, 149-150; cf. 140-1. '"Proc. London Math. Soc, (2), 13, 1914, Records of meetings, xxxi.x. Bull. Amer. Math. Soc, 20, 1913^, 531. Sphinx-Oedipe, 1914, 103-8. 20<>Sphinx-Oedipe, June, 1914, 85; I'interm^diaire des math., 24, 1917, 33. -"Proc Indiana Acad. Science, 1914, 429-431. 2«Proc London Math. Soc, (2), 15, 1916, Records of meetings, Feb. 10, 1916, xxii. »"Sphinx-Oedipe, 1909, 1-26. Chap. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 33 G. Wertheim, Anfangsgriinde der Zahlentheorie, 1902. G. Giraud, Periodico di Mat., 21, 1906, 124-9. F. Ferrari, Suppl. al Periodico di Mat., 11, 1908, 36-8, 53, 75-6 (Cipolla). P. Bachmann, Niedere Zahlentheorie, II, 1910, 97-101. A. Aubry, Assoc, frang. avanc. sc, 40, 1911, 53-4; 42, 1913; Tenseignement math., 1911, 399; 1913, 215-6, 223. *M. Kiseljak, Beitrage zur Theorie der vollkommenen Zahlen, Progr. Agram, 1911. *J. Vaes, Wiskundig Tijdschrift, 8, 1911, 31, 173; 9, 1912, 120, 187. J. Fitz-Patrick, Exercices Math., ed. 3, 1914, 55-7. Multiply Perfect Numbers. A multiply perfect or pluperfect number n is one the sum of whose divisors, including n and 1, is a multiple of n. If the sum is mn, m is called the multiplicity of n. For brevity, a multiply perfect number of multi- plicity m shall be designated by P^. Thus an ordinary perfect number is a P2. Although Robert Recorde^^ in 1557 cited 120 as an abundant number, since the sum of its parts is 240, such numbers were first given names and investigated by French writers in the seventeenth century. As a P3 equals one-half of the sum of its aliquot divisors or parts (divisors KPs), it was called a sous-double; a P4 equals one-third of the sum of its aliquot parts and was called a sous-triple; a P5 a sous-quadruple; etc. F. Marin Mersenne proposed to R. Descartes^°^ the problem to find a sous-double other than P^^^^ = 120 = 2^3-5. The latter did not react on the question until seven years later. Mersenne^"^ mentioned (in the Epistre) the problem to find a P4, a P5 or a P^, a P3 besides 120, and a rule to find as many as one pleases. He remarked (p. 211) that the P3 120, the P4 240 [for 30240?] and all other abundant numbers can signify the most fruitful natures. Pierre de Fermat^"- referred in 1636 to his former [lost] letter in which he gave "the proposition concerning aliquot parts and the construction to find an infinitude of numbers of the same nature." He^^^ found the second P3, viz., P3<2) = 672 = 2^3-7. Mersenne^°* stated that Fermat found the 1 3 7 15 . . . P3 672 and knew infallible rules and analysis 2 4 8 16 . . . to find an infinitude of such numbers. He^°^ 3 5 9 17 . . . later gave [Fermat's] method of finding such P3: Begin with the geometric '""Oeuvres de Descartes, 1, Paris, 1897, p. 229, line 28, letter from Descartes to Mersenne, Oct or Nov., 1631. ^"iLes Preludes de I'Harmonie Universelle ou Questions Curiouses, Utiles aux Predicateurs, aux Theologiens, Astrologues, Medecins, & Philosophes, Paris, 1634. '•"Oeuvres de Fermat, 2, Paris, 1894, p. 20, No. 3, letter to Mersenne, June 24, 1636. "^Oeuvres de Fermat, 2, p. 66 (French transl. 3, p. 288), 2, p. 72, letters to Mersenne and Roberval, Sept., 1636. '"Harmonic Universelle, Paris, 1636, Premiere Preface Generale (preceded by a preface of two pages), imnumbered page 9, remark 10. Extract in Oeuvres de Fermat, 2, 1894, 20-21. "'Mersenne, Seconde Partie de I'Harmonie Universelle, Paris, 1637. Final subdivision: Nou- velles Observations Physiques et Math^matiques, p. 26, Observation 13. Extract in Oeuvres de Fermat, 2, 1894, p. 21. , 34 History of the Theory of Numbers. [Chap. I progression 2, 4, 8, ... . Subtract unity and place the remainders above the former. Add unity and place the sums below. Then if the quotient of the (n+3)th number of the top line by the nth number of the bottom line is a prime, its triple multiplied by the (n+2)th number of the middle line is a P3. Thus if ?i = l, 15/3 is a prune and 3-5-8 = 120 is a P3. For n = 3, 63/9 is a prime and 3-7-32 = 672 is a P3. [This rule thus states in effect that 3-2"+2p is a P3 if p = (2'»+3-l)/(2"+l) is a prune.] The third P3, discovered by Andr6 Jumeau, Prior of Sainte-Croix, is P3^'^ = 523776 = 29311-31. In April, 1638, he communicated it to Descartes^°^ and asked for the fourth P3 (the fifth and last of St. Croix's challenge problems). Descartes^*^^ stated that the rule ^^^ of Fermat furnishes no P3 other than 120 and 672 and judged that Fermat did not find these numbers by the formula, but accommodated the formula to them, after finding them by trial. Descartes^^^ answered the challenge of St. Croix with the fourth P3, P3^'^ = 1476304896 = 2^33.1143.127. Soon afterwards Descartes^ °^ announced the following six P4: P4^i) = 30240 = 2^335.7, P4(2) =32760 = 2^325.7.13, P^^^^ = 23569920 = 2^335. 1 1 -31 , P4(*) = 142990848 = 2^327.11.13.31, P4(^> = 66433720320 = 2^^335. 1 1 .43. 1 27, P4^®^ =403031236608 = 2^3327.11.13.43.127, and the sous-quadruple Ps^^^ = 14182439040 = 2^3^5.7-11217.19. He stated that his analysis had led him to a method which would require time to explain in the form of a rule, but that he could find, for example, a sous-centuple, necessarily very large. Fermat apparently responded to the fifth challenge problem of St. Croix on the fourth P3. Without warrant, Descartes^^" suspected that Fermat had not found independently the fourth P3, but had learned from some one in Paris of its earlier discovery by Descartes. Fermat^^^ indicated that he possessed an analytic method by which he could solve all questions con- '"^Oeuvres de Descartes, 2, Paris, 1898, p. 428, p. 167 (latter without name of St. Croix); cf. Oeuvres de Fermat, 2, 1894, pp. 63-64. "'Oeuvres de Descartes, 2, 1898, p. 148, letter to Mersenne, May 27, 1638. ^osQeuvres de Descartes, 2, 1898, 167, letter to Mersenne, June 3, 1638. '"'Oeuvres de Descartes, 2, 1898, 2.50-1, letter to Mersenne, July 13, 1638. In June, 1645, Descartes, 4, 1901, p. 229, again mentioned the first two of these Pt. ""Oeuvres de Descartes, 2, 1898, 273, letter to Mersenne, July 27, 1638. "^Oeuvres de Fermat, 2, 1894, p. 165, No. 4; p. 176, No. 1; letters to Mersenne, Aug. 10 and Dec. 26, 1638. Chap. I] PERFECT, MULTIPLY PERFECT, AND AmICABLE NUMBERS. 35 cerning aliquot parts, apart from the testing of the primaUty of a number n, knowing no method except the trial of each number < \/n as a divisor. Descartes"^ gave the following rules for multiply perfect numbers: I. If n is a P3 not divisible by 3, then 3n is a P4. II. If a P3 is divisible by 3, but by neither 5 nor 9, then 45P3 is a P4. III. If a P3 is divisible by 3, but not by 7, 9 or 13, then 3-7-13 P3 is a P4. IV. If n is divisible by 2^ but by no one of the numbers 2^°, 31, 43, 127, then 31n and 16-43-127n are proportional to the sums of their aliquot parts. V. If n is not divisible by 3 and if 3n is a P^k, then n is a Ps^. By applying rule II to P3^^\ Ps^^\ P3^*^ Descartes obtained his P^^^^ P4<3^ P^^^K By applying rule III to Ps^'\ Ps^^\ Ps^''\ he obtained his p (2) p (4) p (6) ^4 ) ^4 5 ^4 • In the same letter, Descartes expressed to Mersenne a desire to know what Frenicle de Bessy had found on this subject. Frenicle wrote direct to Descartes, who in his reply^^^ expressed his astonishment that Frenicle should regard as sterile the above rules for finding P4, since Descartes had deduced by them six P4 from four P3, at a time when Mersenne had stated to Descartes that it was thought to be impossible to find any at all. Des- cartes stated that, since one can find an infinity of such rules, one has the means of finding an infinitude of P^- From one of Frenicle's P5 (com- municated to Descartes by Mersenne), Ps^^) = 30823866178560 = 2i°3^5-72l3- 19-23-89, Descartes (p. 475) derived the smaller P5: Pg^^) = 31998395520 = 2^3^5.72.13.17.19. Mersenne^^^ listed various P^ due to his correspondents, without cita- tion of names. He listed the above Ps^'^ (^ = 1, 2, 3, 4) and remarked that "un excellent esprit "^^^ found that when P3^'^ = 459818240 = 2^5.7.19.37.73 is multiplied by 3, the product is a P4: P4(^ = 2^3.5.7.19.37.73, attributed to Lucas^^^ by Carmichael.^^^ "^Oeuvres, 2, 1898, 427-9, letter to Mersenne, Nov. 15, 1638. "'Oeuvres de Descartes, 2, 1898, 471, letter to Frenicle, Jan. 9, 1639. '"Les Nouvelles Pensees de Galilei, traduit d'ltalien en Frangois, Paris, 1639, Preface, pp. 6-7. Quoted in Oeuvres de Descartes, 10, Paris, 1908, pp. 564-6, and in Oeuvres de Fermat, 4, 1912, pp. 65-66. '"Frenicle de Bessy, according to the editors of the Oeuvres de Fermat, 2, 1894, p. 255, note 2; 4, 1912, p. 65, note 2 (citing Oeuvres de Descartes, 2, letter Descartes to Mersenne, Nov. 15, 1638, pp. 419-448 [p. 429]). It is clear that the discoverers Fermat, St. Croix, and Descartes of the P|^*) (i = 2, 3,4) are not meant. It is attributed to Legendre"" by Carmichael.*'* 36 History of the Theory of Numbers. [Chap, i There are listed Descartes' six P^ and P^^^^ Frenicle's Ps^\ and also P4^«) =45532800 = 2^3^5217.31, P4^^^ = 43861478400 = 2^°3^5223.31.89, and the erroneous P5 508666803200 (not divisible by 5^+5-|-l), probably a misprint for the correct P5 (in the list by Lehmer^'^) : P5^'^ = 518666803200 = 2^^3'5-72l3- 19-31 . A part of these Pm, but no new ones, were mentioned by Mersenne^" in 1644; the least P3 is stated to be 120. (Oeuvres de Fermat, 4, 66-7.) In 1643 Fermat^ ^^ cited a few of the P^ he had found: Ps^^^ = 51001 180160 = 2^*5.7.19.31.151 , P4^'^) = 14942123276641920 = 2^3^5.17.23.137.547.1093, P5(^) = 1802582780370364661760 = 22°335.72l32l9.31.61. 127.337, Ps^®^ = 87934476737668055040 = 2^^3^5.7313. 19-37.73. 127, Pg(i) = 223375374113133^7231.41.^1.241.307.467.2801, Pg(2) = 22735537.11.13219.29.31.43.61.113.127. He stated that he possessed a general method of finding all P^n- Replying to Mersenne's query as to the ratio of Pg(3) = 22^3^5^11.13219.31243.61.83.223.331.379.601.757 X 1201.7019.823543-616318177:100895598169 to the sum of its aUquot parts, Fermat^^^ stated that it is a Pq, the prime factors of the final factor being 112303 and 898423 [on the finding of these factors, see Ch. jXIV, references 23, 92, 94, 103]. Note that 823543 = 7^ Descartes^^^ constructed P3^2) = 572 = 21.32 by starting with 21 and noting that (r(21) =32, o-(32) =63 = 3.21, for a defined as on p. 53. Mersenne^^ noted that if a P3 is not divisible by 3, then 3P3 is a P4 [rule I of Descartes^^-]; if a P5 is not divisible by 5, then 5P5 is a Pq, etc. He stated that there had been found 34 P4, 18 P5, 10 Pg, 7 P7, but no Pgso far. In 1652, J. Broscius (Apologia,^* p. 162) cited the P4^'^ [of Descartes^"^]. The P3 120 and 672 are mentioned in the 1770 edition of Ozanam's'^ Recreations, I, p. 35, and in Hutton's translation of Montucla's^^ edition, I, p. 39. A. M. Legendre^^^ determined the Pm of the form 2"a/37 . . . , where a, /3, 7, . . .are distinct odd primes, for 7n = 3, n^8; ?n=4, n = 3, 5; m = 5, n = 7. No new P„ were found. "*Oeuvres, 2, 1894, p. 247 (261), letter to Carcavi; Varia opera, p. 178; Pr^cia des oeuvres math, de Fermat, par E. Brassinne, Toulouse, 1853, p. 150. 3'^Oeuvers de Fermat, 2, 1894, 255, letter to Mersenne, April 7, 1643. The editors (p. 256, note) explained the method of factoring probably used by Fermat. The sum of the aliquot parts of 23« is 223iV, where N = 616318177, and the sum of the aliquot parts of N is 2-7? M iVf = 898423. As M does not occur elsewhere in Pe., it is to be expected as a factor of the final factor of Pe. "8Manuscript published by C. Henry, Bull. Bibl. Storia Sc. Mat. e Fis., 12, 1879, 714. '^•Thor^ie des nombres, 3d ed., vol. 2, Paris, 1830, 146-7; German transl. by H. Maser, Leipzig, 2, 1893, 141-3. The work for n? =3 was reproduced by Lucas'^" without reference. Chap. I] PERFECT, MULTIPLY PERFECT, AND AmICABLE NUMBERS. 37 E. Lucas^^o ga^g a table of P^ of the form 2''-\2''-l)N which includes only 15 of the 26 Pm given above and no additional P^, m>2, except two erroneous P5 : 2^°3^5-72.1l2.19.23.89, 2i^5-72l3.19237-73-127, attributed elsewhere^^^ by him to Fermat. If we replace 7^ by 7 in the former, we obtain a correct P5 listed by Carmichael 'P"^ P5(7> = 2'°3*5-7. 11219.23-89. If in the second, we replace 5-7^ by 3^-5-7^ we obtain Fermat's P^-^K A. Desboves^^^ noted that 120 and 672 are the only P3 of the form 2"-3-p, where p is a prime. D. N. Lehmer^^^ gave the additional P„: p^(i2) ^22325.7213.19, P^(i3) = 2«3272l3.19237.73-127, P5® =2213^527.19.23231.79.89.137.547.683.1093, Pg(4) =2193^5^7211. 13.19.23.31.41. 137.547.1093, Pg(5) =22^3^5.7211.13.17.19231.43.53.127.379.601.757.1801. He readily proved that a P3 contains at least 3 distinct prime factors, a P4 at least 4, a P5 at least 6, a Pq at least 9, a P^ at least 14. J. Westlund324 proved that 2^3.5 and 2^3.7 are the only P3 of the form V\'V2Vzy where the p's are primes and Pi<P2<P3. He^^^ proved that the only P3 = Pi"p2P3P4, Pi<P2<P3<P„ is P3^'^ =293.11.31. A. Cunningham^^^ considered P^ of the form 2^ \2^—l)F, where F is to be suitably determined. There exists at least one such P^ for every q up to 39, except 33, 35, 36, and one for g = 45, 51, 62. Of the 85 P^ found, the only one published is the largest one, viz., for q = Q2, giving Pq^^^ with F = 3^5'72ll. 13.19223.59.71.79.127.157.379.757.43331.3033169; while none have m>6, and for m = 3 at most one has a given q. He found in 1902 (but did not publish) the two P7 = 2^H2^^-1)P, where F = C.192127 or 0.19^51-911, C = 3i^-5^.7^.1M3.17-23.31-37-41.43.61.89.97-193.442151. R. D. CarmichaeP^^ has shown that there exists no odd P^ with only three distinct prime factors; that 2^3-5 and 2^3.7 are the only P^ with only '20Bull. Bibl. e Storia Mat. e Fis., 10, 1877, 286. In 253-5-7, listed as a Pt, 3 is a misprint for 3». '"Lucas, Theorie des Nombres, 1, Paris, 1891, 380. Here the factor 11^ IS^ of Fermat's P«(') is given erroneously as lllS^, while the Pe^i^ of Descartes is attributed to Fermat. '^Questions d'Algebre, 2d ed., 1878, p. 490, Ex. 24. '^'Annals of Math., (2), 2, 1900-1, 103-4. "^Annals of Math., (2), 2, 1900-1, 172-4. '"Annals of Math., (2), 3, 1901-2, 161-3. '"British Association Reports, 1902, 528-9. '"American Math. Monthly, 13, Feb., 1906, 35-36. 717 8 5 38 History of the Theory of Numbers. [Chap, i three distinct prime factors ;^^^ that those with only four distinct prime fac- tors are^^" the P^^^^ of St. Croix^"^ and the P4^'^ of Descartes f°^ and that the even P„ with five^^^ distinct prime factors are P3^^\ Pi^\ Pi^^ of Des- cartes^"^' ^"^ and P^^^^ of Mersenne.^" CarmichaeP^^'* stated and J. Westlund proved that if n>4, no P„ has only n distinct prime factors. Carmichael's^^^ table of multiply perfect numbers contains the misprint 1 for the final digit 0 of Descartes' Pi^\ and the erroneous entry 919636480 in place of its half, viz., P^-^^ of Mersenne.^^^ The only new P^ is Pg(7) = 2^^3^527211. 13-17.19-3143.257. All P,;,< 10^ were determined; only known ones were found. CarmichaeP^^ gave an erroneous P5 and the new P^: p^(i4)^2"3272l3-1923M27.151, p^(i5)^2253^52l9-31.683.2731-8191, p^(i6)^225365-19-23-137-547-683-1093-2731-8191. Carmichael and T. E. Mason^^ gave a table which includes the above hsted 10 P2, 6 P3, 16 P4, 8 P5, 7 Pq, together with 204 new multiply perfect numbers P, (i = 3, . . . , 7) . Of the latter, 29 are of multiplicity 7, each having a very large number of prime factors. No P7 had been previously published. [As a generalization, consider numbers n the sum of the kth. powers of whose divisors < n is a multiple of n. For example, n = 2p, where p is a prime 8/1 ±3 and k is such that 2*^+1 is divisible by p; cases are p = 3, k = l; p = 5, k = 2; p = ll, k = 5; p = 13, A; = 6.] Amicable Numbers. Two numbers are called amicable* if each equals the sum of the aliquot divisors of the other. According to lamblichus^ (pp. 47-48), "certain men steeped in mistaken opinion thought that the perfect number was called love by the Pythago- reans on account of the union of different elements and affinity which exists in it; for they call certain other numbers, on the contrary, amicable num- bers, adopting virtues and social quahties to numbers, as 284 and 220, for the parts of each have the power to generate the other, according to the rule of friendship, as Pythagoras affirmed. WTien asked what is a friend, he replied, 'another I,' which is shown in these numbers. Aristotle so defined a friend in his Ethics." «»Aimalsof Math., (2), 7, 1905-6, 153; 8, 1906-7, 49-56; 9, 1907-8, 180, for a simpler proof that there is no Pa = Pi^p^Vi^t c> 1. ""Annals of Math., (2), 8, 1906-7, 149-158. "'Bull. Amer. Math. Soc, 15, 1908-9, pp. 7-8. Fr. transl., Sphinx-Oedipe, Nancy, 5, 1910, 164-5. wi^Amer. Math. Monthly, 13, 1906, 165. »«Bull. Amer. Math. Soc, 13, 1906-7, 383-6. Fr. transl., Sphinx-Oedipe, Nancy, 5, 1910, 161-4. »"Sphinx-Oedipe, Nancy, 5, 1910, 166. »"Proc. Indiana Acad. Sc, 1911, 257-270. *Amiable, agreeable, befreundete, verwandte. Chap. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 36 In the ninth century the Arab Thabit ben Korrah^° (prop. 10) noted that 2'*/i^ and 2"s are amicable numbers if (1) /i = 3-2"-l, f=3-2"-i-l, s = 9'2^''-'-l are primes > 2, literally, if /l = 2+2^ t = z-2''-\ 3 = H-2+...+2^ s = (2"+^+2'*-2)2''+^-l. The term used for amicable numbers was se invicem amantes. In the article in which F. Woepcke^'^ translated this Arabic manuscript into French, he noted that a definition of these numbers, called congeneres, occurs in the 51st treatise (on arithmetic) of Ikhovan Algafa, manuscript 1105, anciens fonds arabes, p. 15, of the National Library of Paris. Among Jacob's presents to Esau were 200 she-goats and 20 he-goats, 200 ewes and 20 rams (Genesis, XXXII, 14). Abraham Azulai^^^ (1570- 1643), in commenting on this passage from the Bible, remarked that he had found written in the name of Rau Nachshon (ninth century A. D.): Our ancestor Jacob prepared his present in a wise way. This number 220 (of goats) is a hidden secret, being one of a pair of numbers such that the parts of it are equal to the other one 284, and conversely. And Jacob had this in mind; this has been tried by the ancients in securing the love of kings and dignatories. Ibn Khaldoun^^° related "that persons who have concerned themselves with talismans affirm that the amicable numbers 220 and 284 have an influence to establish a union or close friendship between two individuals. To this end a theme is prepared for each individual, one during the ascend- ency of Venus, when that planet is in its exaltation and presents to the moon an aspect of love or benevolence; for the second theme the ascendency should be in the seventh. On each of these themes is written one of the specified numbers, the greater (or that with the greater sum of its aliquot parts?) being attributed to the person whose friendship is sought." The Arab El Madschriti,^^! or el-Magriti, (flOOT) of Madrid related that he had himself put to the test the erotic effect of ''giving any one the smaller number 220 to eat, and himself eating the larger number 284." Ibn el-Hasan^"'' (tl320) wrote several works, including the "Memory of Friends," on the explanation of amicable numbers. Ben Kalonymos^^^^ discussed amicable numbers in 1320 in a work written for Robert of Anjou, a fragment of which is in Munich (Hebr. MS. 290, f. 60). A knowledge of amicable numbers was considered necessary by Jochanan Allemanno (fifteenth century) to determine whether an aspect of the planets was friendly or not. ^^^Baale Brith Abraham [Commentary on the Bible], Wilna, 1873, 22. Quotation suppUed by Mr. Ginsburg. '^oProlegomenes hist. d'Ibn Khaldoun, French transl. by De Slane, Notices et Extraits des Manuscrits de la Bibl. Imperiale, Paris, 21, I, 1868, 178-9. "^'Manuscript Magriti; Steinschneider, Zur pseudoepigraphischen Literatur inbesondere der geheimen Wissenschaften des Mittelalters, Berlin, 1862, p. 37 (cf. p. 41). '""H. Suter, Abh. Gesch. Math. Wiss., 10, 1900, 159, § 389. »"*Hebr. Bibl., VII, 91. Steinschneider, Zeitschrift der Morgenlandischen Ges., 24, 1870, 369. 5 11 23 47 2 4 8 16 6 12 24 48 71 287 1151 40 History of the Theory of Numbers. [Chap, i Alkalacadi,^" a Spanish Arab (tl486), showed the method of finding the least amicable numbers 220, 284. Nicolas Chuquet^^ in 1484 and de la Roche^^ in 1538 cited the amicable numbers 220, 284, "de merueilleuse familiarite lung auec laultre." In 1553, Michael StifeP^ (folios 26v-27v) mentioned only this pair of amicable num- bers. The same is true of Cardan, ^^ of Peter Bungus^- (Mysticae numerorum signif., 1585, 105), and of TartagUa.^^ Reference may be made also to Schwenter." In 1634 Mersenne^"^ (p. 212) remarked that "220 and 284 can signify the perfect friendship of two persons since the sum of the aliquot parts of 220 is 284 and conversely, as if these two numbers were only the same thing." According to Mersenne's^"^ statement in 1636, Fermat^^ found the second pair of amicable numbers 17296 = 2'.23.47, 18416 = 2^-1151, and communicated to Mersenne^°^ the general rule: Begin with the geo- metric progression 2, 4, 8, ... , write the prod- ucts by 3 in the line below; subtract 1 from the products and enter in the top row. The bottom row is 6-12-1, 12-24-1,. . .When a mmiber of the last row is a prime (as 71) and the one (11) above it in the top row is a prime, and the one (5) preceding that is also a prime, then 71.4 = 284, 5-11-4 = 220 are amicable. Similarly for 1151-16 = 18416, 23-47-16 = 17296, and so to infinity. [The rule leads to the pair 2"/i<, 2'*s, where h, t, s are given by (1).] Descartes^^^ gave the rule: Take (2 or) any power of 2 such that its triple less 1, its sextuple less 1, and the 18-fold of its square less 1 are all primes;* the product of the last prime by the double of the assumed power of 2 is one of a pair of amicable numbers. Starting with the powers 2, 8, 64, we get 284, 18416, 9437056, whose aliquot parts make 220, etc. Thus the third pair is 9363584 = 2^-191-383, 9437056 = 2"-73727. Descartes^^^ stated that Fermat's rule agrees exactly with his own. Although we saw that Mersenne quoted in 1637 the rule in Fermat's form and expressly attributed it to Fermat, curiously enough Mersenne^ ^^ gave in 1639 the rule in Descartes' form, attributing it to "un excellent Geometre" (meaning without doubt Descartes, according to C. Henry^"), ''^Manuscript in Biblioth^que Nationale Paris, a commentary on the arithmetic Talkhys of Ibn Albanna (13th cent.). Cf. E. Lucas, L'arithm^tique amusante, Paris, 1895, p. 64. '"Quesiti et Inventione, 1554, fol. 98 v. '^♦Oeuvres de Fermat, 2, 1894, p. 72, letter to Roberval, Sept. 22, 1636; p. 208, letter to Frenicle, Oct. 18, 1640. »^euvre8 de Descartes, 2, 1898, 93-94, letter to Mersenne, Mar. 31, 1638. •Evidently the numbers (1) if the initial power of 2 be 2""^ "•Oeuvres de Descartes, 2, 1898, 148, letter to Mersenne, May 27, 1638. "'Bull. Bibl. Storia So. Mat. e Fis., 12, 1879, 523. Chap. I) PERFECT, MULTIPLY PERFECT, AND AmICABLE NUMBERS. 41 and derived as did Descartes the first three pairs of amicable numbers from 2, 8, 64. We shall see that various later writers attributed the rule to Descartes. Mersenne^^ again in 1644 gave the above three pairs of amicable num- bers, the misprints in both^^^ of the numbers of the third pair being noticed at the end of his book, and stated there are others innumerable. Mersenne®^ in 1647 gave without citation of his source the rule in the form 2-2% 2-2"/i<, where 1 = 3-2'' -1, h = 2t+l, s = ht+h-i-t are primes [as in (1)]. Frans van Schooten,^^^ the younger, showed how to find amicable numbers by indeterminate analysis. Consider the pair 4a:, 4yz [x, y, z odd primes]; then 7+3a: = 4i/0, 7-\-7y+7z+Syz = 4x. Eliminating x, we get 2 = 34-16/(2/ — 3). The case ^ = 5 gives 2;= 11, ^' = 71, yielding 284, 220. He proved that there are none of the type 2x, 2yz, or 8x, Syz, and argued that no pair is smaller than 284, 220. For 16.x, 16yz, he found 2=15-l-256/(i/ — 15), which for y = 47 yields the second known pair. There are none of the type 32a;, S2yz, or type 64a:, Myz. For 128a:, 1281/2, he got 2= 127-^16384/(1/ -127), which for 2/ = 191 yields the third known pair. Finally, he quoted the rule of Descartes. W. Leyboum^^ stated in 1667 that ''there is a fine harmony between these two numbers 220 and 284, that the aliquot parts of the one do make up the other . . . and this harmony is not to be found in many other numbers." In 1696, Ozanam''^ gave in great detail the derivation of the three known pairs of "amiable" numbers by the rule as stated by Descartes, whose name was not cited. Nothing was added in the later editions.'^' ^^ Paul Halcke^^" gave Stifel's^^ rule, as expressed by Descartes.^^^ E. Stone^^^ quoted Descartes' rule in the incorrect form that 2^''pq and 3-2"p are amicable if p = 3-2'*— 1 and g = 6-2'*— 1 are primes. Leonard Euler^^^ remarked that Descartes and van Schooten found only three pairs of amicable numbers, and gave, without details, a fist of 30 pairs, all included in the later paper by Euler.^^^ G. W. Kraft^^^ considered amicable numbers of the type APQ, AR, where P, Q, R are primes not dividing A. Let a be the sum of all the divi- sors of A . Then R+1 = {P+1){Q+1), {R-{-l)a = APQ-^AR. Assuming prime values of P and Q such that the resulting R is prime, he sought a number A for which A /a has the derived value. For P = 3, Q = 1 1 , "'Not noticed in the correction (left in doubt) in Oeuvres de Fermat, 4, 1912, p. 250 (on pp. 66-7). One error is noted in Broscius*^, Apologia, 1652, p. 154. "'Exercitationum mathematicarum libri quinque, Ludg. Batav., 1657, liber V : sectiones triginta miscellaneas, sect. 9, 419-425. Quoted by J. Landen.*^ ""Deliciae Mathematicae, oder Math. Sinnen-Confect, Hamburg, 1719, 197-9. *"New Mathematical Dictionary, 1743 (under amicable) . "'De numeria amicabilibus. Nova Acta Eruditorum, Lipsiae, 1747, 267-9; Comm. Arith. Coll., II, 1849, 637-8. -•»Novi Comm. Ac. Petrop., 2, 1751, ad annum 1749, Mem., 100-18. 42 History of the Theory of Numbers. [Chap, i then A:a = S:5; he took A = 3B, S^B, 3^5, but found no solution. For p = 5, Q = 41, we have 72 = 251, 38A = 21a; set A = 495, whence 3-576 = 38-75, where b is the sum of the divisors of B; set B = 9C, whence C:c = 13:14, C=13, yielding the amicable numbers 5-41A., 251A, where A = 3^-7213 = 5733 [the pair VII in Euler's^'^^ ijgt and (7) in the table below]. Again, to make A/a = 3/8, set A = 35, whence a = 46 and the condition is 6 = 25, whence 5 is a perfect number prime to 3. Using 5 = 28, we get A = 84. For use in such questions, Kraft gave a table of the sum of the divisors of each number^ 150. He quoted the rule of Descartes. L. Euler^*^^ obtained, in addition to two special pairs, 62 pairs [including two false pairs] of amicable numbers of the type am, an, in which the common factor a is relatively prime to both vi and n. He wrote jm for the sum of all the divisors of m. The conditions are therefore jm=jn, fa-jm = a{m-{-n). If m and n are both primes, then 7n = n and we have a repeated perfect number. Euler treated five problems. (1) Euler's problem 1 is to find amicable numbers apq, ar, where p, q, r, are distinct primes not dividing the given number a. From the first con- dition we have r = xy — \, where x — p-\-\, y = q-\-\. From the second, xyi a = a{2xy —x — y). Let a/{2a— \a) equal 6/c, a fraction in its lowest terms. Then y = bx/{cx — b), {cx — b){cy — b)=b^. Thus x and y are to be found by expressing 6^ as a product of two factors, increasing each by 6, and dividing the results by c. (li) Fu-st, takea = 2". Then6 = 2^ c=l, x, 2/ = 2"**+2^ Letri-A; = m. Then p = 2"»(22*=+2*)-l, g = 2'"(l+2*)-l, r = 2^"'(2^''+^+2^''+2'')-l. When these three are primes, 2"*"^^'^^ and 2'"''"^ are amicable. Euler noted that the rule communicated by Descartes to van Schooten is obtained by taking A:= 1, and stated that 1, 3, 6 are the only values ^ 8 of m which yield amicable numbers (above^^^). For k = 2 or 4, Euler remarked that r is divisible by 3; for k = 3, vi<Q, and for k = 5, mS2, p, q, or r is composite. (I2) Take a = 2J, where /=2"+^+e is a prune. Then 2a-fa = e+l. If e+1 divides a, we have c = l. Set e+ 1 = 2^, A;^?n, n = m+A:. Then /=2*(2'"+^ + l)-l, a = 2"'+i, 6 = 27, b""={x-b){y-b). For k = l, /=2'"+2+l is to be a prime, whence m+2 is a power of 2. If w = 0, 6=/=5, and either x = y, p = q; or x, y = Q,30; p, q = 5, 29, whereas p and q are to be distinct and prime to 10. If m = 2, /=17, 68^ is to be resolved into distinct even factors; in the four resulting cases, p, q, r are '"De numeris amicabilibus, Opuscula varii argumenti, 2, 1750, 23-107, Berlin; Comm. Arith., 1, 1849, 102-145. French transl. in Sphinx-Oedipe, Nancy, 1, 1906-7, Supplement I-LXXVI. Chap. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 43 not all prime. In the next case w = 6, /=257, Euler examined only the case* x — 6 = 2^-257, finding q composite. For A; = 2, Euler excluded m = 1, 3 [m = 4 is easily excluded]. (13) For k^n in (I2), 0 = 2*", where m = k—n. Then 6 = 2"+i+2"'+"-l=/ must be a prime. Thus we must take as the factors of 6^ 2'"x-6 = l, 2"'2/-fe = 62, whence z = 2" +2"+^"'", y-=hx. If m = 1, one of /=2'»+2-l, p = 2^+'-l has the factor 3 and yet must be a prime; hence n=l, g = 27. If m = 2, Euler treated the cases n^5 and found (for n = 2) the pair (4) of the table. [For 6^n^l7, / or p is composite.] For m odd and >1, / or p has the factor 3. For m = 4, n^ 17, no solution results. (14) For a = 2"(gr— l)(/i — 1), where the last two factors are prime, set d = 2a-fa. Then ig - 2^*+^) {h - 2^^+^) =d- 2"+^ + 2^""+^ Euler treated the cases n^3, d = 4, 8, 16, finding only the pair (9). (15) Special odd values of a led (§§56-65) to seven pairs (5)-(8), (11)-(13). The cases a = 3^.5, 32-72-13-19 were unfruitful. (2) Euler's problem 2 is to find amicable numbers apq, ars, where p, q, r, s are distinct primes not dividing the given number a. Since jP'j2 — ['>''] s, we may set p = ax — l, q = ^y — l, r = /3x — 1, s = ay — l. We set fa:a = 2b—c:h, where b and c are relatively prime. The second condition fa- fpq = a{pq+rs) gives ca/3x2/ = 6(a+iS) (x+2/) -26. Multiply it by ca^. Then [ca^x-h{a+^)][ca^y-b(a+^)] = h\a+^y-2hca^. Given a, ^, a and hence b, c, we are to express the second member as a prod- uct of two factors and then find x, y. For a = l, i3 = 3, 0 = 2^ Euler obtained the pairs (a), (28). For a = 2, ^ = 3, a = 32-5-13, he got (32); for a = l, /3 = 4, a = 3^-5, (30). The ratio a:^ may be more complex, as 5 :21 or 1 :102, in (7). As noted by K. Hunrath,^^^* the numbers (7) are not amicable. Nor are the ratios as given, although these ratios result if we replace 8563 by 8567 = 13-659. This false pair occurs as XIII in Euler's^^^ list. (3) Problem 3 is derived from problem 2 by replacing s by a number / not necessarily prime. Let h be the greatest common divisor of ff=hg andp+l = ^x. Thenr-{-l=xy, q-{-l = gy. Also ghxyfa=f{afr)=a(pq+fr)^a\{hx-l){gy-l)+f{xy-l)\. *A11 the remaining cases are readily excluded. "♦"BibUotheca Math., (3), 10, 1909-10, 80-81. 44 History of the Theory of Numbers. [Chap, i Multiply by 6/a and replace bja by 2ab — ac [see case (1)]. Thus exy — bhx — hgy = b{f—l), e=bf—bgh-{-cgh. Thus L^b'^gh+be{f—l) is to be expressed as the product PQ of two factors and they are to be equated to ex—bg, ey — bh. The case a = 2 is unfruitful. (3i) Let a = 4. Then 6 = 4, c = l, e = 4/— 3^/i. The case/= 3 is excluded since it gives e = 0. For/ =5, g = 2, h = S, we again get (a) and also (j3). For/=5, ^ = 1, h = 6, we get only the same two pairs. For a prime /^ 7, no new solutions are found. For /= 5-13, (51) results. (32) Let a = 8, whence 6 = 8, c = L The cases /= 11, 13 are fruitless, while /= 17 yields (16). The least composite/ yielding solutions is 11-23, giving (44), (45), (46). This fruitful case led Euler to the more convenient notations (§88) M = hP, N = gQ, L = PQ. The problem is now to resolve L ff into two factors, Af , N, such that M+bff N+bff are integers and primes, while in r+1 = {p+l){q+l)/jf, r is a prime. (33) Let a = 16. For/=17, we obtain the pairs (21), (22); for/=19, (23); for/=23, (17), (19), (20); for/=47, (18); for/= 17-167, (49). Cases /=31, 17-151 are fruitless [the last since 129503 has the factor 11, not noticed by Euler]. (34) For a = 3^-5 or 32.7-13, 6 = 9, c = 2; the first a with/=7 yields (30). (4) Problem 4 relates to amicable numbers agpq, ahr, where p, q, r are primes. Eventually he took also g and h as primes. We may then set g-\-l — km, h-\-\ = kn. For m = \, n = 3, a = 4 or 8, no amicables are found. Form = 3, n = l, the cases a =10, A: = 8 and a = 3^-5, ^' = 8, yield (38), (55). (5) Euler's final problem 5 is of a new type. He discussed amicable numbers zap, zbq, where a and 6 are given numbers, p and q are unknown primes, while z is unknown but relatively prime to a, 6, p, q. Set JoM 6 = m:n, where m and n are relatively prime. Since(p-fl)j a = (54-l)j6, we may set p-\-\=^nx, q-{-\=mx. The usual second condition gives r r / K 7 / ^ — nxfa nx\a'{z = za{nx — l)-j-zb{mx — l), C —-, 1^ r* ■^ -^ j^ {na-\-mb)x — a — b Let the latter fraction in its lowest terms be r/s. Then z = kr, jz = k$. Since f{kr)'^kCr, we have s'^ff. Hence we have the useful theorem: if z:Cz = r':s', s'<\r', then r' and s' have a common factor > 1. (5i) The unfruitful case a = 3, 6 = 1, was treated like the next. (52) Let 0 = 5, 6=1, whence w = 6, n = l, 2:J z = 6a::llx — 6. By the theorem in (5), x must be divisible by 2 or 3. Euler treated the cases x = 3(3^+1), x = 2{2t+\). But this classification is both incomplete and Chap. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 45 overlapping. Since p = x — l is to be prime, x is even (since a; = 3 makes z divisible by p = 2) . Hence x = 2P,z:fz = QiP:llP—S. By the theorem in (5) , QP and IIP — 3 have a common factor 2 or 3, so that P is either odd or divis- ible by 6. For P = Ql, the ratio is that of 126 to 22Z — 1 , which as before must have the common factor 3, whence l = 3t-\-l. Then z:fz = 4:{St-{-l) :22t-\-7, a ratio of relatively prime numbers, whence 22t+7'^f 4(31+1), and hence t = 2k, k = 0 or k>3. For A; = 0, we obtain the pair 220, 284. The next value >3 of A; for which p = x — l and q = Qx—l are primes is k = Q, giving p = 443, g = 2663, numbers much larger than those in the (unnecessary) cases treated by Euler. Then z:jz = 4-37:271; set z = S7''d, d not divisible by 37; the cases e = 1, 2, 3 are excluded by the theorem in (5). For the remaining case P odd, P = 2Q-\-l, Euler treated those values ^100 of Q, and also Q = 244, for which p and q are primes and obtained the pair in (I3), two pairs in (I5), and (14), (15). (53) Euler treated in §§112-7 various sets a, b, and obtained (a) and nine new pairs given in the table. In the following table of the 64 pairs of amicable numbers obtained by Euler, the numbering of any pair is the same as in Euler's list, but the pairs have been rearranged so that it becomes easy to decide if any proposed pair is one of Euler's. As noted by F. Rudio,^^^'' (37) contained the mis- print 3^ for 3^, w^hile (7) and (34) are erroneous, 220499 being composite (311-709); he checked that all other entries are correct. (4) 22-23{^27^^ (9) 2M3.17{3||09 (47) 23/11-29-239 ^*'^ "^ 1191449 ^4»; ^ \29-47-59 (21) 2417'5119 (^)2^{83lo39(f-l««) (18) 2^{i:?^ (27) o«/79- 11087 ^^^^ ^ \383-2309 (37) s^-sgi'ig'' (15) 32.7-13-4M63< (35) 32.5.19{7^227 5-977 5867 (38) 2-5{7: (a)22 60659 23-29-673 5-131 17-43 ^•' ^ 1647-719 (43) 2' (60) 11-59-173 47-2609 2^-19-41 26-199 (oo\ 94/17-10303 (2) 2^ [23-47 \1151 (19) 2^{i:t2? (25) 2»P-12671 K^Q) z ^227-2111 ,3. 27/191-383 (5) 32-7-13[ (14) 32-72-13-97 5-17 107 5193 1163 (8) 3^-5-7{i?2S 251 107 (1) 22{^jll i3) 2^(^32 ,45. 23(ll-23-1871 V4^) ^ \467-1151 .4Q. 23/11-163-191 ^4u; z 131.11807 ,r.,. /23-41-467 ^^^1 \25-19-233 (23) 2^{}9gl439 (50) 2423-47-9767 \tM) z |i583.7io3 ,17s 24/23-1367 ^^'^ "^153-607 (26) 2^{^|JJ^f /90X r,8/383-9203 K^-6) z |ii5i.3067 (7) 32-72-13{|5Y (10) 32-5- 19-37 (710^ (6) 3^-5-13gl9l9 (51) 2^^^^ •131187 -2267 (29) 2^-ll{17:263 ,,,. 23/11-23-2543 ^**^ ^ 1383-1907 (16) 2^(M:^^ (49) 24/17-167-13679 KV6) z I809.51071 (36) 2^-67{|72411 .24) 26/59-1103 (.Z4J L J79.827 {b2)Z^-l-\Z^^lll]f '"^Bibliotheca Math., (3), 14, 1915, 351-4. 46 History of the Theory of Numbers. [Chap, i (31) S^-5'13{ll:]f (54) 3«.5^{}1:^9.179 ^g^ ^,_^_^^,^^l29.5m (33) 33.5.13.19^/2711 (32) 3^-5-13{^^:^J (12) S^.r-lVlsl^lf^^' (41)33.7.13.23|JJ;1%367 ^g^^ 3,.-,. ^g.^gjl 1-220499 ^^^^^^ (30) 33.5{[7^1j (55) ■S^-5{',lll%^ (42) 33.5.23(11 -.J^SIT (11) 3^.5.1l{29g89 (56) 3^.7.11M9{g97019 (57)3^.7.11M9{^3^6959^ (53) 3».7M3.53{ll4211 (53) 3s.7M3.19{4™19 (59^ 3^.7M3.19{53g6959^ Euler's final list of 61 pairs did not include the pairs a, jS, 7, although he had obtained a four times in the body of his paper, viz., in (2), (3i), (63); jS twice in (3i); 7 in (2). Moreover, these three unlisted pairs occur as VIII, IX, and XIII among the 30 pairs in Euler's^^^ earlier list, a fact noted on p. XXVI and p. LVIII of the Preface by P. H. Fuss and N. Fuss to Euler's Comm. Arith. Coll., who failed to observe that these three pairs occur in the text of Euler's present paper. Nor did these editors note that the fourth mentioned case of divergence between the two lists is due merely to the misprint^^^'' of 57 for 47 in (43) of the present list, so that the correctly printed pair XXVIII of the list of 30 is really this (43) and not a new pair, as supposed by them. From the fact that Euler obtained in his posthumous tract®^ on amicable numbers the pairs a, jS (once on p. 631 and again on p. 633 and finally on p. 635), the editors inferred, p. LXXXI of the Preface, that the tract differs in analysis from the long paper just discussed. But no new pairs are found, while the cases treated on pp. 631-2 are merely problems 1 and 2 of Euler's preceding paper. It is different with p. 634, where Euler started with two numbers like 71 and 5-11 which, by his table, have the same sum, 72, of divisors, and required a number a relatively prime to them such that 71a and 55o are amicable. The single condition is 72J a=(71+55)o, whence ja:a = 7A. Thus a has the factor 4. If a = 46, where b is odd, then Ch = h = l, and the pair 284, 220 results. The case a = 86 is impossible. This method was used in a special way by Kraft^^^ who limited the numbers from which one starts to a prime and a product of two primes. In the Encyclopedie Sc. Math., I, 3i, p. 59, note 320, it is stated that this posthumous tract contains four pairs not in Euler's list of 61, two pairs being those of Fermat^^^ and Descartes.^^^ But these were fisted as (2) and (3) by Euler and were obtained by him in case (li) and attributed to Descartes. E. Waring^^^ noted that 2'*x, 2''yz are amicable if 2"'?/z_2"+^ + l 2^" 2"-l ' i/-2"+l where x^ y, z are primes and ?/ — 2"H-1 divides 2^**. He cited the first two such pairs of amicable numbers. 3"«G. Enestrom, Bibliotheca Math., (3), 9, 1909, 263. 3«*Meditationes algebraicae, 1770, 201; ed. 3, 1782, 342-3. Chap. I] PERFECT, MULTIPLY PERFECT, AND AmICABLE NUMBERS. 47 The first three pairs were given in an anonymous work.^^® In 1796, J. P. Gruson^°° (p. 87) gave the usual rule (1) leading to the three first known amicable pairs (verwandte Zahlen). A. M. Legendre^^^ attributed the rule (1) to Descartes. G. S. KliigeP^^ gave a process leading to the choice of P and Q, left arbitrary by Kraft.^^^ ^^ ^ave A:a = R+l:PQ-\-R = 2R-P-Q. Thus P-(-Q= \R{2A—a)—a\ /A, while PQ is given by Kraft's second equation. Hence P and Q are the roots of a quadratic equation. For example, if A = 4, then 8P, SQ = R-7±VR^-Q2R-QS. The positive root of a;^ — 62a: — 63 = 0 lies between 60 and 61. Thus we try primes ^ 61 for R, such that i^ — 7 is divisible by 8. The first available R is 71, giving P=ll, Q — 5 and the amicable pair 220, 284. In general, the quantity a^R^+2^R-\-y under the radical sign can be made equal to the square of ai^+P ip arbitrary) by choice of R. John Gough^^^ considered amicable numbers ax, ayz, where x, y, z are distinct primes not dividing a. Let q be the sum of the aliquot divisors of a. Then a+q-\-qx = ayz, x+l = {y+l){z+l). If q^a/i, the first gives ayz< (l+a:)a/4, while 2y-2z>x-\-l by the second, Thusg'>a/4. Let a = r", where r is a prime > 1 . Then Q'=(a — l)/(r' — 1), which with g>a/4 implies a(5— r)>4, r = 2 or 3. He proved that Tt^S. whence r = 2, the case treated by van Schooten.^^^ J. Struve^^^ cited his Osterprogramm, 1815, on amicable numbers. A. M. Legendre^^° discussed the amicable numbers of the type (li) of Euler^^^ (with Euler's m, k replaced by m—iJi,,fx). Legendre noted that r = 2^'""''^ (2*^+1)^ — 1 is of the form s^ — 1 and hence composite, if k is even; also that, if A: = 3, p = 9-2'"+3-l, g = 9-2"'-l, one of which is of the form s^ — 1. He considered the new case k = 7 and found for m = l that p = 33023, q = 257, r = 8520191, stating that if r be a prime we have the amicable num- bers 2^pq, 2V. This is in fact the case.^^^ For ^ = 1, we have the ancient rule (1); he proved that for n^l5 it gives only the known three pairs of amicable numbers. Paganini^'^^, at age 16, announced the amicable numbers 1184 = 2^37, 1210 = 2.5.11"^, not in the list by Euler^^'*, but gave no indication of the method of discovery. ^^'EncyclopMie methodique. . .Amusemens des Sciences Math, et Phys., nouv. 6d., Padoue, 1793, I, 116. Cf. Les amusemens math., Lille, 1749, 315. »"Th6orie des nombres, 1798, 463. «8Math. Worterbuch, 1, 1803, 246-252 [5, 1831, 55]. «»New Series of the Math. Repository (ed., Th. Leyboum), vol. 2, pt. 2, 1807, 34-39. He cited Button's Math. Diet., article Amicable Numbers, taken from van Schooten^^'. ""Theorie des nombres, ed. 3, 1830, II, §472, p. 150. German transl. by H. Maser, Leipzig, 1893, II, p. 145. "iTchebychef, Jour, de Math., 16, 1851, 275; Werke, 1, 90. T. Pepin, Atti Ace. Pont. Nuovi Lincei, 48, 1889, 152-6. Kraitchik, Sphinx-Oedipe, 6, 1911, 92. Also by Lehmer's Fac- tor Table or Table of Primes. '"B. Nicol6 I. Paganini, Atti deUa R. Accad. Sc. Torino, 2, 1866-7, 362. Cf. Cremona's Ital. transl. of Baltzer's Mathematik, pt. III. 48 History of the Theory of Numbers. (Chap, i P. Seelhoff^^^ treated Euler's^^ problems 1 and 2 by Euler's methods (though the contrary is implied), and gave about 20 pairs of amicable numbers due to Euler, with due credit for only three pairs. The only new- pairs (pp. 79, 84, 89) are Q2721Q in oQf83-1931 ^ef 139-863 6 i A^-Ay'^'^|i62287 "^1167.719. E. Catalan^^^ stated empirically that if ??i is the sum of the divisors <n of n, and n2 is the sum of the divisors <ni of ?ii, etc., then n, nj, n2, . . . have a limit X, where X is unity or a perfect number. J. Perrott"^ [Perott] noted that there is no limit for n = 220, since ^1 = 713= . . . =284, n2 = n4= . . . =220. H. LeLasseur^^*^ found that for n< 35 the numbers (1) are all odd primes, and hence give amicable numbers, only when n = 2, 4, 7. Josef Bezdicek^^^ gave a translation into Bohemian of Euler,^^ without credit to Euler, and a table of 65 pairs of amicable numbers. Aug. Haas"^ proved that, if M and A^ are amicable numbers, l/S-+l/si=l, m n where m and n range over all divisors of M and iV, respectively. For, 2w = Sn = M+iV, so that m~ M ~ M ' n~ N ~ N If M = N, N is perfect and the result becomes that of Catalan. ^"'^ A. Cunningham^'^ considered the sum s{n) of the divisors <n of ?i and wrote s^{n) for s]s(??)}, etc. For most numbers, s^(n) = l when A; is suffi- ciently large. There is a small class of perfect and amicable numbers, and a small class of numbers n (even when n< 1000) for which s''{n) increases beyond the practical power of calculation [cf. Catalan^'^]. A. Gerardin^^" proved that the only pairs 2^-5a;, 2-yz of amicable num- bers, where x, y, z are odd primes, are Euler's (a), (^3) ; the only pairs 2*-23x, 2^yz are Euler's (17), (19), (20). He cited the Exercices d'arithm^tique of Fitz-Patrick and Chevrel; also Dupuis' Table de logarithmes, which gives 24 pairs of amicable numbers. G^rardin^^^ proved that the only pair Sxy, S2z is Euler's (60). He made an incomplete examination of 16-53a;, IQyz, but found no new pairs. 3"Archiv Math. Phys., 70, 1884, 75-89. "*Bull. Soc. Math. France, 16, 1887-8, 129. Mathesis, 8, 1888, 130. '"76id., 17, 1888-9, 155-6. "•Lucas, Theorie dcs nombres, 1, 1891, 381. "'Casopis mat. a fys., Praze (Prag), 25, 1896, 129-142, 209-221. "8/6id., 349-350. "«Proc. London Math. Soc, 35, 1902-3, 40. ""Matheshs, 6, 1906, 41^4. "'Sphinx-Oedipe, Nancy, 1906-7, 14-15, 53. Chap. I] PERFECT, MULTIPLY PERFECT, AND AmICABLE NUMBERS. 49 G^rardin^^^ proved that the three numbers (1) with n = m-f 2 are not all primes if 34< m^ 60, the cases m = 38 and 53 not being decided. Replacing m by m+1 and A; by 2^+1 in case (li) of Euler^^'*, we get the pair 2"pg, 2"r, where n = m+2g-]-2, p = 2"*+2''+^P-l, 5 = 2"*+^P-l, ^ = 22'"+2''+3p2_i^ with P = 2^^'*"^ + l. For 9^ = 0, we have the case (1) just mentioned; all values m^200 are excluded except m = 38, 74, 98, 146, 149, 182, 185, 197. The case gr= 1 is excluded since i/ or 2 is a difference of two squares. For g = 2, all values m ^ 60 are excluded except m= 29, 34, 37, 49. For g = 3, all values <100 are excluded except m = 8, 15, 23, 92. 0. Meissner,^^^ using the notation of Cunningham,^^^ noted that n and s{n) are amicable if s^(n)=n and raised the question of the existence of numbers n for which s''{n)—n for k^S, so that n, s{n),. . .,s*~^(n) would give amicable numbers of higher order. He asked if the repetition of the operation s, a finite number (k) of times always leads to a prime, a perfect or amicable number; also if k increases with n to infinity. On these ques- tions, see Dickson^^^ and Poulet.^^^ A. Gerardin^^^ stated that the only values n<200 for which the numbers (1) are all primes are the three known to Descartes. L. E. Dickson^^^ obtained the two new pairs of amicable numbers 2*-12959-50231, 2*- 17- 137-262079; 2*- 10103-735263, 2^-17-137-2990783, by treating the type IQpq, 16-17-137r, where p, q, r are distinct odd primes. These are amicable if and only if p = m+9935, g = w+9935, r = 4(w+n) +88799, wn = 2^3*7-23-73. Although Euler^^ mentioned this type (33) in §95, he made no discussion of it since r always exceeds the limit 100000 of the table of primes accessible to him. An examination of the 120 distinct cases led only to the above two amicable pairs. Dickson^^® proved that there exist only five pairs of amicable numbers in which the smaller number is <6233, viz., (1), (a), (^), (60) in Euler's^^* table, and Paganini's^^^ pair. In the notation of Cunningham,^^^ the chain n, s{n), s^{n), . . .is said to be of period k if s^(n) =n. The empirical theorem of Catalan'^* is stated in the corrected form that every non-periodic chain contains a prime and verified for a wide range of values of n. In particular, if n<6233, there is no chain of period 3, 4, 5, or 6. For k odd and > 1, there is no chain arii, an2, . . . , aUk of period k in which /ii, . . . , n^ have no common factor and each rij is prime to a> 1. '^^Sphinx-Oedipe, 1907-8, 49-56, 65-71 ; some details are inaccurate, but the results correct. 'S'Archiv Math. Phys., (3), 12, 1907, 199; Math.-Naturw. Blatter, 4, 1907, 86 (for k=3). '** Assoc, frang. avanc. sc, 37, 1908, 36-48; I'intenn^diaire des math., 1909, 104. '8*Amer. Math. Monthly, 18, 1911, 109. "•Quart. Jour. Math., 44, 1913, 264-296. 50 History of the Theory of Numbers. [Chap, i P. Poulet'^' discovered the chain of period five, 71 = 12496 = 24-11.71, s(n)= 2^- 1947, s\n) = 2^-967, s\n) =2^-23-79, sHn) =2^.1783, with s^{n) =n; and noted that 14316 leads a chain of 28 terms. Generalizations of Amicable Numbers. Daniel Schwenter^^ noted in 1636 that 27 and 35 have the same sum of ahquot parts. Kraft^^^ noted in 1749 that this is true of the pairs 45, 3-29; 39, 55; 93, 145; and 45, 13-19. In 1823, Thomas Taylor^^^ called two such numbers imperfectly amicable, citing the pairs 27, 35; 39, 55; 65, 77; 51, 91; 95, 119; 69, 133; 115, 187; 87, 247. George Peacock^°o used the same term. E. B. Escott^"^ asked if there exist three or more numbers such that each equals the sum of the [aliquot] divisors of the others. A. G^rardin^°^ called numbers with the same sum of aliquot parts nombres associes, citing 6 and 25; 5-19, 7-17, and 11-13, and many more sets. An equivalent definition is that the n numbers be such that the product of n — 1 by the sum of the aliquot divisors of any one of them shall equal the sum of the aliquot divisors of the remaining n — 1 numbers. L. E. Dickson^°^ defined an amicable triple to be three numbers such that the sum of the aliquot divisors of each equals the sum of the remaining two numbers. After developing a theory analogous to that by Euler^®* for amicable numbers, Dickson obtained eight sets of amicable triples in which two of the numbers are equal, and two triples of distinct numbers: 293-3370, 5- 16561a, 99371o (a = 25.3-13), 3-896, 11-296, 3596 (6 = 2i*.5-19-31-151). ^L'intermddiaire des math., 25. 1918, lOO-l. ♦""Encyclopaedia Metropolitana, London, I, 1845, 422. «"L'interm6diaire des math., 6, 1899, 152. ♦""Sphinx-Oedipe, 1907-8, 81-83. ♦MAmer. Math. Monthly, 20, 1913, 84-92. CHAPTER II. FORMULAS FOR THE NUMBER AND SUM OF DIVISORS. PROBLEMS OF FERMAT AND WALLIS. Formula for the Number of the Divisors of a Number. Cardan^ stated that a product P oi k distinct primes has 1+2+2^+ . . -\-2^~'^ aUquot parts (divisors <P). Michael StifeP proved this rule and found^ the number of divisors of 2*3^52p, where P = 7-11.13-17-19-23-29, by first noting that there are 1+2+ . . .+64 divisors <P oi P according to Cardan's rule and hence 128 divisors of P. The factor 5^ gives rise to 128+128 more divisors, so that we now have 384 divisors. The factor 3^ gives 3.384 more, so that we have 1536. Then the factor 2* gives 4.1536 more. Mersenne* asked what number has 60 divisors; since 60 = 2-2-3-5, sub- tract unity from each prime factor and use the remainders 1, 1, 2, 4 as exponents; thus 3^-2*-7-5 = 5040 (so much lauded by Plato) has 60 divisors. It is no more difficult if a large number of aliquot parts is desired. I. Newton^ found all the divisors of 60 by dividing it by 2, the quotient 30 by 2, and the new quotient 15 by 3. Thus the prime divisors are 1, 2, 2, 3, 5. Their products by twos give 4, 6, 10, 15. The products by threes give 12, 20, 30. The product of all is 60. The commentator J. Castillionei, of the 1761 edition, noted that the process proves that the number of all divisors of a'"6'*. . .is (m+l)(n+l) . . .if a, 6, . . .are distinct primes. Frans van Schooten^ devoted pp. 373-6 to proving that a product of k distinct primes has 2'''— 1 aliquot parts and made a long problem (p. 379) of that to find the number of divisors of a given number. To find (pp. 380-4) the numbers having 15 aliquot parts, he factored 15+1 in all ways and subtracted unity from each factor, obtaining abed, a^bc, a%^, a^b, a^^. By comparing the arithmetically least numbers of these various types, he found (pp. 387-9) the least number having 15 aliquot parts. John Kersey'^ cited the long rule of van Schooten to find the number of aliquot parts of a number and then gave the simple rule that Oi" . . . a^" has (6i+l) . . . (e„+l) divisors in all if ai, . . . , a„ are distinct primes. John Wallis^ gave the last rule. To find a number with a prescribed number of divisors, factor the latter number in all possible ways; if the iPractica Arith. & Mensurandi, Milan, 1537; Opera, IV, 1663. *Arithmetica Integra, Norimbergae, 1544, lib. 1, fol. 101. 'Stifel's posthumous manuscript, fol. 12, preceding the printed text of Arith. Integra; cf. E. Hoppe, Mitt. Math. Gesell. Hamburg, 3, 1900, 413. *Cogitata Physico Math., II, Hydravhca Pnevmatica, Preface, No. 14, Paris, 1644. (Quoted by Winsheim, Novi Comm. Ac. Petrop., II, ad annum 1749, Mem., 68-99). Also letter from Mersenne to Torricello, June 24, 1644, Bull. Bibl. Storia Sc. Mat., 8, 1875, 414-5. •Arithmetica UniversaUs, ed. 1732, p. 37; ed. 1761, I, p. 61. De Inventione Divisorum. •Exercitationum Math., Lugd. Batav., 1657. ^The Elements of Algebra, London, vol. 1, 1673, p. 199. •A Treatise of Algebra, London, 1685, additional treatise, Ch. III. 61 52 History of the Theory of Numbers. [Chap, ii factors are r, s, . . ., the required number is p''~^q'~^ . . ., where p, q,. . are any distinct primes. WTien the number of divisors is odd, the number itself is a square, and conversely. The number of ways A^ = a^b^ . . . can be expressed as a product of two factors is A = |(a+l)(j3+l) . . .or \-\-k, according as N is not or is a square. Jean Prestet^ noted that a product of k distinct primes has 2'' divisors, while the ?ith power of a prime has n+1 divisors. The divisors of a^h^c^ are the 12 divisors of or}?, their products by c and by c^, the general rule not being stated explicitly. Pierre R^mond de Montmort^° stated in words that the number of divisors of Oi**. . .a„*" is (ci+l) . . .(e„+l) if the a's are distinct primes. Abb^ Deidier^^ noted that a product of k distinct primes has ^+^+(2) + (3)+ • divisors, treating the problem as one on combinations (but did not sum the series and find 2*"). To find the number of divisors of 2*3^5^ he noted that five are powers of 2 (including unity). Since there are three divisors of 3^, multiply 5 by 3 and add 5, obtaining 20. In view of the two divisors of 5^, multiply 20 by 2 and add 20. The answer is 60. E. Waring^- proved that the number of divisors of a"'?)". . .is (m+1) (n+1) . . .if a, 6, . . are distinct primes, and that the number is a square if the number of its divisors is odd. E. Lionnet^^ proved that if a, b, c, . . .are relatively prime in pairs, the number of divisors of abc. . .equals the product of the number of divisors of a by the number for b, etc. According as a number is a square or not, the number of its divisors is odd or even. T. L. Pujo^^ noted the property last mentioned. Emil Hain^^ derived the last theorem from a"* = (<i . . . t„y, where <i, . . . , <„ denote the divisors of a. A. P. Minin^^ determined the smallest integer with a given number of divisors. G. Fontene'" noted that, if 2"3^. . .mV (a^/S^ . . . ^n^v) is the least number with a given number of di\4sors, then I'+l is a prime, and /x+1 is a prime except for the least number 2^3 ha\'ing eight di\'isors. Formula for the Sum of the Divisors of a Number. ' R. Descartes,^^ in a manuscript, doubtless of date 1638, noted that, if p is a prim6, the sum of the aliquot parts of p" is (p"— l)/(p — 1). If 6 is the "Nouv. Elemens des Math., Paris, 1689, vol. 1, p. 149. loEssay d'analyse sur les jeux de hazard, ed. 2, Paris, 1713, p. 55. Not in ed. 1, 1708. "Suite de I'arithm^tique des g^om^tres, Paris, 1739, p. 311. i^Medit. Algebr., 1770, 200; ed. 3, 1782, 341. "Nouv. Ann. Math., (2), 7, 1868, 68-72. "Les Mondes, 27, 1872, 653-4. "Archiv Math. Phys., 55, 1873, 290-3. "Math. Soc. Moscow (in Russian), 11, 1883-4, 632. "Nouv. Ann. Math., (4), 2, 1902, 288; proof by Chalde, 3, 1903, 471-3. *'"De partibus ahquotis numerorum," Opuscula Posthuma Phys. et Math., Amstelodami, 1701, p. 5; Oeuvres de Descartes (ed. Tannery and Adams, 1897-1909), vol. 10, pp. 300-2. 23 3 13 Chap. II] FORMULAS FOR NUMBER AND SUM OF DiVISORS. 53 sum of the aliquot parts of a, the sum of the ahquot parts of ap is 6p+a+6. If b is the sum of the ahquot parts of a and if x is prime to a, the sum of the aUquot parts of ax"" is — ^n — [=^'+^K-j^)-^^\' Descartes^^ stated a result which may be expressed by the formula (1) <j{nm)=(T{n)(T{m) (n, m relatively prime), where (T{n) is the sum of the divisors (including 1 and n) of w. Here he solved n : (T(n) = 5 : 13. Thus n must be divisible by 5. Enter 5 in column A and (r(5) = 6 in column B. Then enter the factor A B 2 in column A and (r(2) =3 in column B. Having two threes in column B, we enter 9 in column A and cr(9) = 13 in B. Every 5 number except 13 in column B is in column A. Hence the 2 product 5-2-9 = 90 is a solution n. Next, to solve n : a- (n) = 5 : 14, 9 we enter also 13 in column A and 14 in B, and obtain the solu- tion 90-13. If ?i is a perfect number, 5n: (7(5n) = 5: 12 and, if n?^6, 15n: (r(15n) = 5:16. Descartes^^ stated that he possessed a general rule [illustrated above] for finding numbers having any given ratio to the sum of their aliquot parts. Fermat^^ had treated the same problem. Replying to Mersenne's remark that the sum of the aliquot parts of 360 bears to 360 the ratio 9 to 4, Fermat^^ noted that 2016 has the same property. John Wallis^^ noted that Frenicle knew formula (1). Wallis^^ knew the formula (2) ^(a•6^...) = 211^1-*^^.... a— 1 0—1 Thus these formulae were known before 1685, the date set by Peano,^^ who attributed them to Wallis.^^ G. W. Kraft^^ noted that the method of Newton^ shows that the sum of the divisors of a product of distinct primes P, . . ., S is (P+1) . . .(S+1). He gave formula (1) and also (2), a formula which Cantor^" stated had probably not earlier been in print. To find a number the sum of whose divisors is a square, Kraft took PA, where P is a prime not dividing A. If (r(A)=a, then (r(PA) = (P-f-l)a will be the square of (P+l)5 if P = ^^"De la fagon de trouver le nombres de parties aliquotes in ratione data," manuscript Fonds- frangais, nouv. acquisitions, No. 3280, ff. 156-7, Bibliothfeque Nationale, Paris. Pub' Ushed by C. Henry, BuU. Bibl. Storia Sc. Mat. e Fis., 12, 1879, 713-5. ^Oeuvres, 2, p. 149, letter to Mersenne, May 27, 1638. K)euvre8 de Fermat, 2, top of p. 73, letter to Roberval, Sept. 22, 1636. "Oeuvres, 2, 179, letter to Mersenne, Feb. 20, 1639. ''^ommercium Epistolicum, letter 32, April 13, 1658; French transl. in Oeuvres de Fermat, 3, 553. "Commercium Epist., letter 23, March, 1658; Oeuvres de Fermat, 3, 515-7. "Formulaire Math., 3, Turin, 1901, 100-1. "Novi Comm. Ac. Petrop., 2, 1751, ad annum 1749, 100-109. "Geschichte Math., 3, 595; ed. 2, 616. 54 History of the Theory of Numbers. [ChapII a/S^ — 1; for A = 14, take B = 2, whence P = 5. Again, the sum of the aUquot parts of 3P~ is (2+-P)^. The numbers AP and BPQ have the same sum of divisors if a(P+l) = 6(P+1)(Q+1), i. e., if Q = a/6-l; takmg a = 24, 6 = 6, we have Q = 3, a prime, .4 = 14, B = 5 (by his table of the sum of the divisors of 1,. . ., 150); this problem had been solved otherwise by Wolff." L. Euler^^ gave a table of the prime factors of o-(p), a(p^), and <t(p^) for each prime p<1000; also those of aip") for various a's for p^23 (for instance, a ^36 when p = 2). He proved formulas (1) and (2) here and in his^ posthumous tract, where he noted (p. 514) all the cases in which a{n) =a-(?70 = 60. E. Waring^2 proved formula (2). He^ noted that if P = arir. . .and Q = a'^h^ . . . , where m — a,n — ^,... are large, then a{PQ)/a{P) is just greater than Q. If A = {1-1)\, <t{IA)/(t{A)^1-^1. If a'br..=A and (x+l) (2/+ 1) . . . is a maximum, then a'"''"^ = 6""^^ = . . . For a, 6, . . . distinct primes, a{A) is not a maximum. He cited numbers with equal sums of divisors: 6 and 11, 10 and 17, 14 and 15 and 23. L. Kronecker^^ derived the formulas for the number and sum of the divisors of an integer by use of infinite series and products. E. B. Escott^^ listed integers whose sum of divisors is a square. Problems of Fer\la.t and Wallis on Sums of Divisors. Fermat'*^ proposed January 3, 1657, the two problems: (i) Find a cube which when increased by the sum of its aUquot parts becomes a square;* for example, 7^ + ( 1 + 7 + 7^) = 20^. (ii) Find a square which when increased by the sum of its aliquot parts becomes a cube. John WalUs^^ replied that unity is a solution of both problems and pro- posed the new problem: (m) Find two squares, other than 16 and 25, such that if each is increased by the sum of its ahquot parts the resulting sums are equal. Brouncker*^ gave 1/n^ and 343/n^ as solutions (!) of problem (i). "Elementa Analyseos, Cap. 2, prob. 87. «Opuscula varii argumenti, 2, Berlin, 1750, p. 23; Comm. Arith., 1, 102 (p. 147 for table to 100). Opera postuma, I, 1862, 95-100. F. Rudio, Bibl. Math., (3), 14, 1915, 351, stated that there are fully 15 errors. "Comm. Arith., 2, 512, 629. Opera postuma, I, 12-13. «Meditationea Algebr., ed. 3, 1782, 343. (Not in ed. of 1770.) "Vorlesungen iiber Zahlentheorie, I, 1901, 265-6. ^Amer. Math. Monthly, 23, 1916, 394. *Erroneou8ly given as "cube" in the French tr., Oeuvres de Fermat, 3, 311. '"OeuvTes, 2, 332, "premier d6fi aux mathdmaticiens;" also, pp. 341-2, Fermat to Digby, June 6, 1657, where 7' is said to be not the only solution. These two problems by Fermat were quoted in a letter by the Astronomer Jean H6v4hus, Nov. 1, 1657, pubhshed by C. Henry, Bull. Bibl. Storia Sc. Mat. e Fis., 12, 1879, 683-5, along with extracts from the Commercium EpistoUcum. Cf. G. Wertheim, Abh. Geschichte Math., 9, 1899, 558-561, 570-2 ( = Zeitschr. Math. Phys., 44, Suppl. 14). "Commercium Epistohcum de Wallis, Oxford, 1658; Walhs, Opera, 2, 1693. Letter II, from WaUis to Brouncker, Mar. 17, 1657; letter XVI, Walhs to Digby, Dec. 1, 1657. Oeuvres de Fermat, 3, 404, 414, 427, 482-3, 503-4, 513-5. **Commercium, letter IX, Wallis to Digby; Fermat'e Oeuvres, 3, 419. Chap. II] PkOBLEMS ON SUMS OF DiVISORS. 55 Frenicle^^ expressed his astonishment that experienced mathematicians should not hesitate to present, for the third time, unity as a solution. Wains'*^ tabulated <r{x^) for each prime a:<100 and for low powers of 2, 3, 5, and then excluded those primes a: for which (T(rc^)has a prime factor not occurring elsewhere in the table. By similar eliminations and successive trials, he was led to the solutions^^ of (i) : a=3^5-lM3-41-47, 6=2-3-5-13-41-47; 7a, 7b, adding that they are identical with the four numbers given by Frenicle.*'' Note that o-(a) is the square of 2^3^5-7-lM3-17-29-61, while a{b) is the square of 2^3V7-13-17-29. Wallis^^ gave the further solutions of a{x^) = y^: a:=17-3147-191, ?/=2^°325-13-17-29-37, 2-3-5-13-17.314M91, 2^^33527. i3.;^7.29237^ 3'5-lM3-17-31-4M91, 2'23^5-7-lM3-17-29237-61, and the products of each x by 7. Wallis^^ gave solutions of his problem (m) : 2^3.37, 2-19-29; 223-1M9-37, 2'7-29.67; 29-67, 2-3-5-37; 2^7.29.67, 3-5-1M9-37. Frenicle*^ gave 48 solutions of WalUs' problem (m), including 2-163 11-37; 3-11-19, 7-107; 2-5-151, 3^-67; also 83 sets of three squares having the same sum of divisors, for example, the squares of 2^11.37-151, 3^67-163, 5-11-37-151, (7 = 3^7^19-31-67-1093; also various such sets of n squares (with prime factors <500) for n^l9, for example, the squares of ac, ad, 4:bd, 46c, 5bd, and 56c, where a = 2-5-29-47-67-139, 6 = 13-37-191-359, c = 7.107, d = 3-ll-19. Frans van Schooten^" made ineffective attempts to solve problems (i), (n). Frenicle^^ gave the solution a: = 225-7.11-37-67-163-191-263-439-499, t/ = 327^3- 19-31^67- 109 of problem (n), a{x^)=y^; also a new solution of a{x^)=y^: a; = 255-7-31-73-241-243-467, i/ = 2i23253ii. 13217.37.41. 113.193.257. ^'Letter XXII, to Digby, Feb. 3, 1658. Cf. Leibnitii et BernouUii Commercium philos. et math., I, 1795, 263, letter from Johann Bernoulli to Leibniz, Apr. 3, 1697. "Letter XXIII, to Digby, Mar. 14, 1658. "The same tentative process for finding this solution a was given by E.Waring, Meditationes Algebraicae, 1770, pp. 216-7; ed. 3, 1782, 377-8. The solution 6 = 751530 was quoted by Lucas, Thiorie des nombres, 1891, 380, ex. 3. **Solutio duorum problematum circa numeros cubos . . . 1657, dedicated to Digby [lost work]. See Oeuvres de Fermat, p. 2. 434, Note; WaUis." "Letter XXVIII, March 25, 1658; WaUis, Opera, 2, 814; Wallia". ««Letter XXIX, Mar. 29, 1658; WaUis^^. "Letter XXXI, Apr. 11, 1658. "Letter XXXIII, Feb. 17, 1657 and Mar. 18, 1658. "Letter XLIII, May 2, 1658. 56 History of the Theory of Numbers. [Chap, ii Wallis^'^ for use in problem (ii) gave a table showing the sum of the divisors of the square of each number < 500. Excluding numbers in whose divisor sum occurs a prime entering the table only once or twice, there are left the squares of 2, 4, 8, 3, 5, 7, 11, 19, 29, 37, 67, 107, 163, 191, 263, 439, 499. By a very long process of exclusion he found only two solutions within the limits of the table, viz., Frenicle's" and (rj(7.11-29.163-191439)2[ = ]3.7-13-19-31-67(^ Jacques Ozanam" stated that Fermat had proposed the problem to find a square which with its aliquot parts makes a square (giving 81 as the answer) and the problem to find a square whose aliquot parts make a square. For the latter, Ozanam found 9 and 2401, whose aliquot parts make 4 and 400, and remarked that he did not believe that Fermat ever solved these questions, although he proposed them as if he knew how. Ozanam^ noted that the sum of 961 = 31^ and its aliquot parts 1 and 31 is 993, which equals the sum of the aliquot parts of 1 156 = 34". As examples of two squares with equal total sums of divisors [WaUis' problem (m)], he cited 16 and 25, 326^ and 407^, while others may be derived by multiplying these by an odd square not divisible by 5. The sum of all the divisors of 9^ is 11^ that of 20^ is 31l The numbers 99 and 63 have the property that the sum 57 of the aliquot parts of 99 exceeds the sum 41 of the aliquot parts of 63 by the square 16; similarly for 325 and 175. E. Lucas^^ noted that the problem to find all integral solutions of (1) l-\-x+z^-\-x^ = y^ is equivalent to the solution of the system (2) l+x = 2w2, l+x2 = 2t;2, y = 2uv, and stated that the complete solution is given by that of 2y^— x^ = l. E. Gerono^^ proved that the only solutions of (1) are (x, 2/) = (-l, 0), (0, ±1), (1, ±2), (7, ±20). E. Lucas^^ stated that there is an infinitude of solutions of Fermat's problem (i); the least composite solution is the cube of 2-3-5-13'41-47, the sum of whose divisors is the square of 2^3^5^7- 13- 17-29. [This solution was given by Frenicle.^®] For the case of a prime, the problem becomes (1). A. S. Bang^^ gave for problem (i) the first of the three answers by Wallis;*' for (it), (7(43098^) = 1729^ for {in), 29-67, 2-3-5-37 of Wallis^* and the first two by Frenicle;^^ all without references. "A Treatise of Algebra, 1685, additional treatises, Ch. IV. "Letter to De Billy, Nov. 1, 1677, published by C. Henry, Bull. Bibl. Storia Sc. Mat. e Fia., 12, 1879, 519. Reprinted in Oeuvres de Fermat, 4, 1912, p. 140. "Recreations Math6matiques et Phys., new ed., 1723, 1724, 1735, etc., Paris, I, 41-43. "Nouv. Corresp. Math., 2, 1876, 87-8. ••Nouv. Ann. Math., (2), 16, 1877, 230-4. "Bull. Bibl. Storia Sc. Mat. e Fis., 10, 1877, 287. "Nyt Tidsskrift for Mat., 1878, 107-8; on problems in 1877, 180. Chap. II] PROBLEMS ON SUMS OF DiVISORS. 57 E. Fauquembergue,^^ after remarking that (1) is equivalent to the sys- tem (2), cited Fermat's^" assertion that the first two equations (2) hold only for a: = 7 [aside from the evident solutions a: = ± 1, 0], which has been proved by Genocchi.^^ H. Brocard^^ thought that Fermat's assertion that 7^ is not the only solution of problem (i) implied a contradiction with Genocchi.^^ G. Vacca {ihid., p. 384) noted the absence of contradiction as (i) leads to equation (1) only if a: be a prime. C. Moreau®^ treated the equation, of type (1), While he used the language of extracting the square root of X = x*+ . . . written to the base x, he in effect put X={x^-\-a)^, 0<a<x. Then a^ = x+1, 2ax^ = x^-\-x^, whence 2a = a^, a = 2, x = 3, y = ll. E. Lucas®^ stated that {x^-\-y^)/{x+y) =^ has the solutions (3,-1, 11), (8, 11, 101), (123, 35, 13361),. . . Moret-Blanc^^ gave also the solutions (0, 1, 1), (1, 1, 1). E. Landau^^ proved that the equation — T=y^ x — 1 is impossible in integers (aside from x = 0, ?/ = =fc 1) for an infinitude of values of n, viz., for all n's divisible by 3 such that the odd prime factors of n/3, if any, are all of the form 6y— 1 (the least such n being 6). For, setting n = 3m, we see that y^ is the product of x^+x+1 and F = x^"'~^-{- . . . +a:^+l. These two factors are relatively prime since x^=l gives F=m (mod x^+ a: + 1 ) . Hence x^+x+lis Si square, which is impossible for a; f^ 0 since it lies between x^ and (a:+l)^. Brocard^^ had noted the solution a: = 1, y=m, if n = mP. A. Gerardin^^ obtained six new solutions of problem (i) : a: = 2.47.193.239.701, 2/ = 2^3l5M3M7.97.149. x = 2.5.23.41.83.239, y = 2\S\5\7.1S\29.53. x = 3.13.23.47.83.239, y = 2^^3^517.13117.53. X = 2.3.13.23.83.193.701, y = 2^3^5^7.13.17.53.97.149. a; = 3.5.13.41.193.239.701, 2/ = 2^3l5l7.13M7.29.97.149. a; = 2.5.13.43.191.239.307, ?/ = 2i^32.5MlM7.29.37.53.113.197.241.257. Also <t{N^)=S^ for Ar = 3-7-ll-29-37, ^ = 3-7-13-19-67. "Nouv. Ann. Math., (3), 3, 1884, 538-9. '"Oeuvres, 2, 434, letter to Carcavi, Aug., 1659. "Nouv. Ann. Math., (3), 2, 1883, 30&-10. Cf. Chapter on Diophantine Equations of order 2. "L'intermMiaire des math., 7, 1900, 31, 84. "Nouv. Ann. Math., (2), 14, 1875, 335. »Ibid., 509. «76id., (2), 20, 1881, 150. "L'interm^diaire des math., 8, 1901, 149-150. "Ibid., 22, 1915, 111-4, 127. 58 History of the Theory of Numbers. [Chap.ii G^rardin^^ gave five new solutions of (i) : X = 3.11.31.443.499, i/ = 2^3.5M3.37.61.157. x = 2.3^31.443.449, 2/ = 2^3.5ni. 13.37.61.157. a; = 1 1 . 17.41 .43.239.307.443.499, 2/ = 2^2 3^5'.7.11. 13^29^.37.61. 157. x = 2.11. 17.23.41.211.467.577.853, t/ = 2^''.3^5l7.13M7.292.53.61. 113.193.197. x = 3ni. 13.23.83.193.701, ?/ = 293'537.11. 13.17.53.61.97.149, the last following from his*^^ fourth pair in \'iew of a{SnV): a{2'S') = 2'3.nm^: 233.52 = 2=11-612; 5=. A. Cunningham and J. Blaikie^^ found solutions of the form x = 2'p of s{x) =g^, where s{n) is the sura of the divisors <n of n. product of aliquot parts. Paul Halcke'^^ noted that the product of the aliquot parts of 12, 20, or 45 is the square of the number; the product for 24 or 40 is the cube; the product for 48, 80 or 405 is the biquadrate. E. Lionnet"^ defined a perfect number of the second kind to be a number equal to the product of its aliquot parts. The only ones are p^ and pq, where p and q are distinct primes. "L'interm^diaire des math., 24, 1917, 132-3. "Math. Quest. Educ. Times, (2), 7, 1905, 68-9. "Dehciae Math, oder Math. Sinnen-Confect, Hamburg, 1719, 197, Exs. 150-2. "Nouv. Ann. Math., (2), 18, 1879, 306-8. Lucas, Th6orie des nombres, 1891, 373, Ex. 6 I CHAPTER III. FERMAT'S AND WILSON'S THEOREMS, GENERALIZATIONS AND CONVERSES; SYMMETRIC FUNCTIONS OF 1,2 P-\ MODULO P. Fermat's and Wilson's Theorems; Immediate Generalizations. The Chinese^ seem to have known as early as 500 B. C. that 2^—2 is divisible by the prime p. This fact was rediscovered by P. de Fermat^ while investigating perfect numbers. Shortly afterwards, Fermat^ stated that he had a proof of the more general fact now known as Fermat's theorem: If p is any prime and x is any integer not divisible by p, then x^~^ — 1 is divisible by p. G. W. Leibniz^ (1646-1716) left a manuscript giving a proof of Fermat's theorem. Let p be a prime and set x = a+6+c+. . .. Then each multi- nominal coefficient appearing in the expansion of x^ — 2a^ is divisible by p. Take a = 6 = c=...=l. Thus a;^ — a: is divisible by p for every integer x. G. Vacca^ called attention to this proof by Leibniz. Vacca^ cited manuscripts of Leibniz in the Hannover Library showing that he proved Fermat's theorem before 1683 and that he knew the theorem now known as Wilson's^^ theorem: If p is a prime, l + (p — 1)! is divisible by p. But Vacca did not explain an apparent obscurity in Leibniz's state- ment [cf. Mahnke'^]. D. Mahnke'' gave an extensive account of those results in the manuscripts of Leibniz in the Hannover Library which relate to Fermat's and Wilson's theorems. As early as January 1676 (p. 41) Leibniz concluded, from the expressions for the ^th triangular and yth. pyramidal numbers, that (2/+l)2/=2/'-2/=0 (mod 2), {y+2){y+l)y=y'-y=0 (mod 3), and similarly for moduU 5 and 7, whereas the corresponding formula for modulus 9 fails for y = 2, — thus forestalling the general formula by Lagrange.^* On September 12, 1680 (p. 49), Leibniz gave the formula now known as Newton's formula for the sum of like powers and noted (by incomplete induction) that all the coefficients except the first are divisible by the exponent p, when p is a prime, so that a''+h''+c''-{- . . . = {a+h+c+ . . .Y (mod p). Taking a = b= . . . =1, we obtain Fermat's theorem as above.'* That the binomial coefficients in (1 + 1)^ — 1 — 1 are divisible by the prime p was ^G. Peano, Formulaire math., 3, Turin, 1901, p. 96. Jeans.^^" ''Oeuvres de Fermat, Paris, 2, 1894, p. 198, 2°, letter to Mersenne, June (?), 1640; also p. 203, 2; p. 209. "Oeuvres, 2, 209, letter to Frenicle de Bessy, Oct. 18, 1640; Opera Math., Tolosae, 1679, 163. *Leibnizens Math. Schriften, herausgegeben von G. J. Gerhardt, VII, 1863, 180-1, "nova algebrae promotio." »Bibliotheca math., (2), 8, 1894, 46-8. •Bolletino di BibUografia Storia Sc. Mat., 2, 1899, 113-6. 'Bibliotheca math., (3), 13, 1912-3, 29-61. 69 60 History of the Theory of Numbers. [Chap, hi proved in 1681 (p. 50). Mahnke gave reasons (pp. 54-7) for believing that Leibniz rediscovered independently Fermat's theorem before he became acquainted, about 1681-2, with Fermat's Varia opera math, of 1679. In 1682 (p. 42), Leibniz stated that (p-2)!=l (mod p) if p is a prime [equivalent to Wilson's theorem], but that {p—2)l=m (mod p), if p is composite, m ha\dng a factor > 1 in common wdth p. De la Hire^ stated that if k-'"^^ is divided by 2(2r+l) we get A; as a remainder, perhaps after adding a multiple of the divisor. For example, if kr' is divided by 10 we get the remainder k. He remarked that Carr^ had observed that the cube of any number /:<6 has the remainder k when divided by 6. L. Euler^ stated Fermat's theorem in the form: If n+1 is a prime divid- ing neither a nor h, then a" — 6" is divisible by n+1. He was not able to give a proof at that time. He stated the generaUzation : If e = p'"~Hp — 1) and if p is a prime, the remainder obtained on dividing a* by p"" is 0 or 1 [a special case of Euler^^]. He stated also that ii m, n, p,. . . are distinct primes not dividing a and if A is the 1. c. m. of m — 1, n — 1, p — 1, . . ., then o"* — 1 is divisible by mnp . . . [and a* — 1 by m'' n\ . .ii k = A rrC~^n^~^ . . .]. Euler^° first published a proof of Fermat's theorem. For a prime p, 2'' = (l + l)^ = l+p-h(^)H-...+p+l = 2+mp, 3P = (l+2)P = l+A:p+2^ 3^-3- (2^-2) = A:p, (1+0)"= 1+np+aP, (l+o)P-(l+a)-(aP-a)=np. Hence if a^—a is divisible by p, also (1+a)" — (1+a) is, and hence also (a+2)''-(a+2),. . ., (a+6)P-(a+6). For a = 2, 2" - 2 was proved divisible by p. Hence, wTiting x for 2+6, we conclude that x^—x is divisible by p for any integer x. G. W. Kraft^^ proved similarly that 2" — 2 = 7np. L. Euler's^- second proof is based, hke his first, on the binomial theorem. If a, 6 are integers and p is a prime, (a+6)"— a" — 6" is divisible by p. Then, if a^ — a and 6^ — 6 are di\'isible by p, also (a+6)" — a — 6 is di\4sible by p. Take h = \. Thus (a+1)"— a— 1 is divisible by p if a^—a is. Taking a = 1, 2, 3, ... in turn, we conclude that 2''— 2, 3"— 3, . . . , c^ — c are divisible by p. L. Euler^^ preferred his third proof to his earlier proofs since it avoids the use of the binomial theorem. If* p is a prime and a is any integer not *Hist. Acad. Sc. Paris, annee 1704, pp. 42-4; ra4m., 358-362. »Comm. Ac. Petrop., 6, 1732-3, 106; Coram. Arith., 1, 1849, p. 2. [Opera postuma, I, 1862, 167-8 (about 1778)]. ^••Comm. Ac. Petrop., 8, ad annum 1736, p. 141; Comm. Arith., 1, p. 21. "Novi Comm. Ac. Petrop., 3, ad annos 1*50-1, 121-2. "Novi Comm. Ac. Petrop., 1, 1747-8, 20; Comm. Arith., 1, 50. Also, letter to Goldbach, Mar. 6, 1742, Corresp. Math. Phys. (ed. Fuss), I, 1843, 117. An extract of the letter is given in Nouv. Ann. Math., 12, 1853, 47. "Novi Comm. Ac. Petrop., 7, 1758-9, p. 70 (ed. 1761, p. 49); 18, 1773, p. 85; Comm. Arith., 1, 260-9, 518-9. Reproduced by Gauss, Disq. Arith., art. 49; Werke, 1, 1863, p. 40. Chap. Ill] FerMAt's AND WiLSON's THEOREMS. 61 divisible by y, at most p — 1 of the positive residues < p, obtained by dividing 1, a, a^, . . . by p, are distinct. Let, therefore; a" and a", where ix^v, have the same residue. Then a""" — 1 is divisible by p. Let X be the least positive integer for which a^ — \ is divisible by p. Then \,a,a^,..., a^~^ have dis- tinct residues when divided by p, so that X^p — L IfX = p — 1, Fermat's theorem is proved. If X<p — 1, there exists a positive integer k ik<p) which is not the residue of a power of a. Then k, ak, a^k, . . ., a^~^k have distinct residues, no one the residue of a power of a. Since the two sets give 2X distinct residues, we have 2X^ p — 1. If X< (p — 1)/2, we start with a new residue s and see that s, as, a^s, . . ., a^~^s have distinct residues, no one the residue of a power of a or of a^k. Hence X^ (p — 1)/3. Proceeding in this manner, we see that X divides p — L Thus d^~^ — 1 is divisible by a'' — 1 and hence by p. L. Euler^^ soon gave his fundamental generalization of Fermat's theorem from the case of a prime to any integer N: Euler's theorem: If n=(f){N) is the number of positive integers not exceeding N and relatively prime to N, then x" — 1 is divisible by N for every integer x relatively prime to N. Let V be the least positive integer for which x" has the residue 1 when divided by N. Then the residues of 1, a:, a:^, ... , x"'^ are distinct and prime to N. Thus v^n. If v<n, there is an additional positive integer a less than A'' and prime to N. Then, when a, ax, ax^, . . ., ax"'^ are divided by N, the residues are distinct from each other and from those of the powers of x. Thus, 2v^n. Similarly, if 2v<n, then Zv^n. It follows in this manner that V divides n. J. H. Lambert^^ gave a proof of Fermat's theorem differing shghtly from the first proof by Euler.^" If h is not divisible by the prime p, 6^"^ — 1 is divisible by p. For, set 6 = c+L Then 6^-^-1 =-l+c''-i + (p-l)c^-2+...+l = -l+c^-'-c^-2 +0^"^- . . . +1+Ap, where A is an integer. The intermediate terms equal Hence c+1 c+1 -fA-/, /=' P p ' P(c+1) The theorem will thus follow by induction if / is shown to be integral. [Take p>2, so that p — 1 is even.] Then c^"^ — 1 is divisible by c+l, and by the hypothesis for the induction, by p. Since c-\-l = h is relatively prime to p, / is an integer. "Novi Comm. Ac. Petrop., 8, 1760-1, p. 74; Comm. Arith., 1, 274-286; 2, 524-6. "Nova Acta Eruditorum, Lipsiae, 1769, 109. 62 History of the Theory of Numbers. [Chap, hi E. Waring^^ first published the theorem that [Leibniz®] l + (p — 1)! is divisible by the prime p, ascribing it to Sir John Wilson^^ (1741-1793). Waring (p. 207; ed. 3, p. 356) proved that if a^ — a is divisible by p, then (a+1)''— a — 1 is, since {a+iy = a^-\-pA-\-l, "a, property first invented by Dom. Beaufort and first proved by Euler." J. L. Lagrange^^ was the first to publish a proof of Wilson's theorem. Let (x+l)(x+2) . . . (x+p-l) =x''-^+AiX^-'+ . . . -h^p-i. R eplace x by x + 1 and multiply the resulting equation by x + 1 . Comparing with the original equation multiplied by x+p, we get {x+p){x^-'+. . . +A,_,) = {x+ir+Ai{x+ir-' + . . . +^p_i(x+l). Apply the binomial theorem and equate coefl5cients of like powers of x. Thus Let p be a prime. Then, for 0<k<p, (j) is an integer divisible by p. Hence Ai, 2A2, . . . , {p—2)Ap_2 are divisible by p. Also, (P-1)4,..= (P + (PI})A:+(P-2)^+... = 1+^+A,+ ...+^,.2. Thus 1+Ap_i is divisible by p. By the original equation, Ap_i = (p — 1)!, so that Wilson's theorem follows. Moreover, if x is any integer, the proof shows that xP-^-l-(x+l)(x+2)...(x+p-l) is divisible by the prime p. If x is not divisible by p, some one of the integers x+1,. . .,x+p — 1 is divisible by p. Hence x''"^ — 1 is divisible by p, giving Fermat's theorem. Lagrange deduced Wilson's theorem from Fermat's. By the formula^* for the differences of order p — 1 of P~\ . . ., n^~^, (1) (p-i)\=p^-'-{p-i){p-iy-'+(^p~^){p-2r-' -(^3^)(p-3)^-^+. . .+(-1)^-^ Dividing the second member by p, and applying Fermat's theorem, we obtain the residue "Meditationes algebraicae, Cambridge, 1770, 218; ed. 3, 1782, 380. "On his biography see Nouv. Corresp. Math., 2, 187.6, 110-114; M. Cantor, Bibliotheca math., (3), 3, 1902, 412; 4, 1903, 91. "Nouv. M6m. Acad. Roy. BerUn, 2, 1773, ann^e 1771, p. 125; Oeuvres, 3, 1869, 425. Cf. N. Nielsen, Danske Vidensk. Selsk. Forh., 1915, 520. "Euler, Novi Comm. Ac. Petrop., 5, 1754-5, p. 6; Comm. Arith., 1, p. 213; 2, p. 532; Opera postuma, Petropoli, 1, 1862, p. 32. Chap. Ill] FeRMAt's AND WiLSON's THEOREMS. 63 Finally, Lagrange proved the converse of Wilson's theorem: If n divides l + (n— 1)!, then n is a prime. For n = 4m+l, w is a prime if (2-3. . .2my has the remainder —1 when divided by n. For n = 4m — 1, if (2m — 1)! has the remainder =±=1. L. Euler^" also proved by induction from a: = n to n+1 that (2) xl = a'-x{a-ir+(^iya-2r-(^f){a-Sr+..., which reduces to (1) ior x = p — l, a = p; and more generally, (3) a^_n{a-ir+(^){a-2y- . . . +(-l)^Q (a-A:r+ . . . =|^, ^^ x<n x = n. D'Alembert^^ stated that the theorem that the difference of order m of a*" is m ! had been long known and gave a proof. L. Euler^^ made use of a primitive root a of the prime p to prove Wilson's theorem (though his proof of the existence of a was defective). When l,a,a^, . . ., oF~^ are divided by p, the remainders are 1, 2, 3, . . . , p — 1 in some order. Hence a(p-i)(p-2)/2 j^^s the same remainder as (p — 1) !. Taking p>2, we may set p = 2n+l. Since a" has the remainder —1, then a"a^"^'*~^\ and hence also (p — 1)!, has the remainder —1. P. S. Laplace^^ proved Fermat's theorem essentially by the first method of Euler^° without citing him: If a is an integer <p not divisible by the prime p, ^l = \a-l^lY = \\{a-iy+p{a-iy-'+ . . . +l[ , a a a aV-^-\=-\{a-lY-\-l-a+hp{a-V)\ =^—^\{a-iy-^-l+hp] . a Cv Hence by induction a^~^ — l is divisible by p. For a>p, set a = np+q and use the theorem for q. He gave a proof of Euler's^^ generalization by the method of powering: if n = p''p{\ . ., where p, Pi,... are distinct primes, and if a is prime to n, then d° — l is divisible by n, where -"(^)(^) ■='^' q = p''-\p-l), r = pr-\Pi-l)P2''-\p2-l).. .. Set a'^ = x. Then a' — l=x'" — 1 is divisible by x — 1. Using the binomial theorem and a^~^ — l = hp, we find that aj — 1 is divisible by p". "Novi Comm. Ac. Petrop., 13, 1768, 28-30. "Letter to Turgot, Nov. 11, 1772, in unedited papers in the Biblioth^que de I'lnstitut de France. Cf. BuU. Bibl. Storia Sc. Mat. e Fis., 18, 1885, 531. "Opuscula analytica, St. Petersburg, 1, 1783 [Nov. 15, 1773], p. 329; Comm. Arith., 2, p. 44; letter to Lagrange (Oeuvres, 14, p. 235), Sept. 24, 1773; Euler's Opera postuma, I, 583. "De la Place, Th^orie abr^g^e des nombres premiers, 1776, 16-23. His proofs of Fermat's and Wilson's theorems were inserted at the end of Bossut's Algdbre, ed. 1776, and reproduced by S. F. Lacroix, Trait6 du Calcul Diff. Int., Paris, ed. 2, vol. 3, 1818, 722-4, on p. 10 of which is a proof of (2) for o=a; by the calculus of differences. 64 History of the Theory of Numbers. [Chap, ill From the (p — l)th order of differences for x""^ — 1, {x+p-ir-'-i-{p-i)\{x-\-p-2r-'-i\ + (^p~^y,{x-{-p-sr-'-i\ Set x = l and use Fermat's theorem. Hence l + (p — 1)! is divisible by p. E. Waring/^ 1782, 380-2, made use of x' = xix-l). . .{x-r-{-l)-\-Pxix-l). . .(x-r+2) +Qxix-1) . . .(x-r+3)+ . . . +Hx(x-l)+Ix, where P = H-2+ • • . -f (r — 1), Q = PA^—B, etc., B denoting the sum of the products of 1, 2,..., r — 1 two at a time, and A^ = l+2+ . . . +(r— 2). Then r+2'+ . . .+x^ = -^{x+l)x{x-l) . . .ix-r+l)+-{x+l)x. ..{x-r+2) r+1 r +-^(x+l)x. ..{x-r+3)+. . . +^ix+l)x{x-l)-\-Ux+l)x. Take r = x and let x+1 be a prime. By Fermat's theorem, V, 2"^, . . ., x' each has the remainder unity when divided by x+1, so that their sum has the remainder x. Thus l+x\ is divisible by x+1. Genty^^ proved the converse of Wilson's theorem and noted that an equivalent test for the primahty of p is that p divide (p— n)!(n — 1)! — ( - 1)". For n = (p+ 1)/2, the latter expression is \ (^zi) !^ 2± i [Lagrange^^]. Franz von Schaffgotsch^^ was led by induction to the fact (of which he gave no proof) that, if n is a prime, the numbers 2, 3, . . . , n — 2 can be paired so that the product of the two in any pair is of the form xn+1 and the two of a pair are distinct. Hence, by multipUcation, 2-3...(n — 2) has the remainder unity when divided by n, so that (n — 1)! has the remainder n — 1. For example, if n = 19, the pairs are 2-10, 4-5, 3-13, 7-11, 6-16, 8-12, 9-17, 14-15. Similarly, for n any power of a prime p, we can so pair the integers <n — l which are not divisible by p. But for n=15, 4 and 4 are paired, also 1 1 and 1 1 . Euler^^ had already used these associated residues (residua sociata). F. T. Schubert^^" proved by induction that the nth order of differences of r, 2",....isn!. A. M. Legendre-^ reproduced the second proof by Euler^^ of Fermat's theorem and used the theory of differences to prove (2) for a = x. Taking a; = p — 1 and using Fermat's theorem, we get (p — 1)!=(1 — 1)" — 1 (mod p). "Histoire et m6m. de I'acad. roy. sc. insc. de Toulouse, 3, 1788 (read Dec. 4, 1783), p. 91. "Abhandlungen d. Bohmischen Gesell. Wiss., Prag, 2, 1786, 134. "Opusc. anal., 1, 1783 (1772), 64, 121; Novi Comm. Ac. Petrop., 18, 1773, 85, §26; Comm. Arith. 1, 480, 494, 519. ""Nova Acta Acad. Petrop., 11, ad annum 1793, 1798, mem., 174-7. "Th6orie des nombres, 1798, 181-2; ed. 2, 1808, 166-7. Chap. Ill] FeRMAt's AND WiLSON's THEOREMS. * 65 C. F. Gauss^^ proved that, if n is a prime, 2, 3, . . . , n— 2 can be associated in pairs such that the product of the two of a pair is of the form xn+1. This step completes Schaffgotsch's^^ proof of Wilson's theorem. Gauss^^ proved Fermat's theorem by the method now known to be that used by Leibniz^ and mentioned the fact that the reputed proof by Leibniz had not then been published. Gauss^*^ proved that if a belongs to the exponent t modulo p, a prime, then a-a^-a^ . . .a^^i — lY'^^ (mod p). In fact, a primitive root p of p may be chosen so that a=p^^~^^'\ Thus the above product is congruent to p*, where Thus p*=(p~2~}''^^ = ( — 1)'"*"^ (mod p). When a is a primitive root, a, a^,. . ., aF~^ are congruent to 1, 2, . . . , p — 1 in some order. Hence (p — 1) != ( — 1)^. This method of proving Wilson's theorem is essentially that of Euler.22 Gauss^^ stated the generalization of Wilson's theorem: The product of the positive integers < A and prime to A is congruent modulo ^ to — 1 if A = 4, p"* or 2p^, where p is an odd prime, but to + 1 if ^ is not of one of these three forms. He remarked that a proof could be made by use of associated numbers^^ with the difference that a;^=l (mod A) may now have roots other than ± 1 ; also by use of indices and primitive roots^° of a composite modulus. S. F. Lacroix^^ reproduced Euler's^^ third proof of Fermat's theorem without giving a reference. James Ivory^^ obtained Fermat's theorem by a proof later rediscovered by Dirichlet.^" Let N be any integer not divisible by the prime p. When the multiples N, 2N, SN, . . ., {p — l)N are divided by p, there result p dis- tinct positive remainders <p, so that these remainders are 1, 2, . . ., p — 1 in some order .^^ By multiplication, N^~^Q = Q-\-mp, where Q = (p — 1)!. Hence p divides iV^~^ — 1 since it does not divide Q. Gauss^^ used the last method in his proof of the lemma (employed in his third proof of the quadratic reciprocity law): If k is not divisible by the odd prime p, and if exactly /x of the least positive residues of k, 2k,. . ., l{p-l)k modulo p exceed p/2, then k^p-'^^^^= ( - 1)" (mod p) . [Cf . Grunert.^^] ''^Disquisitiones Arith., 1801, arts. 24, 77; Werke, 1, 1863, 19, 61. 2*Disq. Arith., art. 51, footnote to art. 50. soDisq. Arith., art. 75. ^iDisq. Arith., art. 78. 32Compl4ment des 416mens d'alglbre, Paris, ed. 3, 1804, 298-303; ed. 4, 1817, 313-7, "New Series of the Math. Repository (ed. Th. Leybourn), vol. 1, pt. 2, 1806, 6-8. "A fact known to Euler, Novi Comm. Acad. Petrop., 8, 1760-1, 75; Comm. Arith., 1, 275; and to Gauss, Disq. Arith., art. 23. Cf. G. Tarry, Nouv. Ann. Math., 18, 1899, 149, 292. ^''Comm. soc. reg. so. Gottingensis, 16, 1808; Werke, 2, 1-8. Gauss' Hohere Arith., German transl. by H. Maser, Berhn, 1889, p. 458. 66 History of the Theory of Numbers. [Chap, hi J. A. Gninert^® considered the series K n] = n-- (^) (71-1)-+ Q (n-2r- . . ., to which Euler's (3) reduces for a = n, x = m, and proved that [m, n]=n\[m — l, n — l] + [w — 1, n]\ . This recursion formula gives [m,n] = 0 (m = 0, l,...,n-l); [72, n]=n\ [cf. (2)], Any [m, n] is di\isible by n\. As by the proof of Lagrange,^^ [m, n] + ( — 1)" is di\isible by w + 1 if the latter is a prime >n. Again, which for x = 0, h=l, gives [m, m]=ml. W. G. Horner^" proved Euler's theorem by generaUzing Ivory's^^ method. If ri, . . . , r^ are the integers <m and prime to m, then riN, . . . , r^N have the r's as their residues modulo m. P. F. Verhulst^^ gave Euler's proof^^ in a sUghtly different form. F. T. Poselger^^ gave essentially Euler's^° first proof. G. L. Dirichlet^° derived Fermat's and Wilson's theorems from a com- mon source. Call m and n corresponding numbers if each is less than the prime p and if mn=a (mod p), where a is a fixed integer not di\dsible by p (thus generahzing Euler's-^ associated numbers). Each number 1, 2, . . ., p — 1 has (5ne and but one corresponding number. If a:"=a (mod p) has no integral solution, corresponding numbers are distinct and (p-l)!=a^-^)''2 (modp). But if A; is a positive integer <p such that ^'^=a (mod p), the second root is p — A', and the product of the numbers 1, . . ., p — 1, other than k and p—k, has the same residue as a^^~^^^^, whence (p-l)!=-a^-i^/2(j^Q^p) The case a = 1 leads to Wilson's theorem. By the latter, we have a(p-i)/2=±i (modp), "Math. Abhandlungen, Erste Sammlung, Altona, 1822, 67-93. Some of the results were quoted by Gnmert, Archiv Math. Phys., 32, 1859, 115-8. For an interpretation in factoring of [m, n], see Minetola'" of Ch. X. "Annals of Phil. (Mag. Chem. . . .), new series, 11, 1826, 81. >8Corresp. Math. Phys. (ed. Qu^telet), 3, 1827, 71. "Abhand. Ak. Wiss. Berhn (Math.), 1827, 21. "Jour, fiir Math., 3, 1828, 390; Werke, 1, 1889, 105. Dirichlet," §34. Chap. Ill] FeRMAT's AND WiLSON's THEOREMS. 67 the sign being + or — according as A;^=a (mod p) has or has not integral solutions (Euler's criterion). Squaring, we obtain Fermat's theorem. Finally, Dirichlet rediscovered the proof by Ivory .^^ [Cf. Moreau.-^^^] J. Binet^^ also rediscovered the proof by Ivory .^^ A. Cauchy^^ gave a proof analogous to that by Euler.^" An anonymous writer^^ proved that if n is a prime the binomial coeffi- cient (n — l)k has the residue ( — 1)*' modulo n, so that {l+xr-'-l=-x+x^- . . .+x''-\ {l+x)\{l+xy-^-l\^x{x''-^-l), modulo n. Thus Fermat's theorem follows by induction on x as in the proof by Euler.^^ V. Bouniakowsky^ gave a proof of Euler's theorem similar to that by Laplace. ^^ If a^h is divisible by a prime p, aP^-'ifo^""' is divisible by p", provided p>2 when the sign is plus. Hence if p, p' ,. . . are distinct primes, o'±6' is divisible by iV = p"p"*'. . . , where t = p''~^p"''~^ . . . , if a=*=6 is divisible by pp' . . . , provided the p's are > 2 if the sign is plus. Replace a by its (p — l)th power and 6 by 1 and use Fermat's theorem; we see that a' — l is divisible by N if e=(f}{N). The same result gives a generalization of Wilson's theorem^ U?)-l)!t^'*"+l=0(modp"). He gave {ibid., 563-4) Gauss'^" proof of Wilson's theorem. J. A. Grunert^^ used the known fact that, if 0<k<p, then k, 2k,. . ., (p — l)k are congruent to 1, 2, . . ., p — 1 in some order modulo p, a prime, to show that kx=l (mod p) has a unique root x. Wilson's theorem then follows as by Gauss. If {ibid., p. 1095) we square Gauss' formula,^^ we get Fermat's theorem. Giovanni de Paoli^® proved Fermat's and Euler's theorems. In (x+iy=x^-\-i-{-pS,, where p is a prime, S^ is an integer. Change x to x — 1, . . . , 2, 1 and add the resulting equations. Thus x-l x^-x^p^S,. Replace x by a:"*, divide by x"^ and set y = x^~'^. Thus r - 1 = pXm, X^=XSz"'/x'' = integer. Replace m by 2m,..., {p — l)m, add the resulting equations, and set Y„=l +Xrn+X2m-\- ■ • . +X(,^i),n. Thus r"-l=p(y"-l)F^ = p^X^7^. "Jour, de I'^cole polytechnique, 20, 1831, 291 (read 1827). Cauchy, Comptes Rendus Paris, 12, 1841, 813, ascribed the proof to Binet. «Exer. de math., 4, 1829, 221; Oeuvres, (2), 9, 263. R^sumg analyt., Turin, 1, 1833, 10. «Jour. fiir Math., 6, 1830, 100-6. "M6m. Ac. Sc. St. P^tersbourg, Sc. Math. Phys. et Nat., (6), 1, 1831, 139 (read Apr. 1, 1829). "Kliigel's Math. Worterbuch, 5, 1831, 1076-9. "Opuscoli Matematici e Fisici di Diversi Autori, Milano, 1, 1832, 262-272. 68 History of the Theory of Numbers. [Chap, hi Change m to mp, . . . , T/zp""^. Thus 2/-^ - 1 = p(^-^ - 1) F^, = p^X^y^F^,, Hence x^^^ — 1 is divisible by N for iV" = p" and so for any N. For X odd, x^ — 1 is divisible by 8, and x*'" — 1 by 2(x^*"— 1). As above, he found that x' - 1 is divisible by 2* for t = m-2'-^, i> 2. Thus, if iV = 2'n, n odd, X*— 1 is divisible by N for A: = 2*"^0(n). A. L. Crelle^^ employed a fixed quadratic non-residue v of the prime p, and set j^=ry, vf=Vj (mod p). By multipUcation of ip-jf=rj, vf=v^ (mod p) (i = 1, • • • »^^) and use of v^^~'^^'^= — \, we get -](p-l)!f2=nr,v,= (p-l)! (modp). F. Minding^* proved the generaUzed Wilson theorem. Let P be the product of the tt integers a, /3, . . ., <A and relatively prime to A. Let A = 2''p"'g"r* . , . , where p, q,r,. . . are distinct odd primes, and m> 0. Take a quadratic non-residue t of p and determine a so that a=t (mod p), o=l (mod 2qr. . .). Then a is an odd quadratic non-residue of A. Let ax=a (mod A). For ^9^x, a, let i3?/=a (mod A). Then y^^a, x, jS. In this way the TT numbers a, j8, . . . can be paired so that the product of the two in any pair is =a (mod A), whence P=a''^^ (mod A). First, let A = 2''p^ Then a'=-l (mod p"^), s = p"*-^(p-l)/2, whence P=-l (mod^) if M = Oor L But, if m>1, a^={-iy =l(modp'"), a^=/ =l(mod2''), P= + l(modA). Next, let m>l, n>l, in A. Raising the above a*=— 1 to the power 2''"V~^(? — !)• • •> we get a'^^=-\-l (mod p"). A like congruence holds moduli g", r^ . . ., and 2", whence P=4-l (mod A). Finally, let A = 2", /x>L Then a=— 1 is a quadratic non-residue of 2" and, as above, P= ( — 1)^ (mod A),l = 2""^. The proof of Fermat's theorem due to Ivory^^ is given by Minding on p. 32. J. A. Grunert*^ gave Horner's^^ proof of Euler's theorem, attributing the case of a prime to Dirichlet instead of Ivory .^^ A part of the generalized Wilson theorem was proved as follows: Let ri,. . ., r, denote the positive integers <p and prime to p. Let a be prime to p. In the table riflVg, r2a^rg, . . . , rqa\ «'Abh. Ak. Wiss. Berlin (Math.), 1832, 66. Reprinted." "Anfangsgriinde der Hoheren Arith., 1832, 75-78. "Math. Worterbuch, 1831, pp. 1072-3; Jour, fiir Math. 8, 1832, 187. Chap. Ill] FeKMAT's AND WiLSON's THEOREMS. 69 a single term of a row is =1 (mod p). If this term be TkO^rk, replace it by (p—rk)a\^-l. Next, if r^^a^^ =f1, r„aVi=±l, then rk-^ri=p and one of the r„ is replaced by p—r^. Hence we may separate riO, . . ., r^a into q/2 pairs such that the product of the two of a pair is = ± 1 (mod p) . Taking a = 1, we get ri . . .rg= ± 1 (mod p). The sign was determined only for the case p a prime (by Gauss' method). A. Cauchy^*^ derived Wilson's theorem from (1), page 62 above. *Caraffa^^ gave a proof of Fermat's theorem. E. Midy^^ gave Ivory's^^ proof of Fermat's theorem, W. G. Horner^^ gave Euler's^^ proof of his theorem. G. Libri^^ reproduced Euler's proof^^ without a reference. Sylvester^^ gave the generalized Wilson theorem in the incomplete form that the residue is ± 1. Th. Schonemann^^ proved by use of symmetric functions of the roots that if s"+6i2;""^+ ... =0 is the equation for the pth powers of the roots of x^+aiX^~^-{- ... =0, where the a's are integers and p is a prime, then hi=af (mod p). If the latter equation is (x — 1)'' = 0, the former is 2'*-(nP+pQ)2"-^+. ..=0, and yet is evidently (2:-l)'* = 0. Hence 71^=71 (mod p). W. Brennecke^^ elaborated one of Gauss'^^ suggestions for a proof of the generalized Wilson theorem. For a>2, x^=l (mod 2°) has exactly four incongruent roots, =•= 1, ='= (l+2"~^), since one of the factors x=^l, of differ- ence 2, must be divisible by 2 and the other by 2""^. For p an odd prime, let ri, . . ., r^ be the positive integers <p" and prime to p", taking ri = l, r^ = p"— 1. For 2^s^)u — 1, the root x of r^x^l (mod p") is distinct from Ti, r^, r^. Thus 7-2, ... , r^_i may be paired so that the product of the two of a pair is =1 (mod p"). Hence ri . . .r^= — 1 (mod p"). This holds also for modulus 2p". For a > 2, (2-i-i)(2»-i+l)=-l, ri. . .r^=-\-l (mod 2"). Finally, let N=p''M, where M is divisible by an odd prime, but not by p. Then m=(f>{M) is even. The integers <N and prime to p are rj>rj+p'^, rj+2p%.. ., r,.+(M-l)p» (i = l,. . ., m). For a fixed j, we obtain m integers <iV and prime to N. Hence if \N\ denotes the product of all the integers < N and prime to iV, \N\^{n. . .r^)'^=l (mod p"). ForiV = pV.-., \N\=1 (mod O,--, whence jiVt^l (modiV). "R6suin6 analyt., Turin, 1, 1833, 35. "Elem. di mat. commentati da Volpicelli, Rome, 1836, I, 89. "De quelques propri^t^s des nombres, Nantes, 1836. "London and Edinb. Phil. Mag., 11, 1837, 456. "M6m. divers savants ac. sc. Institut de France (math.), 5, 1838, 19. "Phil. Mag., 13, 1838, 454 (14, 1839, 47); Coll. Math. Papers, 1, 1904, 39. "Jour, fiir Math., 19, 1839, 290; 31, 1846, 288. Cf. J. J. Sylvester, Phil. Mag., (4), 18, 1859, 281. "Jour, fiir Math., 19, 1839, 319. 70 History of the Theory of Numbers. [Chap. HI A. L. Crelle^^ proved the generalized Wilson theorem. By pairing each root <T of x-=l (mod s) with the root s—a, and each integer a<s, prime to s and not a root, with its associated number a', where aa'=\ (mod s), we see that the product of all the integers <s and prime to s is = + 1 or —1 (mod s) according as the number n of pairs of roots o-, s—o- is even or odd. To find n, express s in every way as a product of two factors u, v, whose g. c. d. is 1 or 2; in the respective cases, each factor pair gives a single root (T or two roots. Treating four subcases at length it is shown that the num- ber of factor pairs is 2^" in each case, where k is the number of distinct odd primes dividing s ; and then that n is odd if s = 4, p"* or 2p", but even if n is not of one of these three forms. A. Cauchy^^" proved Fermat's theorem as had Leibniz.'* V^^ (S. Earnshaw?) proved Wilson's theorem by Lagrange's method and noted that, if Sr is the sum of the products of the roots of AqX"'+Aix"'~^-\- . . . = 0 (mod p) taken r at a time, then AoSi — { — iyAi is divisible by p. Paolo Gorini^" proved Euler's theorem 6'=1 (mod A), where t=(f>(A), by arranging in order of magnitude the integers (A) p', p", . . . , p^'^ which are less than A and prime to A. After omitting the numbers in (A) which are di\'isible by h, we obtain a set (B) q',. . ., q^^\ Let 5^"^ be the least of the latter which when increased by A gives a multiple of h : (C) g(-)+A = p^'^^6. V (a-1) i(0 The numbers* (A) coincide with those in sets (B) and (D) : (D) p%p"b,...,p^-%. Hence by multiplication and cancellation of p', (F) q'...qH''-^ = p^''\..p^ To each number (B) add the least multiple of A which gives a sum divisible by b, say (G) q'+g'A,..., q^^+g^^A. The least of these is q^''^-\-A = p^^^h, by (C). Each number (G) is <6A and all are distinct. The quo- tients obtained by di\'iding the numbers (G) by h are prime to A and hence included among the p^^V-j P^'\ whose number is t—a-\-l=l, so that each arises as a quotient. Hence (H) n(g«+^«A)=PA+g'. . .g^'^ = p' Combine this with (F) to eliminate the p's. pW -a+l q'. . .g«6''-W-"+i = PA+5'. . .q (0 ■Q We get 6'-l = QA. "Jour, fur Math., 20, 1840, 29-56. Abstract in Bericht Akad. Wiss. Berlin, 1839, 133-5. »8aM6m. Ac. Sc. Paris, 17, 1840, 436; Oeuvres, (1), 3, 163-4. "Cambr. Math. Jour., 2, 1841, 79-81. "Annali di Fisica, Chimica e Mat. (ed., G. A. Majocchi), Milano, 1, 1841, 255-7. *To follow the author's steps, take A = 15, 6 = 2, whence « = 8, i = 4, (A) 1, 2, 4, 7, 8, 11, 13, 14; (B) 1, 7, 11, 13; (C) 1 + 15 = 8-2, ?(<>) =8, a = 5; (D) 2, 4, 8, 14; (F) 171113 2« = 8111314; (G) 1 + 15,7 + 15, 11+15;13 + 15, each g = l; the quotients of the latter by 2 are 8, 11, 13, 14, viz., last four in (A); (H) P.15 + 1.7.11.13=8.11.13.14.2«; the second member ifl 1-71113 2" by (F). Hence 171113 (28-l) = 15P. Chap. Ill] FeRMAT's AND WiLSON's THEOREMS. 71 E. Lionnet^^ proved that, if p is an odd prime, the sum of the mth powers of 1,. . ., p — 1 is divisible by p for 0<m<p — l. Hence the sum P^ of the products of 1, ... ,p — 1 taken w at a time is divisible by p [Lagrange^^]. Since (l + l)(l+2)...(H-p-l) = l+Pi+P2+...H-Pp-2+(p-l)!, l + (p — 1)! is divisible by p. E. Catalan^^ gave the proofs by Ivory^^ and Horner.^^ C. F. Arndt^^ gave Horner's proof; and proved the generalized Wilson theorem by associated numbers. O. Terquem^^ gave the proofs by Gauss^^ and Dirichlet.^° A. L. Crelle^^ republished his proof'*'' of Wilson's theorem, as well as that by Gauss^° and Dirichlet.^° Crelle^^ gave two proofs of the generalized Wilson theorem, essentially that by Minding^^ and that given by himself.^^ If fj, is the number of distinct odd prime factors of z, and 2^" is the highest power of 2 dividing z, and r is a quadratic residue of z, then (p. 150) the number n of pairs of roots ±x of x^=r (mod z) is 2""^ if m = 0 or 1, 2" if m = 2, 2"'^^ if w>2. Using the fact (p. 122) that the quadratic residues of z are the e=(f){z)/(2n) roots of r*=l (mod z), it is shown (p. 173) that, if v is any integer prime to z, y*'^^^'''*=l (mod z), "a, perfection of the Euler- Fermat theorem." L. Poinsot^^ failed in his attempt to prove the generalized Wilson theorem. He began as had Crelle.^^ But he stated incorrectly that the number n of pairs of roots =^x of x^^l (mod s) equals the number v of ways of expressing s as a product of two factors P, Q whose g. c. d. is 1 or 2. For each pair =^x, it is implied that x—1 and x+1 uniquely determine P, Q. For s = 24, n = y = 4; but for the root x = 7 (or for x = 17), a: ± 1 yield P, Q = 3, 8, and 6, 4. To correct another error by Poinsot, let n be the number of distinct odd prime factors of s and let 2"* be the highest power of 2 dividing s; then y = 2''-^ 2", 3-2''-^ or 2"+^ according as w = 0, 1, 2, or ^3, whereas [Crelle*^^] n = 2''-\ 2''-\ 2^ 2"+^ No difficulty is met (pp. 53-5) in case the modulus is a power of a prime. He noted (p. 33) that if Vi, r2, . . . are the integers <N and prime to N, and tt is their product, they are congruent modulo N to tt/ti, Tr/ra, ..., whence T=Tr''~^ (mod N), where v=(f){N). Thus, by Euler's theorem, 7r^= 1. This does not imply that 7r= =*= 1 as cited by Aubry,!" pp. 30O-I. Poinsot (p. 51) proved Euler's theorem by considering a regular polygon of N sides. Let x be prime to N and < N. Join any vertex with the xth ver- tex following it, the new vertex with the a:th vertex following it, etc., thus defining a regular (star) polygon of N sides. With the same x, derive "Nouv. Ann. Math., 1, 1842, 175-6. «/6td., 462-4. «Archiv Math. Phys., 2, 1842, 7, 22, 23. "Nouv. Ann. Math., 2, 1843, 193; 4, 1845, 379. «Jour. fur Math., 28, 1844, 176-8. «8/bid., 29, 1845, 103-176. «^Jour. de Math., 10, 1845, 25-30. German exposition by J. A. Grunert, Archiv Math. Phys., 7, 1846, 168, 367. 72 History of the Theory of Numbers. [Chap. hi similarly a new A^-gon, etc., until the initial polygon is reached.®^ The number )U of distinct polygons thus obtained is seen to be a divisor of <t>{N), the number of polygons corresponding to the various a:'s. If in the initial polygon we take the x^th vertex following any one, etc., we obtain the initial polygon. Hence of and thus also x"^^^ has the remainder unity when divided by N. [When completed this proof differs only shghtly from that by Euler."] E. Prouhet^^ modified Poinsot's method and obtained a correct proof of the generalized Wilson theorem. Let r be the number of roots of x^=l (mod N), and w the number of ways of expressing iV as a product of two relatively prime factors. If AT = 2'"pi" . . . p/", where the p's are distinct odd primes, evidently w; = 2*' if m>0, 1^ = 2""^ if m = 0. By considering divisors of a: =*= 1, it is proved that r = 2u' if ttz = 0 or 2, r = w; if w = 1, r = 4iy if m>2. Hence r = 2" if m = 0 or 1, 2"+^ if m = 2, 2"+^ if m> 2. By Crelle,^^ the product P of the integers <A'' and prime to N is =( — 1)''''^ (mod N). Thus for jLt>0, P= + l unless m = 0 or l,/i = l, viz., N = p^ or 2p'; while, for ^ = 0, N = 2"', m>2, we have r = 4, P=-\-l. Friderico Arndt'^° elaborated Gauss'^^ second suggestion for a proof of the generalized Wilson theorem. Let gf be a primitive root of the modulus p" or 2p", where p is an odd prime. Set y=0(p"). Then g, g^,. . ., g" are congruent to the numbers less than the modulus and prime to it. If P is the product of the latter, P^g''-'^^^'^ But g"^=-l. Hence P=-l. Next, if n>2, the product of the incongruent numbers belonging to an exponent 2"""* is =1 (mod 2"). Next, consider the modulus M = AB, where A and B are relatively prime. The positive integers < M and prime to M are congruent modulo M to Ayi-\-Bxj, where the 0:^ are <A and prime to A, the yi are <B and prime to B. But, if a=0(A), a 7ri = 'n.{Ayi+BXj)=B''xi. . .Xa=Xi. . .x^ (mod A), 3 = 1 P=riTr2. ..^{x^.. .xJ'^^^Hniod A). By resolving M into a product of powers of primes and applying the above results, we determine the sign in P=±l (mod M). J. A. Grunert^^ proved that if a prime n+l>2 divides no one of the integers ai, . . ., a„, nor any of their differences, it divides aia2. . .a„+l, and stated that this result is much more general than Wilson's theorem (the case aj=j). But the generalization is only superficial since ai,. . ., a„ are congruent modulo n+1 to 1,..., n in some order. His proof employed Fermat's theorem and certain complex equations involving products of differences of n numbers and sums of products of n numbers taken m at a time. J. F. Heather^^ gave without reference the first results of Grunert.^^ osCf. P. Bachmann, Die Elemente der Zahlentheorie, 1892, 19-23. «»Nouv. Ann. Math., 4, 1845, 273-8. "Jour, fvir Math., 31, 1846, 329-332. "Archiv Math. Phys., 10, 1847, 312. "The Mathematician, London, 2, 1847, 296. Chap. Ill] FeRMAT's AND WiLSON's THEOREMS. 73 A. Lista'^^ gave Lagrange's proof of Wilson's theorem. V. Bouniakowsky'^^ gave Euler's^^ proof. P. L. Tchebychef^^ concluded from Fermat's theorem that (a:-l)(x-2). . .(x-p+l)-a;^-i+l=0 (mod p) is an identity if p is a prime. Hence if Sj is the sum of the products of 1, . . . , p — 1 taken j at a time, Sj=0 (i<p — 1), Sp_i= — 1 (mod p), the last being Wilson's theorem. Sir F. PoUock'^^ gave an incomplete statement and proof of the general- ized Wilson theorem by use of associated numbers. Likewise futile was his attempt to extend Dirichlet's^" method [not cited] of association into pairs with the product = a (mod m) to the case of a composite m. E. Desmarest" gave Euler's^^ proof of Fermat's theorem. 0. Schlomilch^^'' considered the quotient {„p_ (») („_i)p+ («) („_2)p- . . . f/n!. J. J. Sylvester'^* took x = l, 2,. . ., p — 1 in turn in {x-l){x-2) . . . (x-p+l) =x^-'+A,x^-^+ . . . +A,_i, where p is a prime. Since x^~^=l (mod p), there result p — 1 congruences linear and homogeneous in Ai, . . . , Ap_2, Ap-i+1, the determinant of whose coefficients is the product of the differences of 1, 2, . . . , p — 1 and hence not divisible by p. Thus Ai=0,..., Ap_i+1=0, the last giving Wilson's theorem. W. Brennecke'^^ proved Euler's theorem by the methods of Horner^^ and Laplace, ^^ noting that {a^-y=l (mod p^), (a^-i)^'=l (mod p^), .... He gave the proof by Tchebychef ^^ and his own proof." J. T. Graves^" employed nx=n+l (mod p), where p is a prime, and stated that, for n = l,..., p — 1, then x=2,..., p in some order. Also x=p ior n = p — l. Hence 2-3. . .(p — l)=p — 1 (mod p). H. Durege^^ obtained (2) for a = x and Grunert's^^ results on the series [m, n] by use of partial fractions for the reciprocal o( x(x — l) . . .{x — n). E. Lottner^^ employed for the same purpose infinite trigonometric and algebraic series, obtaining recursion formulae for the coefficients. "Periodico Mensual Cienciaa Mat. y Fis., Cadiz, 1, 1848, 63. T*BuU. Ac. Sc. St. P6tersbourg, 6, 1848, 205. "Theorie der Congruenzen, 1849 (Russian); in German, 1889, §19. Same proof by J. A. Serret, Cours d'algebre sup^riem-e, ed. 2, 1854, 324. ^«Proc. Roy. Soc. London, 5, 1851, 664. "TMorie des nombres, Paris, 1852, 223-5. ""Jour, fur Math., 44, 1852, 348. "Cambridge and Dublin Math. Jour., 9, 1854, 84; Coll. Math. Papers, 2, 1908, 10. "Einige Satze aus den Anfangsgriinden der Zahlenlehre, Progr. Realschule Posen, 1855. soBritish Assoc. Report, 1856, 1-3. "Archiv Math. Phys., 30, 1858, 163-6. "/bid., 32, 1859, 111-5. 74 History of the Theory of Numbers. [Chap, hi J. Toeplitz^ gave Lagrange's proof of Wilson's theorem. M. A. Stern^ made use of the series for log (1 — x) to show that 1+x+xH. . . =-^ = e'+*'/2+^/3+...^ l—x Multiply together the series for e', e'*^^, etc. By the coeflBcient of x^. p! P' (p-2)!' ••• Take p a prime. No term of s has a factor p in the denominator. Hence (1-s) • (p-l)! = ^-tfcli^ = integer. P V. A. Lebesgue^^ obtained Wilson's theorem by taking x = p — l in p X Hk+l) . . . {k-\-p-2) =x(x+l) . . . (x+p-1). k=i If P is a composite number ?^4, (P — 1)! is di\'isible by P. He (p. 74) attributed Ivory's^^ proof of Fermat's theorem to Gauss, without reference. G. L. Dirichlet^^ gave Horner's" and Euler's^^ proof of Euler's theorem and derived it from Fermat's by the method of powering. His proof (§38) of the generalized Wilson theorem is by associated numbers, but is some- what simpler than the analogous proofs. Jean Plana" used the method of powering. Let N = p^pi' .... For M prime to N, M^~'^ = 1 +pQ. Hence Thus for e = (p{p^pi'), M" — ! is divisible bj' p'' and p/' and hence by their product, etc. Plana gave also a modification of Lagrange's proof of Wilson's theorem by use of (2) ; take x=a = p — l, subtract the expansion of (1 — 1)""^ and write the resulting series in reverse order: (p-l)!+l = (^2^)(2^-^-l)-(V)(3^-^-l)+... -(^:D](p-2)^-^-lt + 1(p-i)''-'-if- H. F. Talbot^^ gave Euler's^^ proof of Fermat's theorem. J. Blissard^^" proved the last statement of Euler.^ C. Sardi^^ gave Lagrange's proof of Wilson's theorem. P. A. Fontebasso^*^ proved (2) for x = a by finding the first term of the ath order of differences ofy'',{y+hy,{y-\-2hy,. . . and then setting y = 0,h = l. ^'Archiv Math. Phys., 32, 1859, 104. "Lehrbuch der Algebraischen Analysis, Leipzig, 1860, 391. "Introd. thdorie des nombres, Paris, 1862, 80, 17. "Zahlentheorie (ed. Dedekind), §§19, 20, 127, 1863; ed. 2, 1871; ed. 3, 1879, ed. 4, 1894. 8'Mem. Acad. Turin, (2), 20, 1863, 148-150. "Trans. Roy. Soc. Edinburgh, 23, 1864, 45-52. ss^'Math. Quest. Educ. Times, 6, 1866, 26-7. "Giomale di Mat., 5, 1867, 371-6. •"Saggio di una introd. arit. trascendente, Treviso, 1867, 77-81. Chap. Ill] FeRMAT's AND WilSON's THEOREMS. 75 C. A. Laisant and E. Beaujeux^^ used the period ai . . .a„ of the periodic fraction to base B for the irreducible fraction pi/q, where q is prime to B. li P2,. . ., Pn are the successive remainders, Bpi = aiq+p2, Bp2 = a2q+P3,. . ., Bpr, = anq+Pi. Starting with the second equation, we obtain the period a2. . .a„ai for P2/q- Similarly for ps/q,. . ., Pn/q- Thus the f=(p{q) irreducible fractions with denominator q separate into sets of n each. Hence /=A;n. Since 5'*=!, B^=l (modg). L. Ottinger^- employed differential calculus to show that, in P={a+d){a+2d).. . \a+ip-l)d\ =aP-i+Ci^~V-2d+C2^-V-3d2_|__ ^ 3=1 q~ri- Cr being the sum of the products of 1, 2, . . . , A; taken r at a time. Hence, if p is a prime, C?~^ (r = 1, . . ., p — 2) is divisible by p, and P=aP-i+c^-2d. ..{p-l)d (mod p). For a = d = l, this gives 0=l + (p — 1)! (mod p). H. Anton^^ gave Gauss' ^^ proof of Wilson's theorem. J. Petersen^^ proved Wilson's theorem by dividing the circumference of a circle into p equal parts, where p is a prime, and marking the points 1, . . ., p. Designate by 12. . .p the polygon obtained by joining 1 with 2, 2 with 3,. . ., p with 1. Rearranging these numbers we obtain new poly- gons, not all convex. While there are p! rearrangements, each polygon can be designated in 2p ways [beginning with any one of the p numbers as first and reading forward or backward], so that we get (p — 1)!/2 figures. Of these ^(p — 1) are regular. The others are congruent in sets of p, since by rotation any one of them assumes p positions. Hence p divides (p — 1)!/2 -(p-l)/2 and hence (p-2)!-l. Cf. Cayley^o^. To prove Fermat's theorem, take p elements from q with repetitions in all ways, that is, in q^ ways. The q sets with elements all alike are not changed by a cyclic permutation of the elements, while the remaining q^ — q sets are permuted in sets of p. Hence p divides q^—q. [Cf. Perott,^^® Bricard.i"] F. Unferdinger^^ proved by use of series of exponentials that 2''-(';')(^-ir+(2)(^-2r-...+(-ir(^)(2-mr ' "Nouv. Ann. Math., (2), 7, 1868, 292-3. "Archiv Math. Phys., 48, 1868, 159-185. ''Ibid., 49, 1869, 297-8. "Tidsskrift for Mathematik, (3), 2, 1872, 64-65 (Danish). "^Sitzungsberichte Ak. Wisa. Wien, 67, 1873, II, 363. 76 History of the Theory of Numbers. [Chap, hi is zero ii n<m, but, if ti^tt^, equals where For n = m, the initial sum equals Em = rn\. P. Mansion^® noted that Euler's theorem may be identified with a property of periodic fractions [cf. Laisant^^]. Let N be prime to R. Taking R as the base of a scale of notation, divide 100. . .by A^ and let gi . . .g„ be the repetend. Then (72" — l)/iV = Q'i. . .5„. Unless the n remainders r^ exhaust the integers <N and prime to A^, we divide r/ 00. . .by A^, where r/ is one of the integers distinct from the r,-, and obtain n new remainders r/. In this way it is seen that n divides (p{N), so that N divides R'^'-^ — l. [At bottom this is Euler's^* proof.] P. Mansion^^ reproduced this proof, made historical remarks on the theorem and indicated an error by Poinsot.^^ Franz Jorcke^^ reproduced Euler's^^ proof of Wilson's theorem. G. L. P. V. Schaewen^^ proved (2) with a changed to —p, by expanding the binomials. Chr. ZeUer^o" proved that, for n ?^ 4, is divisible by n unless n is a prime such that n — 1 divides x, in which case the expression is = — 1 (mod n) . A. Cayley^°^ proved Wilson's theorem as had Petersen.^^ E. Schering^^^ took a prime to m = 2'pi''\ . .p'", where the p's are dis- tinct odd primes and proved that x^=a (mod m) has roots if and only if a is a quadratic residue of each Pi and if a = l (mod 4) when 7r = 2, a=l (mod 8) when 7r>2, and then has \l/{m) roots, where \p{m)=2'', 2""^^ or 2"''"^, according as 7r<2, 7r = 2, or 7r>2. Let a be a fixed quadratic residue of m and denote the roots by ^aj (j = l,. . ., 4'/2). Set a.- =m—aj. The <}>{m)—\f/{ni) integers <m and prime to m, other than the ay, a/, may be denoted by aj, a/ (i = |iA+lj- • •? 20)j where aja'j=a (mod m). From the latter and —aja/=a (i = l, . . ., ^/2), we obtain, by multiplication, ^i^(m) = (_j)}*(m)^^ .r^ (mod w), ••Messenger Math., 5, 1876, 33 (140); Xouv. Corresp. Math., 4, 1878, 72-6. "Th6orie des nombres, 1878, Gand (tract). •*Uber Zahlenkongruenzen, Progr. Fraustadt, 1878, p. 31. "Die Binomial Coefficienten, Progr. Saarbriicken, 1881, p. 20. looBull. des sc. math, astr., (2), 5, 1881, 211^. "'Messenger of Math., 12, 1882-3, 41; CoU. Math. Papers, 12, p. 45. iwActa Math., 1, 1882, 153-170; Werke, 2, 1909, 69-86. Chap. Ill] FeRMAt's AND WiLSON's THEOREMS. 77 where the Vj are the integers <m and prime to m. Taking a=l, we have the generahzed Wilson theorem. Applying a like argument when a is a quadratic non-residue of m [Minding^^], we get ^J^(m)^^^_ .r^=(_l)^'^('») (mod m). This investigation is a generaUzation of that by Dirichlet.*" E. Lucas^"^ wrote Xp for x{x-\-l) . . .{x-\-p — l), and F^ for the sum of the products of 1, . . . , p taken g at a time. Thus ^ ~rA p-i^ ~r • • • ~rA p-i^ — Xp. Replacing p by 1, . . ., n in turn and solving, we get where (-l)"-^+'A„-p+i = •plp2 pn— p+1 1 r^ Y'*~p 0 o...iri the subscript p — 1 on the F's being dropped. After repeating the argument by Tchebychef^^, Lucas noted that, if p is an odd prime, A„_p+i=l or 0 (mod p), according as p — 1 is or is not a divisor of n. G. Wertheim^°^ gave Dirichlet's^^ proof of the generalized Wilson theorem; also the first step in the proof by Arndt.^° W. E. HeaP''^ gave without reference Euler's^'* proof. E. Catalan^°^ noted that if 2n+l is composite, but not the square of a prime, n\ is divisible by 2n+l; if 2n+l is the square of a prime, (n!)^ is divisible by 2n+l. C. Garibaldi^"^ proved Fermat's theorem by considering the number N of combinations of ap elements p at a time, a single element being selected from each row of the table en ^12. . .eia ^pl ^p2 • • • ^pa. From all possible combinations are to be omitted those containing elements from exactly n rows, for n = l, . . ., p — 1. Let An denote the number of combinations p at a time of an elements forming n rows, such that in each combination occur elements from each row. Then --iV-tO^- "'BuU. Soc. Math. France, 11, 1882-3, 69-71; Mathesis, 3, 1883, 25-8. i^Elemente der Zahlentheorie, 1887, 186-7; Anfangsgriinde der Zahlenlehre, 1902, 343-5 (331-2). "'Annals of Math., 3, 1887, 97-98. »««M6m. soc. roy. so. LiSge, (2), 15, 1888 (Melanges Math., Ill, 1887, 139). "^Giornale di Mat., 26, 1888, 197. 78 History of the Theory of Numbers. [Chap, hi Take each e^ = 1 ; then N = a^ smce the number of the specified combina- tions becomes the sum of all products of p factors unity, one from each row of the table. Thus 0"= ( ^j=a (mod p). p-i R. W. Genese^°^ proved Euler's theorem essentially as did Laisant.'^ M. F. Daniels^^^ proved the generahzed Wilson theorem. If ^(n) denotes the product of the integers <n and prime to n, he proved by induc- tion that ^(p')= — 1 (mod p') for p an odd prune. For, if pi, . . ., p„ are the integers < p' and prime to it, then pi +jp', . . ., Pn +ip' (i = 0, 1 , . . . , p — 1) are the integers < p'"^^ and prime to it. He proved similarly by induction that 1/^(2') = + ! (mod 2') if 7r>2. Evidently iA(2) = l (mod 2), »//(4)=-l (mod 4). If m = a''b^ . . . and n = l^, where I is a new prime, then \p(m)=e (mod m), \l/{n) = r) (mod n) lead by the preceding method to \l/{mn) = e'^^"^ (mod m), viz., 1, unless n = 2. The theorem now follows easily. E. Lucas^^^ noted that, if x is prime to n = AB . . ., where A, B,. . . are powers of distinct primes, and if 0 is the 1. c. m. of (f){A), (l>{B),. . ., then x'^= 1 (mod n). In case A = 2^', A:> 2, we may replace (j){A) by its half. To get a congruence holding whether or not x is prime to n, multiply the former congruence by x", where a is the greatest exponent of the prime factors of n. Note that <}>-\-<T<n [Bachmann^^^' "^]. CarmichaeP^^ wrote X(n) for 0. E. Lucas^^^ found A^~^x^~^ in two ways by the theory of differences. Equating the two results, we have (p-i)!=(p-ir^-(^7^)(p-2ri+...-(^:^)i Each power on the right is = 1 (mod p) . Thus (p-l)!=(l-l)p-i-l=-l (modp). P. A. MacMahon^^^ proved Fermat's theorem by showing that the number of circular permutations of p distinct things n at a time, repetitions allowed, is h<l>(d)p^^', ft where d ranges over the divisors of n. For n a prime, this gives p"+(n — l)p=0, p"=p (mod n). ^^ Another specialization led to Euler's generalization. E. Maillet^^^ applied Sylow's theorem on subgroups whose order is the highest power p'' of a prime p dividing the order m of a group, viz., "^British Association Report, 1888, 580-1. ""Lineaire Congruenties, Diss. Amsterdam, 1890, 104-114. ""BuU. Ac. Sc. St. P6tersbourg, 33, 1890, 496. ">Mathesis, (2), 1, 1891, 11; Th^orie des nombres, 1891, 432. ""Proc. London Math. Soc, 23, 1891-2, 305-313. "'Recherches sur les substitutions, Th§se, Paris, 1892, 115. Chap. Ill] FeRMAT's AND WiLSON's THEOREMS. 79 'm = pN{l+np), when h = l. For the sjonmetric group on p letters, m = p\ and N = p — \, so that (p — 1)!= — 1 (mod p). There is exhibited a special group for which m = pa^, N = a, whence a^=a (mod p). G. Levi"* failed in his attempt to prove Wilson's theorem. Let b and a={p — l)h have the least positive residues ri and r when divided by p. Then r+ri = p. Multiply h/p = q+ri/p by p — 1. Thus ri(p — 1) has the same residue as a, so that Ci T ri(p-l)=r+mp, - = q{p-l)+m+- He concluded that ri(p — l)=r, falsely, as the example p = 5, 6 = 7, shows. He added the last equation to r+ri = p and concluded that ri = l, r = p — l, so that (a+l)/p is an integer. The fact that this argument is independent of Levi's initial choice that 6 = (p — 2) ! and his assumption that p is a prime shows that the proof is fallacious. Axel Thue"^ obtained Fermat's theorem by adding a''-{a-iy = l+kp, {a-iy-{a-2y = l+hp, ..., P-0^=1 [PaoU*^]. Then the differences A^i^(j) of the first order of F{x)=x^~'^ are divisible by p for J = 1,. . ., p-2; likewise A^/^(l),. . .,^''-^F{l), By adding A^+i/r(o)=A^F(l)-A^(0) (i = l,. . ., p-2), we get -A^-^/?'(0) = 1+AV(1)-A2/?'(1)+. . .+A^-2F(1), (p-l)!+l=0(modp). N. M. Ferrers"® repeated Sylvester's"^^ proof of Wilson's theorem. M. d'Ocagne"^ proved the identity in r: (r+l)^+i+^^i^SPfcl^PV(^+l)'^'"''(-0*'=r'+' + l, where g = [(A;+l)/2] and P^"^ is the product of n consecutive integers of which m is the largest, while P^ = L Hence if /c+1 is a prime, it divides (j,_[_j)A:+i_^fc+i_2^ and Fermat's theorem follows. The case k = p — l shows that if p is a prime, q={p — l)/2, and r is any integer, S P%-.^i PV(^+l)''"''(-^)*=0 (mod g!). t=i T. del Beccaro"^ used products of linear functions to obtain a very com- pUcated proof of the generalized Wilson theorem. A. Schmidt"^ regarded two permutations of 1, 2, . . ., p as identical if one is derived from the other by a cyclic substitution of its elements. From one of the (p — 1)! distinct permutations he derived a second by adding "*Atti del R. Istituto Veneto di Sc, (7), 4, 1892-3, pp. 1816-42. "'Archiv Math, og Natur., Kristiania, 16, 1893, 255-265. '"Messenger Math., 23, 1893-4, 56. "^Jour. de I'^cole polyt., 64, 1894, 200-1. "8Atti R. Ac. Lincei (Fis. Mat.), 1, 1894, 344-371. ""Zeitschrift Math. Phys., 40, 1895, 124. 80 History of the Theory of Numbers. [Chap, hi unity to each element and replacing p+1 by 1. Let m be the least number of repetitions of this process which will yield the initial permutation. For p a prime, m = l or p. There are p — 1 cases in which m = \. Hence (p — 1) ! — (p — 1) is divisible by p. Cf . Petersen.^^ Many proofs of (3), p. 63, have been given. ^^° D. von Sterneck^-^ gave Legendre's proof of Wilson's theorem. L. E. Dickson^-^ noted that, if p is a prime, p(p — 1) of the p! substitu- tions on p letters have a linear representation x'=ax-\-h, a^O (mod p), while the remaining ones are represented analytically by functions of degree > 1 which fall into sets of p^(p — 1) each, viz., aJ{x-^h)-\-c, where a is prime top. Hencep!—p(p — l) is a multiple of p^(p — l), and therefore (p — 1)!+1 is a multiple of p. C. Moreau^-^ gave without references Schering's^°^ extension to any modulus of Dirichlet's^° proof of the theorems of Fermat and Wilson. H. Weber^'^ deduced Euler's theorem from the fact that the integers Km and prime to m form a group under multiplication, whence every integer belongs to an exponent dividing the order 0(m) of the group. E. Cahen^-^ proved that the elementary symmetric functions of 1,. . ., p — 1 of order <p — 1 are divisible by the prime p. Hence (a:-l)(a:-2). . .(a:-p+l)=xP-^+(p-l)! (modp), identically in x. The case x = l gives Wilson's theorem, so that also Fer- mat's theorem follows. J. Perott^-^ gave Petersen's^^ proof of Fermat's theorem, using q^ ''con- figurations" obtained by placing the numbers 1, 2,..., q into p cases, arranged in a line. It is noted that the proof is not vaUd for p composite; for example, if p = 4, g = 2, the set of configurations derived from 1212 by cyclic permutations contains but one additional configuration 2121. L. Kronecker^^^ proved the generalized Wilson theorem essentially as had Brennecke.^^ G. Candido^^^ made use of the identity aP+6P= (a+6)P-pa6(a+6)P-2+ . . . ^^_^).p(p-2r+10...(p-r-l) .,,_^ ^^^ 1-2. . .r Take p a prime and 6= —1. Thus a^ — a=(a — 1)^— (a — 1) (mod p). "«L'mterm6diaire des math., 3, 1896, 2&-28, 229-231; 7, 1900, 22-30; 8, 1901, 164. A. Capelli Giornale di Mat., 31, 1893, 310. S. Pincherle, ibid., 40, 1902, 180-3. "iMonatshefte Math. Phys., 7, 1896, 145. »»Annals of Math., (1), 11, 1896-7, 120. i"Nouv. Ann. Math., (3), 17, 1898, 296-302. i^Lehrbuch der Algebra, II, 1896, 55; ed. 2, 1899, 61. ^"£l6ment3 de la throne des nombres, 1900, 111-2. >»'Bull. des Sc. Math., 24, I, 1900, 175. i"Vorlesungen uber Zahlentheorie, 1901, I, 127-130. "'Giomale di Mat., 40, 1902, 223. Chap. Ill] FeRMAT's AND WiLSON's THEOREMS. 81 P. Bachmann^^^ proved the first statement of Lucas."*' He gave as a "new" proof of Euler's theorem (p. 320) the proof by Euler,^^ and of the generaUzed Wilson theorem (p. 336) essentially the proof by Arndt.^° J. W. Nicholson^^^ proved the last formula of Grunert.^^ Bricard^^^ changed the wording of Petersen's^^ proof of Fermat's theorem. Of the q^ numbers with p digits written to the base q, omit the q numbers with a single repeated digit. The remaining q^—q numbers fall into sets each of p distinct numbers which are derived from one another by cyclic permutations of the digits. G. A. Miller^^^ proved the generalized Wilson theorem by group theory. The integers relatively prime to g taken modulo g form under multiplica- tion an abelian group of order (f){g) which is the group of isomorphisms of a cyclic group of order g. But in an abelian group the product of all the ele- ments is the identity if and only if there is a single element of period 2. It is shown that a cyclic group is of order p", 2p* or 4 if its group of isomor- phisms contains a single element of period 2. V. d'Escamard^^^ reproduced Sylvester's''^ proof of Wilson's theorem. K. Petr^^* gave Petersen's®^ proof of Wilson's theorem. Prompt^^® gave an obscure proof that 2^~^ — 1 is divisible by the prime p. G. Arnoux^^® proved Euler's theorem. Let X be any one of the v=4>{m) integers a, /3, 7, . . ., prime to m and <m. We can solve the con- gruences aa'=/3/3'=77'= . . . =\ (mod m). Here a', jS', . . .form a permutation of a, |S, . . . . Thus In particular, for X = l, we get (a/3. . .)^=1. Hence for any X prime to m, V=\ (mod m). [Of. Dirichlet/" Schering,i°2 C. Moreau.i^^] R. A. Harris^^^" proved that (aj8 . . .)^ = 1 as did Arnoux^^^, but inferred falsely that a./3 . . . = ± 1. A. Aubry^^'^ started, as had Waring in 1782, with where yp = a:(a; — 1). . .(x— p+1). Then x'»+i-a;'^= F„+i+A7„+ . . . -I-MF3+F2. Summing for x = 1, . . . , p — 1 and setting Sk = l*+2*+ . . . + (p — 1)*, we get _\n±l\_ \n\ , ,MJ3J , \2\ n+2 w+1 «»Niedere Zahlentheorie, I, 1902, 157-8. ""Amer. Math. Monthly, 9, 1902, 187, 211. "iNouv. Ann. Math., (4), 3, 1903, 340-2. "2Annals of Math., (2), 4, 1903, 188-190. Cf. V. d'Escamard, Giornale di Mat., 41, 1903, 203-4; U. Scarpis, ihid., 43, 1905, 323-8. »"Giomale di Mat., 43, 1905, 379-380. i=^Casopis, Prag, 34, 1905, 164. ^"Remarques sur le theorSme de Fennat, Grenoble, 1905, 32 pp. "'Arithm^tique Graphique; Fonctions Arith., 1906, 24. i»««Math. Magazine, 2, 1904, 272. "'L'enseignement math., 9, 1907, 434-5, 440. 82 History of the Theory of Numbers. [chap. hi where \k\ =p{p — l) . . .(p — k). Hence, if p is a prime and n<p — 1, Sn+i—Sn—0. But Si^O. Hence s„=0(n<p — 1), Sp_i= — (p — 1)!. Thus Wilson's theorem follows from Fermat's. Without giving references, Aubry (p. 298) attributed Horner's" proof of Euler's theorem to Gauss; the proof (pp. 439-440) by Paoli^^ (and Thue^^^) of Fermat's theorem to Euler^^; the proof (p. 458) by Laplace^^ of Euler's theorem by powering to Euler. R. D. CarmichaeP^^ noted that, if L is the 1. c. m. of all the roots z of 0(2) = a, and if a: is prime to L, then a:°= 1 (mod L). Hence except when n and n/2 are the only numbers whose </)-function is the same as that of n, ^•pM = ^ holds for a modulus M which is some multiple of n. A practical method of finding M is given. R. D. CarmichaeP^^ proved the first result by Lucas.^^° J. A. Donaldson^^° deduced Fermat's theorem from the theory of periodic fractions. W. A. Lindsay"^ proved Fermat's theorem by use of the binomial theorem. J. I. Tschistjakov"^ extended Euler's theorem as had Lucas. ^^° P. Bachmann^^^ proved the remarks by Lucas,^^° but replaced <f>+(r<n by 71^0 +(7, stating that the sign is > if n is divisible by at least two distinct primes. A. Thue^^ noted that a different kinds of objects can be placed into n given places in o" ways. Of these let 11'^ be the number of placings such that each is converted into itself by not fewer than n applications of the operation which replaces each by the next and the last by the first. Then U2 is divisible by n. If n is a prime, 1/1 = a""— a and we have Fermat's theorem. Next, a''=2[/a, where d ranges over the divisors of n. Finally, if p, 5, . . . , r are the distinct prime factors of n, C/^=S(-l)?a"/^=0 (mod n), where D ranges over the distinct divisors oi pq. . .r, while 0 is the number of prime factors of D. Euler's theorem is deduced from this. H. C. PockUngton^^^ repeated Bricard's^^^ proof. U. Scarpis^^^ proved the generalized Wilson theorem by a method similar to Arndt's.^° The case of modulus 2^" (X>2) is treated by induction. Assume that Ilr=l (mod 2^), where ri, . . ., r„ are the v=4>{2^) odd integers <2^. Then rj, . . ., r„ ri+2^, . . ., r„+2'' are the residues modulo 2^"+^ and their product is seen to be =1 (mod 2''"''^). Next, let the modulus be "«BuU. Amer. Math. Soc, 15, 1908-9, 221-2. "'/bid., 16, 1909-10, 232-3. ""Edinburgh Math. Soc. Notes, 1909-11, 79-84. "i/bwf., 78-79. "^Tagbl. XII Vers. Russ. Nat., 124, 1910 (Russian). i«Niedere Zahlentheorie, II, 1910, 43-44. '*^Skrifter Videnskaba-Selskabet, Christiania, 1910, No. 3, 7 pp. >«Nature, 84, 1910, 531. i«Periodico di Mat., 27, 1912, 231-3. Chap. Ill] FeEMAT's AND WiLSON's ThEOKEMS. 83 n = pi' . . .p^'* {h>2), n9^2p^. Then a system of residues modulo n, each h prime to n, is given by S A,r„ with i=l A ' w) n \</>(pi°0 } where r^ ranges over a system of residues modulo pi"', each prime to p,. Let P be the product of these SA,rj. Since AiAj is divisible by n if iT^j, h <p(n/pi°-i\ p_2Ai^W(nr,)^ ^ (modn). t=i Thus P— 1 is divisible by each p°^ and hence by n. *Illgner^^^ proved Fermat's theorem. A. Bottari^^^ proved Wilson's theorem by use of a primitive root [Gauss^°]. J. Schumacher^^^ reproduced Cayley's^°^ proof of Wilson's theorem. A. Arevalo^^° employed the sum /S„ of the products taken n at a time of 1, 2, . . . , p — 1. By the known formula it follows by induction that Sn is divisible by the prime p if n<p — 1. In the notation of Wronski, write a^^*" for a{a+r).. . \a+ip-l)r\ =a''-\-Sia''-''r+ . . .+Sp_iar^-\ For a = r = l, we have p! = l+*Si+. . .-\-Sp^i, whence >Sp_i= — 1 (mod p), giving Wilson's theorem. Also, a^^''=a^—a'r^~^. Dividing by a and taking r = l, we have (a+l)(^-^^/^=a^-i-l (modp). The left member is divisible by p if o is not. Hence we have Fermat's theorem. Another proof follows from Vandermonde's formula (x+ay^'= S (J)x^p-''^^'a^^'=x''^'+a''^' (mod p), (xi + . . . +xy'-=x,^/'+ . . . +a^/^ a^/''=a-P^ Remove the factor a and set r = 0; we obtain Fermat's theorem. Prompt^^^ gave Euler's^^ proof of his theorem and two proofs of the type sketched by Gauss of his generalization of Wilson's theorem; but obscured the proofs by lengthy numerical computations and the use of unconven- tional notations. F. Schuh^^^ proved Euler's theorem, the generalized Wilson theorem, and discussed the symmetric functions of the roots of a congruence for a prime modulus. "^Lehrsatz uber x"— x, Uaterrichts Blatter fiir Math. u. Naturwisa., Berlin, 18, 1912, 15. '"II Boll. Matematica Gior. Sc.-Didat., 11, 1912, 289. '"Zeitschrift Math.-naturwiss. Unterricht, 44, 1913, 263-4. ""Revista de la Sociedad Mat. Espafiola, 2, 1913, 123-131. "'Demonstrations nouvelles des th^orlmes de Fermat et de Wilson, Paris, Gauthier-Villars, 1913, 18 pp. Reprinted in Tinterm^diaire des math., 20, 1913, end. "^Suppl. de Vriend der Wiskunde, 25, 1913, 33-59, 143-159, 228-259. 84 History of the Theory of Numbers. [Chap, hi G. Frattini^^ noted that, if F{a, j3, . . . ) is a homogeneous symmetric poljTiomial, of degree g with integral coefficients, in the integers a, /3, . . . less than m and prime to m, and if F is prime to 7??, then k°= 1 (mod m) for ever>' integer A- prime to m. In fact, F(a, jS,. . .) = F{ka, k^,.. .) = k^F{a, 0,...) (mod m). Taking F to be the product a^. . ., we have Euler's theorem. Another corollary is u\l+j)=l + (p-l)l (modp), for p a prime, which implies Wilson's theorem. *J. L. Wildschlitz-Jessen^^^ gave an historical account of Fermat's and Wilson's theorems. E. Piccioh^" repeated the work of Dirichlet.'*° The Generalization F{a,N) = 0 (mod N) of Fermat's Theorem. C. F. Gauss^^° noted that, if N=pi^ . . .p/* (p's distinct primes), »=1 »■<;■ x<i<k is divisible by N when a is a prime, the quotient being the number of irre- ducible congruences modulo a of degree N and highest coefficient unity. He proved that (1) a^=2F(a, d), F{a,\)=a, where d ranges over all the divisors of N, and stated that this relation read- ily leads to the above expression for F (a, N). [See Ch. XIX on inversion.] Th. Schonemann^^^ gave the generalization that if a is a power p" of a prime, the number of congruences of degree A^ irreducible in the Galois field of order a is N~'^F{a, N). An account of the last two papers and later ones on irreducible con- gruences will be given in Ch. VIII. J. A. Serret^^^ stated that, for any integers a and iV, F{a, N) is divisible by N. For N=p\ p a prime, this implies that a</>(pO = l (modpO, when a is prime to p, a case of Euler's theorem. S. Kantor^^^ showed that the number of cycHc groups of order N in any birational transformation of order a in the plane is N~^F{a, N) . He obtained (1) and then the expression for F(a, N) by a lengthy method completed for special cases. iwPeriodico di Mat., 29, 1913, 49-53. i^Nyt Tidsskrift for Mat., 25, A, 1914, 1-24, 49-68 (Danish). i"Periodico di Mat., 32, 1917, 132-4. "oPosthumous paper, Werke, 2, 1863, 222; Gauss-Maser, 611. "iJour. fiir Math., 31, 1846, 269-325. Progr. Brandenburg, 1844. i"Nouv. Ann. Math., 14, 1855, 261-2. "'AnnaU di Mat., (2), 10, 1880, 64-73. Comptes Rendus Paris, 96, 1883, 1423. Chap. Ill] GENERALIZATIONS OF FeRMAT's THEOREM. 85 Ed. Wey^^^^ E. Lucas^^^ and Pellet^^^ gave direct proofs that F{a, N) is divisible by N for any integers a, N. H. Picquet^^^ noted the divisibility of F{Zm — l, N) by iV in an enumera- tion of certain curvilinear polygons of N sides, at the same time inscribed and circumscribed in a given cubic curve. He. gave a proof of the divisi- bility of F{a, N) by N, requiring various subcases. He stated that the function F{a, N) is characterized by the two relations (2) F{a, np") ^F{a''\ n) -F{a''"\ n), F^a, p')=a^'-a''"\ where a is any integer, n an integer not divisible by the prime p. A. Grandi^®^ proved that F{a, N) is divisible by N by writing it as ^N _ ^N/p, _ j ^^N/p, _ ^Nlp,p,-^ j^ ^^N/p, _ ^N/p,v,^ + . . . [ + \ {a^/p^P'-a^/PiP^p>) -f- ... J -I- ... . Each of these binomials is divisible by pi' since G. Koenigs^^* considered a uniform substitution z' =4>{z) and its nth power z" =4)n{z). Those roots of 2— 0^(2) =0 which satisfy no like equation of lower index are said to belong to the index n. If x belongs to the index n, so do also (^i{x) for i—\,..., n — 1. Thus the roots belonging to the index n are distributed into sets of n. If a is the degree of the polynomials in the numerator and denominator of 0(s), the number of roots belonging to the index n is F{a, n), which is therefore divisible by n. MacMahon's^^^ paper contains in a disguised form the fact that F(a, N) is divisible by N. Proofs were given by E. Maillet"^ by substitution groups, and by G. Cordone.^^^ Borel and Drach^'^" made use of Gauss' result that F{p, N) is divisible by N for every prime p and integer N, and Dirichlet's theorem that there exist an infinitude of primes p congruent modulo N to any given integer a prime to N, to conclude that F(a, N) is divisible by N. L. E. Dickson^^^ proved by induction (from k to k-{-l primes) that F{a, N) is characterized by properties (2) and concluded by induction that F{a, N) is divisible by N. A like conclusion was drawn from \F{a, N)\'-F{a, N)=Fia, qN) (mod q), where g is a prime. He gave the relations F{a, nN) = Fia"", n) - i F(a^/^-, n) + S Fia"^^"'"', n)-... + {-TyF{a^^'''-'", n)l F{a,N)=i:<l>id), "Casopis, Prag, 11, 1882, 39. ""Comptes Rendus Paris, 96, 1883, 1300-2. »«/6td., p. 1136, 1424. Jour, de I'^cole polyt., cah. 54, 1884, 61, 85-91. i"Atti R. Istituto Veneto di Sc, (6), 1, 1882-3, 809. i"Bull. des sciences math., (2), 8, 1884, 286. "•Rivista di Mat., Torino, 5, 1895, 25. ""Introd. th^orie dea nombres, 1895, 50. "^Annals of Math., (2), 1, 1899, 35. Abstr. in Comptes Rendus Paris, 128, 1899, 1083-6. 86 History of the Theory of Numbers. [Chap, hi where d ranges over those divisors of a^ — 1 which do not divide d° — l for 0<t;<iV; while, in the former, Pi,. . ., Pa are the distinct prime factors of N, and n is prime to N. L. Gegenbauer^^- wrote F(o, n) in the form S/i(c^)a"'''', where d ranges over the divisors of n, and nid) is the function discussed in Chapter XIX on Inversion. As there shown, 2ju(d) =0 if n> 1. This case fix) =ijl{x) is used to prove the generaUzation : If the function /(x) has the property that 2/((i) is divisible by n, then for every integer a the function l!>f{d)a'''^ is divisible by n, where in each sum d ranges over the divisors of n. Another special case, f{x) =4>{x), was noted by MacMahon.^^^ J. Westlund^^^ considered any ideal Am. o. given algebraic number field, the distinct prime factors Pi, . . . , P^ of ^, the norm n(^) of ^, and proved that if a is any algebraic integer, ^nU) _^gn{A)ln{Pi) _|_2^n(^)/n(PiPj) _ 4- ( — n'^'>(^)/«(Pi. . -Pi) is always divisible by A. J. Vdlyi^^^ noted that the number of triangles similar to their nth pedal but not to the dih. pedal {d<n) is Xin) =^P{n) -^^(^) +2'A(-^) - ■ ■ ., Vp/ ^PlP2^ if Pi> P2, • • • are the distinct prime factors of n, and yp{k)=2^{2^ — \). He proved that x(^) is divisible by n, since if the nth pedal to ABC is the first one similar to ABC, a like property is true of the first pedal, . . ., (n — l)th pedal, so that the x(^) triangles fall into sets of n each of period n. [Note that x(n)=P(4, n)-F(2, n).] A. Axer^^^ proved the following generalization of Gegenbauer's"^ theorem: If G(ri, . . . , r/i) is any polynomial with integral coefficients, and if, when d ranges over all the divisors of n, 2/(rf)G(ri"^. . ., rC'^)=Q (mod n) for a particular function G = Gq and a particular set of values Txq, . . . , r/,o, not a set of solutions of Gq, and for which Go is prime to n, then it holds for every G and every set ri, . . . , r^. Further Generalizations of Fermat's Theorem. For the generalization to Galois imaginaries, see Ch. VIII. For the generalization by Lucas, see Ch. XVII, Lucas,^^ Carmichael.^' On :x^= 1 (mod n) for x prime to n, see Cauchy,^^ Moreau,^^ Epstein,"' of Ch. VII. 0. H. MitchelP'^ considered the 2* products s of distinct primes dividing k = pi...pf and denoted by r/A;) the number of positive integers X,<k which are divisible by s but by no prime factor of k not dividing s. i^Monatshefte Math. Phys., 11, 1900, 287-8. i"Proc. Indiana Ac. Sc, 1902, 78-79. "«Monatshefte Math. Phys., 14, 1903, 243-2.53. »"MonatBhefte Math. Phys., 22, 1911, 187-194. "»Amer. Jour. Math., 3, 1880, 300; Johns Hopkins Univ. Circular, 1, 1880-1, 67, 97. Chap. Ill] GENERALIZATIONS OF FeRMAt's THEOREM. 87 The products of the various X^ by any one of them are congruent modulo k to the Xg in some order. Hence X/^w=ie^ (modifc), where R^ is the corresponding one of the 2' roots of x^^x (mod k). The analogous extension of Wilson's theorem is HXs^^Rs (mod k), the sign being minus only when k/a = p'', 2p' or 4 and at the same time a/s is odd. Here <r = np/^ if s = np,. Cf. Mitchell,^" Ch. V. F. RogeP^^ proved that, if p is a prime not dividing n, n-i = l+(f)(7i-l) + (|)(n-l)2+... + (f)(n-l)Hp, A: = ^, where p is divisible by every prime lying between k and p+l. Borel and Drach^^° investigated the most general polynominal in x divis- ible by m for all integral values of x, but not having all its coefficients divisible by m. If m = p''q^, . . . , where p, q,. . .are distinct primes, and if P{x), Q{x),. . . are the most general polynomials divisible by p", q^,. . ., respectively, that for m is evidently {P{x)+p'^f{x)\\Q{x)+q'g{x)\.... For a<p+l, the most general P{x) is proved to be iMx)Mx), Mx) =p''-\x^-x)\ where the/'s are arbitrary polynomials. For a<2(p+l), the most general P{x) is s/,<A,+ ste, rp,=ct>{x){x^-xy-Y-''-\ k=l k=l where 4>{x) = {x^—xy—p^~^{x^—x), and the/'s, g's are arbitrary poly- nomials. Note that ^^(x) -p'''-'^<f>lx) is divisible by p^'+^+K Cf. Nielsen.^^^ E. H. Moore^^^ proved the generalization of Fermat's theorem: Xi''"*-^ X p*""^ XiP Xi m p— 1 p— 1 = n H . . . H {Xk+Ck+iXk+i+ . . . +c^xj (mod p). F. Gruber^^^ showed that, if n is composite and ai, . . . , a< are the ^=0(n) integers < n and prime to n, the congruence (1) x' — l = (x— fli). . .(x— a«) (mod n) is an identity in x if and only if n = 4 or 2p, where p is a prime 2*+l. "•Archiv Math. Phys., (2), 10, 1891, 84-94 (210). ""Introduction th^orie des nombres, 1895, 339-342. ">BuU. Amer. Math. Soc, 2, 1896, 189; cf. 13, 1906-7, 280. "»Math. Nat. Berichte aus Ungarn, 13, 1896, 413-7; Math, term^s ertesito, 14, 1896, 22-25. 88 History of the Theory of Numbers. [Chap. Ill E. Malo^^ employed integers -4/ and set u = afz, Since f^ d='Eu''/k {k = n, m+n, 2m-\-n, e= w l-u" =ScOpX''-Ma:. 2^x^=2^^'^' P k p k p-ltkt where k takes the values n, w+n, . . .which are ^p/fi. If no prime factor of such a k occurs in the denominator of the expansion of cop/p, the latter is an integer; this is the case if p is a prime and /x= 2. For w = n = l, ix = 2, ^(»-)-(2)-(3)- + ■ax a-3. .0-2 we get o3p = a^—a and hence Fermat's theorem. L. Kronecker^^ generalized Fermat's and Wilson's theorems to modular systems. R. Le Vavasseur^^ obtained a result evidently equivalent to that by Moore^^^ for the non-homogeneous case Xm = l' M. Bauer^^^ proved that if n = p'm, where m is not divisible by the odd prime p, and Oi, . . ., a< are the t=<l>{n) integers <n and prime to n, {x-ai) . . .(a:-a,) = (xP-i-l)'/<P-'Hmod p'), identically in a:. If p = 2 and 7r> 1, the product is identically congruent to {x'^ — iy^^. Hence he found the values of d, n for which (1) holds modulo d, when d is a divisor of n. If p denotes an odd prime and q a prime 2*+ 1, the values are d 2q 4 P 2 n 2q 4 p-, 2p» 2^2"5lg2... M. Bauer^*'' determined how n and N must be chosen so that x" — 1 shall be congruent modulo A^ to a product of linear functions. We may restrict N to the case of a power of a prime. If p is an odd prime, a;" — ! is congruent modulo p° to a product of linear functions only when p=l (mod n), a arbitrary, or when n = p'm, a = l, p = l (mod m). For p = 2, only when n = 2^, a = l, or n = 2, a arbitrary. For the case n a prime, the problem was treated otherwise by Perott.^^^ M. Bauer^^^ noted that, if n = 'p'm, where m is not divisible by the odd prime p, n(a:-i) = (xP-x)"/P (mod p'). t=i >8»L'interm6diaire des math., 7, 1900, 281, 312. ^"Vorlesungen uber Zahlentheorie, I, 1901, 167, 192, 220-2. '"Comptes Rendus Paris, 135, 1902, 949; Mdm. Ac. Sc. Toulouse, (10), 3, 1903, 39-48. •"Nouv. Ann. Math., (4), 2, 1902, 256-264. "'Math. Nat. Berichte aus Ungarn, 20, 1902, 34-38; Math. 6s Phys. Lapok, 10, 1901, 274-8 (pp. 145-152 relate to the "theory of Fermat's congruence"; no report is available). "8Amer. Jour. Math., 11, 1888; 13, 1891. '"Math. 68 Phys. Lapok, 12, 1903, 159-160. Chap. Ill] GENERALIZATIONS OF FeRMAT's ThEOREM. 89 Richard Sauer^^'^ proved that, \i a,h, a — h are prime to k, a^+a^-^6+a^-V+ • • • +?)*'=1 (mod k), <P = (p{k), since a*""^^— 6*"^^=a— 6. Changing alternate signs to minus, we have a congruence valid if a, 6 are prime to k, and if 0+6 is not divisible by k. If p is an odd prime dividing a=F6, is divisible by p, but not by p^. A. Capelli^^^ showed that, if a, 6 are relatively prime. ah =[V]+[V]+i' where [x] is the greatest integer ^ x. M. Bauer^^^ proved that, if p is an odd prime and m = p" or 2p^, every integer x relatively prime to m satisfies the congruence (a;p-i _ l)p"- = (x+A^i) . . . {x-\-ki) (mod m), where /bi, . . ., ki denote the l=<l){m) integers <m and prime to m>2. If m is not 4, p" or 2p", every integer a: prime to m satisfies the congruence (x^^-^y^-iy^(x+ki) . . . (x+ki) (mod m). L. E. Dickson^^^ proved Moore's^^^ theorem by invariantive theory. N. Nielsen^^^ proved that, if ^{x) is a polynomial with integral coeffi- cients not having a common factor > 1, and if for every integral value of x the value of ^{x) is divisible by the positive integer m, then p-i ^{x) = (f>{x) o)p{x)+ S rrip-s A, cos(x), o)n{x)=x{x-\-l) . .{x+n-1), 8= 1 where 4>(x) is a polynomial with integral coefficients, the Ag are integers, p is the least positive integer for which p ! is divisible by m, and mp_s is the least positive integer I for which s\l is divisible by m. Cf. Borel and Drach.i8° H. S. Vandiver^^^ proved that, if V ranges over a complete set of incon- gruent residues modulo m = pi . . .pl^, while U ranges over those F's which are prime to m, A; ll{x-V)^^tXx''^-xT"'% n(a;-C7)=Si,(a;P«-'-l)*'('")/(p»-^>, modulo w, where t^ = {mlpg^^y, e = </)(p/») . For w = p", the second congruence is due to Bauer.^^^' ^^^ ""Eine polynomische Verallgemeinerung des Fermatschen Satzes, Diss., Giessen, 1905. »"Dritter Internat. Math. Kongress, Leipzig, 1905, 148-150. »»Archiv Math. Phys., (3), 17, 1910, 252-3. Cf. Bouniakowsky^s of Ch. XI. "'Trans. Amer. Math. Soc, 12, 1911, 76; Madison Colloquium of the Amer. Math. Soc, 1914, 39-40. "<Nieuw Archief voor Wiskunde, (2), 10, 1913, 100-6. i»Annals of Math., (2), 18, 1917, 119. 90 History of the Theory of Numbers. [Chap, hi Further Generalizations of Wilson's Theorem; Related Problems. J. Steiner-°° proved that, if Ak is the sum of all products of powers of Gi, 02,..., Op-it of degree k, and the o's have incongruent residues p^O modulo p, a prime, then A^,. . ., Ap_2 are divisible by p. He first showed by induction that z =2i.p^i-\-AiA.p^2~T' ■ ■ ■ \Ap-2-^i\Ap^i, Xk={x-ai). . .{x-ttk), ^i = Oi+. . .+ap_i, ^2 = 01^ + 0102+ . . . +0iOp_2 + O2^+O2O3+ . . . +oJ_2, For example, to obtain x^ he multipUed the respective tenns of V=(X — Oi)(x — 02) + (01+02) (x — 0i)+0i^ by X, (a:-03)+a3, (x— 02)+02, (x — oO+Oi. Let Oi,..., Op_i have the residues 1,. . ., p — l in some order, modulo p. For x — 02 divisible by p, x^~^=Ap_i = a{~^ (mod p), so that Ap_2Xi and hence also Ap_2 is divisible by p. Then for x=a3, ^p_3X2 and ^p_3 are divisible by p. For x = 0, ai = l, the initial equation yields Wilson's theorem. C. G. J. Jacobi"''^ proved the generahzation : If Oi,. . ., a„ have distinct residues f^ 0, modulo p, a prime, and Pr^m is the sum of their multipUcative combinations with repetitions m at a time, Pnm is divisible by p for w = p— n, p-n+1,..., p-2. Note that Steiner's Ah is Pp-k.k- We have ][ J^ I Pfil ■ Pn2 (x-oi) . . . (x-o„) "x""^x"+i^x"+2"^ • • •' ' """,=; (1) 7:r^7v^^^^.=i+9h+^2+--: P..= :^a;^-'/Dj, Dj = {cLj - Oi) . . . (Oj - a,_i) (oy - Oy+i) . . . (o, - 0 J , 0=2 Oy /D, {k<n-l). j=i Let n+m-l = k+^{p-\). Then o^+^-^^o/ (mod p). Hence if A;<Cn — 1 ' Di . . .Dr,Pn,„=D, . . .Dj:a)/D„ P^^^O (mod p). The theorem follows by taking /3 = 1 and ^' = 0, 1, . . ., n— 2 in turn. H. F. Scherk^°- gave two generaUzations of Wilson's theorem. Let p be a prime. By use of Wilson's theorem it is easily proved that n! where x is an integer such that px=l (mod n!). Next, let C/ denote the sum of the products of 1, 2, . . . , ^ taken r at a time with repetitions. By use of partial fractions it is proved that (p-r-l)!C;_,_i+(-l)'-=0(modp) (r<p-l). It is stated that ""Jour, fiir Math., 13, 1834, 356; Werke 2, p. 9. ""/bid., 14, 1835, 64-5; Werke 6, 252-3. '"Bericht iiber die 24. Versammlung Deutscher Naturforscher und Aerzte in 1846, Kiel, 1847, 204-208. (p-n-l)!^(-ir^^(modp), Chap. Ill] Genekalizations OP Wilson's Theorem. 91 C;.r-iCr-' + {-iy^O, C:,-m\^0 (modp), ^ = ^- H. F. Scherk^''^ proved Jacobi's theorem and the following: Form the sum Pnh of the multiplicative combinations with repetitions of the hth class of any n numbers less than the prime p, and the sum of the combinations without repetitions out of the remaining p—n — 1 numbers <p; then the sum or the difference of the two is divisible by p according as h is odd or even. Let Cl denote the sum of the combinations with repetitions of the hth class oi 1, 2,. . ., k; Al the sum without repetitions. If 0<h<p — l, Ci^O (mod p), j = p-k,.. ., p-2; Cl+,^Cl For h = p-l, Ci;ik=n+1 for k = l,..., p. For h = m{p-l)+t, Cl=Ci when k<p+l. For l<h<k, the sum of Cl and A^ is divisible by A;^(fc+1)^; likewise, each C and A if /i is odd. For h<2k, Cl—Al is divisible by 2A:H-1. The sum of the 2nth powers of 1, . . . , /c is divisible by 2k-\-l. K. HenseP'^ has given the further generalization: If ai, . . . , a,^, 61, . . . , 6, are n-\-v = p — l integers congruent modulo p to 1, 2, . . . , p — 1 in some order, and ^l/{x) = (x-b,)... (rr-6J =x'-B,x'-'+ . . .=^B„ then, for any j, Pnj^i — iy'Bj^ (mod p), where jo is the least residue of j mod p — 1 and Bh = 0 {k>v). For Steiner's Z„, Z„^(a;)=a:^-i-l (mod p). Multiply (1) by a:"(x^-^-l). Thus X''rP{x)^X'-'+PnlX^-'+ . . .+Pnp-2X + Pnp-l-l + ^"^~^"^ X +^"^^^"^"'+... (modp). X Replace \f/{x) by its initial expression and compare coefficients. Hence p^j^i-iyBj{j=i,...,v). Taking v=j = p — 2 and choosing 2,..., p — 1 for 61,..., 6„ we get 1= — (p — 1)! (mod p). Converse of Fermat's Theorem. In a Chinese manuscript dating from the time of Confucius it is stated erroneously that 2""^ — 1 is not divisible by n if n is not prime (Jeans^^^). Leibniz in September 1680 and December 1681 (Mahnke,^ 49-51) stated incorrectly that 2'*— 2 is not divisible by n if n is not a prime. If n = rs, where r is the least prime factor of n, the binomial coefficient (") was shown to be not divisible by n, since n — 1,..., n— r+1 are not divisible by r, whence not all the separate terms in the expansion of (1 + 1)" — 2 are ^o^Ueber die Theilbarkeit der Combinationssummen aus den natiirlichen Zahlen durch Prim- zahlen, Progr., Bremen, 1864, 20 pp. ^"Archiv Math. Phys., (3), 1, 1901, 319; Kronecker's Zahlentheorie 1, 1901, 503. 92 History of the Theory of Numbers. [Chap, hi divisible by n. From this fact Leibniz concluded erroneously that the expression itself is not divisible by n. Chr. Goldbach^^° stated that {a-\-by — a^ — b^ is divisible by p also when p is any composite number. Euler (p. 124) points out the error by noting that 2^^ — 2 is divisible by neither 5 nor 7. In 1769 J. H. Lambert^'* (p. 112) proved that, if ^"•-l is divisible by a, and d" — 1 by 6, where a, b are relatively prime, then d' — l is divisible by ab if c is the 1. c. m. of m, n (since divisible by d"* — 1 and hence by a) . This was used to prove that if g is odd [and prime to 5] and if the decimal fraction for l/g has a period oi g — 1 terms, then ^ is a prime. For, if ^ = a6 [where a, b are relatively prime integers > 1], 1/a has a period of m terms, m^a — \, and 1/6 a period of n terms, n^b — l, so that the number of terms in the period for 1/^ is ^ (a — 1)(6 — 1)/2<(7 — 1. Thus Lambert knew at least the case /c = 10 of the converse of Fermat's theorem (Lucas^"' ^^'). An anonymous writer ^^^ stated that 2n+l is or is not a prime according as one of the numbers 2"=*= 1 is or is not divisible by n. F. Sarrus^^^ noted the falsity of this assertion since 2^'^° — 1 is divisible by the composite num- ber 341. In 1830 an anonymous writer^ noted that a"~^ — 1 may be divisible by n when n is composite. In a^~^ = /cp+1, where p is a prime, set k = \q. Then ^(P-i)«=l (jjjojj pqy Ti^us oP«-i = l if a«-i = l (mod pq), and the last will hold if g — 1 is a multiple of p — 1 ; for example, ifp = ll,g' = 31,a = 2, whence 2340=1 (mod 341). V. Bouniakowsky^^3 proved that if A^ is a product of two primes and if iV— 1 is divisible by the least positive integer a for which 2"=1, whence 2^~^=1 (mod N), then each of the two primes decreased by unity is divisible by a. He noted that 3^=1 (mod 91 = 7-13). E. Lucas^^^ noted that 2''~^=1 (mod n) for 71 = 37-73 and stated the true converse to Fermat's theorem: If a"" — ! is divisible by p for x = p — l, but not for x<p — lf then p is a prime. F. Proth^^* stated that, when a is prime to n, n is a prime if a*= 1 (mod n) for a:= (n — 1)/2, but for no other divisor of (n — 1)/2; also, if a''=l (mod n) for x = n — l, but for no divisor <\/n of n — 1. If 71 = 7^-2*^+1, where m is odd and < 2*", and if a is a quadratic non-residue of n, then n is a prime if and only if a^"~^^/^= — 1 (mod n). If p is a prime >^\/n, n = mp+l is a prime if a"~^ — 1 is divisible by n, but a^'^l is not. *F. Thaarup^^' showed how to use a"~^=l (mod ti) to tell if n is prime. E. Lucas^^^ proved the converse of Fermat's theorem: If a^=l (mod ti) for a: = 71 — 1, but not for x a proper divisor oi n — 1, then n is a prime. ""Corresp. Math. Phys. (ed. Fuss), I, 1843, 122, letter to Euler, Apr. 12, 1742. "'Annales de Math. (ed. Gergonne), 9, 1818-9, 320. ^"Ibid., 10, 1819-20, 184-7. "»M6m. Ac. Sc. St. P6tersbourg (math.), (6), 2, 1841 (1839), 447-69; extract in Bulletin, 6, 97-8. "*Assoc. frang. avanc. sc, 5, 1876, 61; 6, 1877, 161-2; Amer. Jour. Math., 1, 1878, 302. "'Comptes Rendus Paris, 87, 1878, 926. "•Nyt Tidsskr. for Mat., 2A, 1891, 49-52. *"Th6orie des nombres, 1891, 423, 441. Chap. Ill] CONVERSE OF FeRMAt's THEOREM. 93 G. Levi^^* was of the erroneous opinion that P is prime or composite according as it is or is not a divisor of 10^"^ — 1 [criticized by Cipolla,^^^ p. 142]. K. Zsigmondy^^^ noted that, if g is a prime =1 or 3 (mod 4), then 2q+l is a prime if and only if it divides (2*+l)/3 or 2^ — 1, respectively; 4g+l is a prime if and only if it divides (2^*+l)/5. E. B. Escott^^^ noted that Lucas'^^^ condition is sufficient but not necessary. J, H. Jeans^^® noted that if p, q are distinct primes such that 2^=2 (mod g), 2^=2 (mod p), then 2^^ =2 (mod pq), and found this to be the case for pg = 11-31, 19-73, 17-257, 31-151, 31-331. He ascribed to Kossett the result 2"-^=l (mod n) for n = 645. A. Korselt^^^ noted this case 645 and stated that a^=a (mod p) if and only if p has no square factor and p — 1 is divisible by the 1. c. m. of pi — 1, . . . , p„ — 1, where pi, . . ., p„ are the prime factors of p. J. FraneP^^ noted that 2^^ =2 (mod pq), where p, q are distinct primes, requires that p — 1 and g — 1 be divisible by the least integer a for which 2''=1 (mod pq). [Cf. Bouniakowsky.^^^] L. Gegenbauer222« noted that 2^''-^=l (mod pq) if p = 2'-l = /cpr+l and q = KT-\-l are primes, as for p = Sl, q = ll. T. Hayashi^^^ noted that 2''— 2 is divisible by n = 11-31. If odd primes p and q can be found such that 2^=2, 2^=2 (mod pq), then 2^'— 2 is divisible by pq. This is the case if p — 1 and q — 1 have a common factor p' for which 2"''=! (mod pq), as for p = 23, g = 89, p' = ll. Ph. Jolivald224 asked whether 2^-^=1 (mod N) if N = 2''-l and p is a prime, noting that this is true if p = ll, whence iV = 2047, not a prime. E. Malo^^^ proved this as follows: AT-l =2(2^-1-1) =2pw, 2^-^ = (2^)2- = (Ar+l)2-=i (modiV). G. Ricalde^^^ noted that a similar proof gives a^~''+^=l (mod N) if N = a^—1, and a is not divisible by the prime p. H. S. Vandiver^^^ proved the conditions of J. FraneP^^ and noted that they are not satisfied if a< 10. Solutions for a = 10 and a = 11 are ^3 = 11-31 and 23-89, respectively. H. Schapira^^^ noted that the test for the primality of N that a^=l »8Monat8hefte Math. Phys., 4, 1893, 79. *i«L'mtenn4diaire des math., 4, 1897, 270. 220Messenger Math., 27, 1897-8, 174. ''"L'interm^diaire des math., 6, 1899, 143. ^Ibid., p. 142. =«""Monatshefte Math. Phys., 10, 1899, 373. '^'Jour. of the Physics School in Tokio, 9, 1900, 143-4. Reprinted in Abhand. Geschichte Math. Wiss., 28, 1910, 25-26. *"L'interm4diaire des math., 9, 1902, 258. ^lUd., 10, 1903, 88. *^Ibid., p. 186. *"Amer. Math. Monthly, 9, 1902, 34-36. »»Tchebychef's Theorie der Congruenzen, ed. 2, 1902, 306. 94 History of the Theory of Numbers. [Chap. Ill (mod N) ioT q = N—l, but for no smaller q, is practical only if it be known that a small number a is a primitive root of N. G. Arnoux--^" gave numerical instances of the converse of Fermat's theorem. M. Cipolla^^^ stated that the theorem of Lucas^^' impUes that, if p is a prime and A: = 2, 4, 6, or 10, then kp-\-l is a prime if and only if 2^'^=1 (mod kp-\-l). He treated at length the problem to find a for which a^~^ = 1 (mod P), given a composite P; and the problem to find P, given a. In particular, we may take P to be any odd factor of (a^" — l)/(a^ — 1) if p is an odd prime not dividing a^ — 1. Again, 2^~^= 1 (mod P) for P = F^F^ ■ . . F„ m>n> . . .>s, if and only if 2'>m, where P, = 2^''4-l is a prime. If p and q = 2p — \ are primes and a is any quadratic residue of q, then a^«~^ = 1 (mod pq) ; we may take a = 3 if p = 4n+3 ; a = 2\i p = 4n+ 1 ; both a = 2 and a = 3if p = 12A:+l;etc. E. B. Escott^^° noted that e"~^ = l (mod n) if e'' — 1 contains two or more primes whose product n is =1 (mod a), and gave a list of 54 such n's. A. Cunningham^^^ noted the solutions n = FsF^F^QF7, n = F^. . .P15, etc. [cf. CipoUa], and stated that there exist solutions in which n has more than 12 prime factors. One with 12 factors is here given by Escott. T. Banachiewicz^^^ verified that 2^—2 is divisible by N for N composite and < 2000 only when N is 341 = 11-31, 561=3-1M7, 1387 = 19-73, 1729 = 7-13-19, 1905 = 3-5-127. Since 2^—2 is evidently divisible by N for every N = Fk = 2^ +1, perhaps Fermat was thus led to his false conjecture that every Fk is a prime. R. D. CarmichaeP^^ proved that there are composite values of n (a product of three or more distinct odd primes) for which e"~^=l (mod n) holds for every e prime to n. J. C. Morehead^^ and A. E. Western proved the converse of Fermat's theorem. D. Mahnke^ (pp. 51-2) discussed Leibniz' converse of Fermat's theorem in the form that n is a prime if a:"~^=l (mod n) for all integers x prime to n and noted that this is false when n is the square or higher power of a prime or the product of two distinct primes, but is true for certain products of three or more primes, as 3-11-17, 5-13-17, 5-17-29, 5-29-73, 7-13-19. R. D. CarmichaeP^^ used the result of Lucas^^" to prove that a^~^ = l (mod P) holds for every a prime to P if and only if P — 1 is divisible by X(P). The latter condition requires that, if P is composite, it be a product of three or more distinct odd primes. There are found 14 products P of «8« Assoc, frang., 32, 1903, II, 113-4. "•Annali di Mat., (3), 9, 1903-4, 138-160. ""Messenger Math., 36, 1907, 175-6; French transl., Sphinx-Oedipe, 1907-8, 146-8. "'Math. Quest. Educat. Times, (2), 1^, 1908, 22-23; 6, 1904, 26-7,55-6. »«Spraw. Tow. Nauk, Warsaw, 2, 1909, 7-10. "'Bull. Amer. Math. Soc, 16, 1909-10, 237-8. ^Ibid., p. 2. "»Amer. Math. Monthly, 19, 1912, 22-7. Chap. Ill] SYMMETRIC FUNCTIONS MoDULO p. 95 three primes, as well as P = 13-37-73457, for each of which the congruence holds for every a prime to P. Welsch^se stated that ifk = 4n-\-l is composite and < 1000, 2^-^ = 1 (mod k) only for A; = 561 and 645; hence 71**= 1 (mod k) for these two k's. P. Bachmann^^^ proved that x^'^~^'= 1 (mod pq) is never satisfied by all integers prime to pq if p and q are distinct odd primes [Carmichael^^^]. Symmetric Functions of 1, 2,. . .p— 1 Modulo p. Report has been made above of the work on this topic by Lagrange, ^^ Lionnet," Tchebychef,^^ Sylvester,'^^ ottmger,^^ Lucas,!"^ Cahen,!^^ Aubry,^" Arevalo,^^^ Schuh,^^^ Frattini,!^^ steiner,^^^ Jacobi,^^! Hensel.^o^ We shall denote l"+2"+ . . . +(p — I)'* by s^, and take p to be a prime. E. Waring^^° wrote a, j3, . . .for 1, 2, . . . , x, and considered s = a^^^y' . . . ^a^^^y' . . . +a"/3^7'' .... If < = a+6+c+ . . .is odd and <a:, andx+1 is prime, s is divisible by (x+l)^. If t<2x and a, 6, . . .are all even and prime to 2a; +1, s is divisible by 2a;-f 1. V. Bouniakowsky^^^ noted that s^ is divisible by p^, if p> 2 and m is odd and not =1 (mod p — 1); also if both w=l (mod p — 1) and m=0 (mod p). C. von Staudt252 proved that, if S,Xx) = 1+2"+ • • • +a;% S^{ah)=bSM+naSn-iia)Si{b-l) (mod a"), 2S2n+i{a)^{2n+l)aS2M (mod a^). If a,h,. . ., I are relatively prime in pairs, S,Xah...l) SM S^il) ah. . .1 a ' ' ' I = integer. A. Cauchy253 proved that 1 + 1/2+ . . . +l/(p-l)=0 (mod p). G. Eisenstein^^^ noted that s^= — 1 or 0 (mod p) according as m is or is not divisible by p — 1. If w, n are positive integers <p — 1, 'iyia+iy^O or - (p_f_ J' (mod p), according as m+n<or^p — 1. L. Poinsot^^^ noted that, when a takes the values 1, . . . , p — 1, then (ax)" has the same residues modulo p as a", order apart. By addition, SnX'*=Sn (mod p). Take x to be one of the numbers not a root of a;"=l. Hence s„=0 (mod p) if n is not divisible by p — 1. ^''L'mtermMiaire des math., 20, 1913, 94. *»'Archiv Math. Phys., (3), 21, 1913, 185-7. ""Meditationes algebraicae, ed. 3, 1782, 382. ">BuIl. Ac. Sc. St. P^tersbourg, 4, 1838, 65-9. "'Jour, fiir Math., 21, 1840, 372-4. "'M^m. Ac. Sc. de I'lnstitut de France, 17, 1840, 340-1, footnote; Oeuvres, (1), 3, 81-2. «"Jour. fiir Math., 27, 1844, 292-3; 28, 1844, 232. »"Jour. de Math., 10, 1845, 33-4. 96 History of the Theory of Numbers. [Chap, hi J. A. Serret^^^ concluded by applying Newton's identities to (x— 1) . . . (x— p+l)=0 that Sn=0 (mod p) unless n is divisible by p — 1. J. Wolstenholme^" proved that the numerators of 1+UU...+ 1 1+1 + . ■ 1 2 ' 3 ' •■• ' p-r - . 22 ' ••• ' (p-l)2 are divisible by p^ and p respectively, if p is a prime >3. Proofs have also been given by C. Leudesdorf^^s, A. Rieke,^^^ E. Allardice,^^" G. Osborn,^" L. Birkenmajer,^" P. Niewenglowski,^^ N. Nielsen,^" H. Valentiner,^®^ and others.^^^ V. A. Lebesgue^®' proved that s^ is divisible by p if w is not divisible by p — 1 by use of the identities (n+1) S k{k+l) . . .ik-\-n-l)=xix+l) . . .(x+n) (n = l,. . ., p-1). k=l P. Frost^^^ proved that, if p is a prime not dividing 2^'' — 1, the numera- tors of a2r, (T2r-i, p(2r — l)o-2r+2(T2r-i are divisible by p, p^, p^, respectively, where 1J_1_L 1 2* ' •" ' (p-1)' The numerator of the sum of the first half of the terms of 0*2, is divisible by p; likewise that of the sum of the odd terms. J. J. Sylvester^^^ stated that the sum S^, m of all products of n distinct numbers chosen from 1,. . ., m is the coefficient of T in the expansion of {l+t){l-\-2t) . . . (1+wO and is divisible by each prime >n+l contained in any term of the set m— n+l,. . ., m, m+1. E. Fergola"" stated that, if (a, 6, . . . , ly represents the expression obtained from the expansion of (a+6H- . . . +0" by replacing each numerical coefficient by unity, then (X, x+1,. . ., x+rr= i CY)^^' 2,. . ., rr-^x^. *"Coiirs d'algSbre sup^rieure, ed. 2, 1854, 324. «'Quar. Jour. Math., 5, 1862, 35-39. "sProc. London Math. Soc, 20, 1889, 207. «»Zeit8chrift Math. Phys., 34, 1889, 190-1. ««oProc. Edinburgh Math. Soc, 8, 1890, 16-19. "'Messenger Math., 22, 1892-3, 51-2; 23, 1893-4, 58. "^Prace Mat. Fiz., Warsaw, 7, 1896, 12-14 (Polish). M'Nouv. Ann. Math., (4), 5, 1905, 103. ««Nyt Tidsskrift for Mat., 21, B, 1909-10, 8-10. ^Ibid., p. 36-7. ^^Math. Quest, Educat. Times, 48, 1888, 115; (2), 22, 1912, 99; Amer. Math. Monthly, 22, 1915, 103, 138, 170. «^Introd. k la thdorie des nombres, 1862, 79-80, 17. »8Quar. Jour. Math., 7, 1866, 370-2. M»Giomale di Mat., 4, 1866, 344. Proof by Sharp, Math. Ques. Educ. Times, 47, 1887, 145-6; 63, 1895, 38. "oibid., 318-9. Cf. Wronski"! of Ch. VIII. Chap. Ill] Symmetric Functions Modulo p. 97 The number (1, 2, . . ., r)'* is divisible by every prime >r which occurs in the series n+2, n+3, . . ., n+r. G. ToreUi"! proved that (ai, . . . , GnY = (ai, . . . , a„_iy+a„{ai, . . . , a^y~\ (fli,. . ., o„, ^y-Cfli,. . ., fln, c)'"=(6-c)(ai,. . ., a„, 6, cy-\ which becomes Fergola's for ai = i (i = 0, . . ., n). Proof is given of Syl- vester's^^^ theorem and the generahzation that >Sy,i is divisible by (}+!). Torelli^^^ proved that the sum o-„, « of all products of n equal or distinct numbers chosen from 1, 2, . . ., m is divisible by (n+T), and gave recursion formulas for o-n, m- C. Sardi^'^^ deduced Sylvester's theorem from the equations Ai — (|),, . . used by Lagrange. ^^ Solving them for Ap = Sp,n, we get pi{-iy^%,,= -1 G) (a) C) 0 -2 i;--i) 0 V 2 ; 0 \ 3 ; 0 ("to ( n-p+2\ /n+1 2 y Vp+^ D If n+l is a prime we see by the last column that >S„_i.„ is divisible by n+1. When p = n — l, denote the determinant by D. Then if n+l is a prime, D is evidently divisible by n+l. Conversely, if D is divisible by n+l and the quotient by (n — 1) !, then n+l is a prime. It is shown that p=l r„ = F+...+n^ Using this for m = 1, . . . , n, we see that Vp is divisible by any integer prime to 2, 3, . . ., p+1 which occurs in n+l or n. Hence if n+l is a prime, it divides ri, . . ., r„_i, while rn=n (mod n+l). If n+l divides r„_i it is a prime. Sardi"^ proved Sylvester's theorem and the formula S ( — l)''>Si,, n+r-lO'k-r, n+r = 0, r-0 stated by Fergola."^ "iQiornale di Mat., 5, 1867, 110-120. "276id., 250-3. "mid., 371-6. "*Ibid., 169-174. "Ubid., 4, 1866, 380. 98 History of the Theory of Numbers. [Chap, hi Sylvester"^ stated that, if pi, p2, . . .are the successive primes 2, 3, 5, . . ., ^'^^ ^T^^A^ ^'-'^''^' where Fk{n) is a polynomial of degree k with integral coefficients, and the exponent e of the prime p is given by E. Ces^ro^" stated Sylvester's^®^ theorem and remarked that <S„.„— n! is divisible by w —n if m—n is a prime. E. Ces^ro"^ stated that the prime p divides /S^.p-2 — 1, *Sp_i.p+l, and, except when m = p — l, S^.p-i- Also (p. 401), each prime p>{n+l)/2 divides *Sp_i.„H-l, while a prime p = (n+l)/2 or n/2 divides »Sp_i.„+2. O. H. Mitchell"^ discussed the residues modulo k (any integer) of the symmetric functions of 0, 1, . . . , /c — 1. To this end he evaluated the residue of (x— a)(x— /3) . . ., where a, /3, . . .are the s-totitives of k (numbers<A: which contain s but no prime factor of k not found in s) . The results are extended to the case of moduli p, f{x), where p is a prime [see Ch. VIII]. F. J. E. Lionnet^^° stated and Moret-Blanc proved that, if p = 2n+l is a prime> 3, the sum of the powers with exponent 2a (between zero and 2n) of 1, 2, . . . , n, and the like sum for n-\-l, n+2, . . . , 2n, are divisible by p. M. d'Ocagne^^^ proved the first relation of Torelli.^^^ E. Catalan^^^ stated and later proved^^^ that s^ is divisible by the prime p>k-\-l. If p is an odd prime and p — 1 does not divide k, Sk is divisible by p; while if p — 1 divides k, Sk= — l (mod p). Let p = a''b^ . . . ; if no one of a — 1, 6 — 1,. . . divides k, Sk is divisible by p; in the contrary case, not divisible. If p is a prime >2, and p — 1 is not a divisor of k-\-l, then ^ = l^(p-l)'+2*(p-2)^+ . . .+{p-iyv is divisible by p; but, if p — 1 divides k+l, S= — { — iy (mod p). If k and I are of contrary parity, p divides S. M. d'Ocagne^^ proved for Fergola's^'^" symbol the relation (a. . .fg. . .1. . .V. . .zr^Xia. . .frig. . .^)^ ..{v. . .z)", summed for all combinations such that X+/i+. . .-\-p = n. Denoting by a^^^ the letter a taken p times, we have i=0 "«Nouv. Ann. Math., (2), 6, 1867, 48. »"Nouv. Correap. Math., 4, 1878, 401; Nouv. Ann. Math., (3), 2, 1883, 240. "8Nouv. Coiresp. Math., 4, 1878, 368. "»Amer. Jour. Math., 4, 1881, 25-38. "ONouv. Ann. Math., (3), 2, 1883, 384; 3, 1884, 395-6. "»/6id., (3), 2, 1883, 220-6. Cf. Ces^ro, (3), 4, 1885, 67-9. 2«BuU. Ac. Sc. Belgique, (3), 7, 1884, 448-9. "'M6m. Ac. R. Sc. Belgique, 46, 1886, No. 1, 16 pp. »**Nouv. Ann. Math., (3), 5, 1886, 257-272. Chap. Ill] SYMMETRIC FUNCTIONS MODULO p. 99 It is shown that (1^^^)" equals the number of combinations of n-\-p — l things p — 1 at a time. Various algebraic relations between binomial coefficients are derived. L. Gegenbauer^^^ considered the polynomial p-2+k f{x)= S hix' {l-p<k^p-l) i=0 and proved that V/(X)/X^-2= -h,.2 (mod p), k<p-l, X=l 'xf{X)/\^-'= -6p_2-fc2p-3 (mod p), k = p-\, X=l and deduced the theorem on the divisibility of s„ by p. E. Lucas^^^ proved the theorem on the divisibility of s„ by p by use of the symbolic expression (s+l)"— s" for x" — 1. N. Nielsen^^^" proved that if p is an odd prime and if k is odd and \<k<p — \, the sum of the products of 1, . . ., p — 1 taken A; at a time is divisible by p^. For k=p—2 this result is due to Wolstenholme.^^^ N. M. Ferrers^^^ proved that, if 2n+l is a prime, the sum of the products of 1, 2, . . . , 2n taken r at a time is divisible by 2n+l if r<2n [Lagrange^^], while the sum of the products of the squares of 1, . . . , n taken r at a time is divisible by 2n+l if r<n. [Other proofs by Glaisher.^^^] J. Perott^^^ gave a new proof that s^ is divisible by p if n<p — l. R. Rawson^^^ proved the second theorem of Ferrers. G. Osborn^^° proved for r<p — l that s^. is divisible by p if r is even, by p^ if r is odd; while the sum of the products of 1, . . ., p — l taken r at a time is divisible by p^ if r is odd and l<r<p. J. W. L. Glaisher^^^ stated theorems on the sum Sriai,..., a,) of the products of fli, . . . , rti taken r at a time. If r is odd, Sr{l, . . . , n) is divisible by n+1 (special case n+1 a prime proved by Lagrange and Ferrers). If r is odd and > 1, and if n+1 is a prime> 3, Sr{l, . . . , n) is divisible by {n-{-iy [Nielsen^^^'']. If r is odd and >1, and if w is a prime >2, Sril,. . ., n) is divisible by n^. If n+1 is a prime, Sr{l^,. • ■, n^) is divisible by n+1 for r = l,...,n — 1, except for r = n/2, when it is congruent to ( — 1)1+"/^ j^odulo n+1. If p is a prime ^n, and k is the quotient obtained on dividing n+1 by p, then aSp_i(1,..., n)=— A; (mod p); the case n = p — 1 is Wilson's theorem. "^Sitzungsber. Ak. Wiss. Wien (Math.), 95 II, 1887, 616-7. "«Th6orie des nombres, 1891, 437. 286aNyt Tidsskrift for Mat., 4, B, 1893, 1-10. "'Messenger Math., 23, 1893-4, 56-58. 288BuU. des sc. math., 18, I, 1894, 64. Other proofs. Math. Quest. Educ. Times, 58, 1893, 109; 4, 1903, 42. "•Messenger Math., 24, 1894-5, 68-69. ""Ibid., 25, 1895-6, 68-69. "'Ibid., 28, 1898-9, 184-6. Proofs"*. 100 History of the Theory of Numbers. [Chap. Ill S. Monteiro'^- noted that 2n+l divides {2n)\Z\n/r. J. Westlund-^^ reproduced the discussion by Serret^^^ and Tchebychef.'^^ Glaisher^^ proved his^^^ earlier theorems. Also, ii p = 27n+l is prime, {m-t)pS2t{l,.. ., 2m)=S2t+i{l,. ■ ., 2m) (mod f) and, if i>l, modulo p'^. According as n is odd or even, >S2t(l,. . ., n)=S2t{l,. . ., n-1) (mod n^ or ^n^). For m odd and >3, S2m-z0-i- . ., 2m — 1) is divisible by m^, and ^„_2(1^..., \m-l\^), ^2n.-4(l,...,2m-l) are divisible by m. He gave the values of Sr{\,. ■ ■, n) and Ar = Sr{l,. . ., n — 1) in terms of n for r = l,. . ., 7; the numerical values of 5^(1,. . ., n) for n^22, and a list of known theorems on the divisors of Ar and Sr. For r odd, 3^r^m — 2, Sr{l, . . ., 2w— 1) is divisible by m and, if w is a prime >3, by m.^ He proved {ibid., p. 321) that, if l^r^ (p-3)/2, and 5, is a BernoulU number, 2.S2.+i(l,. . ., p-l)_-{2r+l)S2r{l,..., p-1) V V ^2.(i,...,p-i)_(-ir5: V 2t (modp). Glaisher^^^ gave the residues of a^ [Frost^^^] modulo p^ and p^ and proved that (72, o'i, •• . , o-p-3 are divisible by p, and 0-3, cs, . . . , o-p_2 by p^, if p is a prime. Glaisher^^® proved that, if p is an odd prime. ■'•"' o2n I r2n I ' (p-2) 2n ^0 or — I (mod p), according as 2n is not or is a multiple of p — 1. He obtained (pp. 154-162) the residue of the sum of the inverses of like powers of numbers in arith- metical progression. F. Sibirani^^^" proved for the Sn,m of Sylvester^^^ (designated Sn,m-\-\) that ^n,n '^n— l,n • • • *^n— Jfe+l.n On+A;— l,n+*— 1 Sn+k—2,n+k—l ■ ■ ■ ^n.n+k—l t = inl)K "'Jornal Sc. Mat. Phys. e Nat., Lisbon, 5, 1898, 224. »»Proc. Indiana Ac. Sc, 1900, 103-4, "^Quar. Jour. Math., 31, 1900, 1-35. »'/Wd., 329-39; 32, 1901, 271-305. "•Messenger Math., 30, 1900-1, 26-31. 'wPeriodico di Mat., 16, 1900-1, 279-284. Chap. Ill] SyMMETKIC FUNCTIONS MODULO p. 101 K. HenseP^^ proved by the method of Poinsot^^^ that any integral sym- metric function of degree v of 1,..., p — 1 with integral coefficients is divisible by the prime p if y is not a multiple of p — 1. W. F. Meyer^^^ gave the generalization that, if ai, . . . , ap_i are incongru- ent modulo p", and each af~^ — 1 is divisible by p", any integral symmetric function of degree voiai,..., ap_i is divisible by p'* if v is not a multiple of p — 1. Of the </)(p") residues modulo p", prime to p, there are p'^Cp — 1)^ for which a^~^ — l is divisible by p"~^~'^, but by no higher power of p, where A; = 1, . . ., n— 1; the remaining p — 1 residues give the above ai,..., a^-i. J. W. Nicholson^^^ noted that, if p is a prime, the sum of the nth powers of p numbers in arithmetical progression is divisible by p if n<p — 1, and = — 1 (mod p) if 71 = p — 1. G. Wertheim^"'' proved the same result by use of a primitive root. A. Aubry^°^ took x = 1, 2, . . . , p — 1 in (a:+l)"-rc" = nx^-i+Ax"-2+ . . . -{-Lx+l and added the results. Thus p'' = ns„_i+As„_2+. . .+Lsi+p. Hence by induction Sn_i is divisible by the prime p if n<p. He attributed this theorem to Gauss and Libri without references. U. Concina^°^ proved that s„ is divisible by the prime p>2 if n is not divisible by p — 1. Let 5 be the g. c. d. of n, p — 1, and set ^i5 = p — 1. The )u distinct residues Ti of nth powers modulo p are the roots of ^"=1 (mod p), whence Sr,=0 (mod p) for n not divisible by p — 1. For each r^-, x'*=rj has 6 incongruent roots. Hence s„=5Sri=0. He proved also that, if p+1 is a prime >3, and n is even and not divisible by p, l''+2''+ . . . +(p/2)'* is divisible by p+1. W. H. L. Janssen van Raay^°^ considered, for a prime p>3, (p-1)! ^ (P-I)I ^''~ h ' ^'~h{v-h) and proved that B^-\-B2-\- . . . +-B(p_i)/2 is divisible by p, and are divisible by p^. U. Concina^o^ proved that ^ = 1+2"+ . . .+/c" is divisible by the odd number A; if n is not divisible by p — 1 for any prime divisor of p of k. Next, let k be even. For n odd > 1, ^ is divisible by k or only by k/2 according "'Archiv Math. Phys., (3), 1, 1901, 319. Inserted by Hensel in Kronecker's Vorlesungen tiber Zahlentheorie I, 1901, 104-5, 504. "sArchiv Math. Phys., (3), 2, 1902, 141. Cf. Meissner'" of Ch. IV. "^Amer. Math. Monthly, 9, 1902, 212-3. Stated, 1, 1894, 188. ^ooAnfangsgninde der Zahlentheorie, 1902, 265-6. '"iL'enseignement math., 9, 1907, 296. »°2Periodico di Mat., 27, 1912, 79-83. "'Nieuw Archief voor Wiskunde, (2), 10, 1912, 172-7. so^Periodico di Mat., 28, 1913, 164-177, 267-270. 102 History of the Theory of Numbers. [Chap. hi as k is or is not divisible by 4. For n even, S is divisible only by k/2 pro- vided n is not divisible by any prime factor, diminished by unity, of k. N. Nielsen^"^ wrote Cp for the sum of the products r at a time of 1, ... , p — 1, and s„(p)=2;s^ (7„(p)=S(-l)''-V. *-l «=1 If p is a prime >2m+1, o-2n(p-l)=S2n(p-l)=0(modp), Sgn+iCp " 1) = O(niod p2). If p = 2n+l is a prime >3, and l^r^n — 1, Cf'^^ is divisible by p^. Nielsen^''^ proved that 2Di^'^^ is divisible by 2n for 2p+l^n, where D\ is the sum of the products of 1, 3, 5, . . . , 2n — 1 taken s at a time; also, 2^"+'s2fl(n - 1) = 2^%,{2n - 1) (mod 4n^), and analogous congruences between sums of powers of successive even or successive odd integers, also when alternate terms are negative. He proved (pp. 258-260) relations between the C's, including the final formulas by Glaisher.29^ Nielsen^°^ proved the results last cited. Let p be an odd prime. If 2n is not divisible by p — 1, S2r.(p — 1) = 0 (mod p), S2„+i(p — 1)=0 (modp^). But if 2n is divisible by p — 1, S2n(p-1)=-1, S2„+i(p-l) = 0 (modp), Sp(p - 1) = 0 (mod p^). T. E. Mason^"^ proved that, if p is an odd prime and i an odd integer > 1, the sum Ai of the products i at a time of 1, . . . , p — 1 is divisible by p^. If p is a prime >3, Sk is divisible by p^ when k is odd and not of the form m(p — 1) + 1, by p when k is even and not of the form 7n{p — l), and not by p if A: is of the latter form. If A; = 7n(p — 1) + 1, s^ is divisible by p^ or p according as k is or is not divisible by p. Let p be composite and r its least prime factor; then r — 1 is the least integer t for which At is not divisible by p and conversely. Hence p is a prime if and only if p — 1 is the least t for which At is not divisible by p. The last two theorems hold also if we replace A's by s's. T. M. Putnam^"^ proved Glaisher's^^^ theorem that s_„ is divisible by p if n is not a multiple of p — 1 , and (p-l)/2 9 — 9P 2 jp-2=f_A(inodp). y-i p W. Meissner^^° arranged the residues modulo p, a prime, of the successive •o'K. Danske Vidensk. Selsk. Skrifter, (7), 10, 1913, 353. "•Annali di Mat., (3), 22, 1914, 81-94. •"Ann. sc. I'^cole norm, sup., (3), 31, 1914, 165, 196-7. "*T6hoku Math. Jour., 5, 1914, 136-141. »"Amer. Math. Monthly, 21, 1914, 220-2. "•Mitt. Math. Gesell. Hamburg, 5, 1915, 159-182. 1 i 1 2 4 8 3 6 12 11 9 5 10 7 Chap. Ill] SYMMETRIC FUNCTIONS MODULO Jp. 103 powers of a primitive root /i of p in a rectangular table of t rows and r col- umns, where ir = p — 1. For p = 13, /i = 2, i = 4, the table is shown here. Let R range over the numbers in any column. Then Si2 and Sl/^R are divisible by p. If Ms even, Sl/i? is divisible by p^ as 1/1 + 1/8+1/12+1/5 = 13^/120. For t = p — \, the theorem becomes the first one due to Wolstenholme.^^^ Generalizations are given at the end of the paper. N. Nielsen^^^ proved his^^^*" theorem and the final results of Glaisher.^^^ Nielsen^^^ proceeded as had Aubry^°^ and then proved (p-l)/2 ^_3 S2„+i^0 (mod f), S r^O (mod p), l^n^ ^ . Then by Newton's identities we get Wilson's theorem and Nielsen's^"^ last result. E. Cahen^^^ stated Nielsen's^^^'' theorem. F. Irwin stated and E. B. Escott^^* proved that if Sj is the sum of the products J at a time of 1, 1/2, 1/3,. ., \/t, where ^= (p — 1)/2, then 2S2—Si^, etc., are divisible by the odd prime p. "iQversigt Danske Vidensk. Selsk. ForhandUnger, 1915, 171-180, 521. 3i276id., 1916, 194-5. 'i^Comptes Rendus Stances Soc. Math. France, 1916, 29. »"Ainer. Math. Monthly, 24, 1917, 471-2. I CHAPTER IV. RESIDUE OF (C/^->-l)/P MODULO P. N. H. Abel^ asked if there are primes p and integers a for which (1) a^-'=l (modp'), l<a<p. C. G. J. Jacobi^ noted that, for p^37, (1) holds only when p = ll, a = 3 or 9; p = 29, a = 14; p = 37, a = 18. Cf. Thibault^^ of Ch. VI. G. Eisenstein^ noted that, for p a prime, the function has the properties (2) quv=qu-\-qv, qu+pv^Qu-- (mod p), 2g2=l-Ki-|+ • • • — ^^^\ Md p), P 1 o where s = (p+l)/2, . , ., p — 1. All solutions of (1) are included in a= u+puqu, 0<u<p. E. Desmarest^ noted that (1) holds for p = 4S7, a = 10, and stated that p = 3 and p = 487 are the only primes < 1000 for which 10 is a solution. J. J. Sylvester^ stated that, if p, r are distinct primes, p>2, then g^ is congruent modulo p to a sum of fractions with the successive denominators p — 1, . . . , 2, 1 and (as corrected) with numerators the repeated cycle of the positive integers ^r congruent modulo r to 1/p, 2/p,. . ., r/p. Thus, for r = 5, p — 1 p — 2 p — 6 p — 4: p — 0 p — 0 j,^^+.-l-+-i-+^+-^+-l-+... (p=10fc+7). p — 1 p—2 p — S p—4: p — 5 p — 0 According as p = 4A;+l or 4A: — 1, ^2 is congruent to 2.22.2. 2 p—S p—4: p-7 p—S p — ll 2 2 2 2 2 p-2 p-3 p-Q p-7 p-10 " [the signs were given + erroneously]. For any p, ?2= 7-\ ^ 5+... (modp). p—1 p—2 p—o iJour. fiir Math., 3, 1828, 212; Oeuvres, 1, 1881, 619. mid., 301-2; Werke, 6, 238-9; Canon Arithmeticus, Berlin, 1839, Introd., xxxiv. 'BerUn Berichte, 1850, 41. *Th6orie des nombres, 1852, 295. 'Comptes Rendus Paris, 52, 1861, 161, 212, 307, 817; Phil. Mag., 21, 1861, 136; Coll. Math. Papers, II, 229-235, 241, 262-3. 105 106 History of the Theory of Numbers. [Chap, iv Jean Plana^ developed j (Af — 1) + 1 } " and obtained M''-M-\{M-iy-{M-l)\=pf{M), Take M = m, 7n — l,. . ., 1 m the first equation and add. Thus =/(!) + . . . 4-/(w) =Si+^^S2-f • • • +Sp-i, where s, = r+2'H- . . . + (m — 1)'. For j> 1, we may replace p by j and get m -w=is,_i+ (0s,_2+ Qjsi-aH- • • • +isi, a result obtained by Plana by a long discussion [Euler"]. He concluded erroneously that each Si is divisible by m (for m = 3, S2 = 5). F. Proth^ stated that, if p is a prime, 2''— 2 is not divisible by p^ [error, see Meissner^^]. M. A. Stern^ proved that, if p is an odd prime, rrf — m_ , , , 1 _ , i , ,1 — - — =Si-§S2+iS3-. . .-— YSp-i=o-p-i+io-p_2+. . • + —^0-1 for Si as by Plana and 0-^ = l'+2*+ • • . +w\ Proof is given of the formula below (2) of Eisenstein^ and Sylvester's formulae for q2 (corrected), as well as several related formulae. L. Gegenbauer^ used Stern's congruences to prove that the coefficient of the highest power of a^ in a polynomial f{x) of degree p — 2 is congruent to {m^—m)/p modulo p if /(a:) satisfies one of the systems of equations J{\) = {-lf^'V-\{m-\), /(X)=X--VxW (X = l,. . ., p-1). E. Lucas^° proved that ^2 is a square only for p = 2, 3, 7, and stated the result by Desmarest.^ F. Panizza^^ enumerated the combinations p at a time of ap distinct things separated into p sets of a each, by counting for each r the combina- tions of the things belonging to r of the p sets : (T)=i.(^>a) CO ■•(")' •Mem. Acad. Turin, (2), 20, 1863, 120. ^Comptes Rendus Paris, 83, 1876, 1288. •Jour, fur Math., 100, 1887, 182-8. •Sitzungsber. Ak. Wiss. Wien (Math.), 95, 1887, II, 616-7. ">Th6orie des nombres, 1891, 423. "Periodico di Mat., 10, 1895, 14-16, 54-58. Chap. IV] RESIDUE OF {U^^ — l)/^ MoDULO y. 107 where ^l^-...^-^r = p, ij>0. The term given by r = p is a^. For p a prime, the left member is =a (mod p) and we have Fermat's theorem. By induction on r, Taking r = p, we have D. Mirimanoff^^ wrote ao for the least positive integer making aop+1 divisible by the prime r<p, and denoted the quotient by r%i, where 6i is prime to r. Similarly, let a^ be the least positive integer such that a,p+6i = r%ij^i. We ultimately find an n for which 6n = l- Then 6„4.j = 6i. By (2), ?b<~r-^i?'-+?6.>" - .^ 7r=gr2ei(modp). Oi »-0 Oj Let r belong to the exponent co modulo p and set 6w = p — 1. Then Sev=co, while 1, 61, . . . , 6„_i are the distinct residues of the eth powers of the integers < r and prime to r. Thus g^=eS -^ (mod p). The formula obtained by taking r a primitive root of p is included in the following, which holds also for any prime r : $,= 2 -* (mod p), ai being the least positive integer for which aip+iSi=0 (mod r). Set /3i = p— 5, p'p=l (mod r), 0<p'<r. Then ai=p'b — \ (mod r), ]A;} being the least positive residue modulo r of k. Whence Sylvester's^ statement. J. S. Aladow^^ proved that (1) has at most (p=f1)/4 roots if p = 4?n=tl. A. Cunningham^^" listed 27 cases in which r^~^=l or r'=l (mod p'), r<p^~^, where Z is a divisor of p — 1. For the 11 cases of the first kind, p = 5, 7, 17, 19, 29, 37, 43, 71, 487. W. Fr. Meyer^^ proved by induction that, if p is a prime, x^~^ — l is divisible by p* {l^k<n), but not by p^'^^, for exactly p"~^~^ (p — 1)^ posi- tive integers a:<p" and prime to p, and is divisible by p" for the remaining p — 1 such integers. Set ^=a+MiP+. ■ ■+MnP" (l^Q<P,0^iU><p), Xp=(a^'-a"' Vp "Jour, fiir Math., 115, 1895, 295-300. "St. Petersburg Math. Soc. (Rusaian), 1899, 40-44. "<»Messenger Math., 29, 1899-1900, 158. See Cuniungham"^, Ch. VI. "Archiv Math. Phys., (3), 2, 1901, 141-6. 108 History of the Theory of Numbers. [Chap, iv If k is the least index for which fXk^Xk, M/»=^/. (mod p) for h<k, then A'~^ — l is divisible by p*", but not by p^"*"^. A. Palmstrom and A. Pollak^^ proved that, if p is a prime and n, m are. the exponents to which a belongs modulo p, p~, respectively, then a"" — ! is divisible by p^, so that m is a multiple of n and a divisor of np, whence m = n OT pn. Thus according as a^~^ is or is not =1 (mod p^), m = n or m = np. Worms de Romilly^^" noted that, if co is a primitive root of p^, the incon- gnient roots of x^^=l (mod p^) are co^^{j = l,. . ., p — 1). J. W. L. Glaisher^^ proved that if r is a positive integer <p, p sl prime, r''-^ = l+^iP+K^i'-^2)p'+|(^i'-3^i^2+2^3)p'+. . ., where Qn is the sum of the nth powers of 1 _2_ r-1 . 1 2 r-1 . 1 a [2(t]'" '' [{r-l)a]' r+cr' r+[2(T]'" '' r+[(r-lV]' 2r+(r'* ' '' a being the least positive residue modulo r of — p. If /i^ is the least positive solution of ani=i (mod r), viz., p/i<H-i=0, then Ml , M2 , I Mr-l , Ml , M2 I , Mr-1 , Ml , 12 r-1 r + 1 r+2 ' ' ' ' ' 2r-l ^ 2r+l ' " Set /Ur = Oj Mi+jr =Mi- Then g^=i\^)\ s'^'^O(modp). Sylvester's corrected results are proved. From (1 + 1)^, op_2 1 / 1 \ =l-^+i- • . • 7^2(l+K . . . + -) (mod p). p p — 1 \ p — 2/ For r' = r+A:p, let m/ be the positive root of p/x/+^=0 (mod r'). Then It is shown that, for some integer t, k k k^ 2k k^ hi-gi+- = tp, h2-g2= -2---2+^t-y9i-:^ (mod p), Glaisher,^^ using the same notations, gave ^'-'-l+p(^+f +...+^l) (modp=). "L'intermddiaire des math., 8, 1901, 122, 205-6 (7, 1900, 357). ^Ibid., 214-5. '•Quar. Jour. Math., 32, 1901, 1-27, 240-251. "Messenger Math., 30, 1900-1, 78. Chap. IV] RESIDUE OF (U^ ^ — l)/p MODULO p. 109 Glaisher^^ considered Qu in connection with BernouUian numbers and gave ^-^^-i(}+l+ +1) (modp = 3ft+l). A. Pleskot^^ duplicated the work of Plana. ^ P. Bachmann^° gave an exposition of the work by Sylvester,^ Stern,* Mirimanoff.-^^ M. Lerch^^ set, for any odd integer p and for u prime to p, P Then,* as a generahzation of (2), qnv =qu+qv, Qu+pv =qu-\ (mod p) , "-I^H- 2g.-2i— 2I (mod p), where v ranges over the positive integers <p and prime to p; X over those >p/2; fi over those <p/2. Henceforth, let p be an odd prime and set N=\{p-l)\+l\/p. Then N^q,+ . . .+q,.^, [P/4]1 [p/3]i [p/5]i [2p/5U 32^-1 si, 3^3= -2 si, 5^5^-2 2^-2 Si »=l^ v=l^ a=l« 5=1 0 modulo p. If \p(n) is the number of sets of positive solutions <p oi ixv = n and hence the number of divisors between n/p and p of n, Employing Legendre's symbol and BernoulHan numbers, we have ^= sY^)g =0 or (-1)"-^2j5„ (mod p), v=l \p/ according as p = 4n+3 or 4n+l. In the respective cases, p-i \{^^Vq.^Cl{-p)oTO{modp), where CZ(— A) is the number of classes of positive primitive forms ax^+bxy+cy^ of negative discriminant 6^— 4ac= —A. Also, modulo p, y^iv'^LpJ' ^ ^aaLpJ' 6 ao Lp J a 0 aabf-pj where a, a are quadratic residues of p, and 6, /3 non-residues. isProc. London Math. Soc, 33, 1900-1, 49-50. "Zeitschrift fiir das Realschulwesen, Wien, 27, 1902, 471-2. "Niedere Zahlentheorie, I, 1902, 159-169. *The greatest integer ^x is denoted by [x]. "Math. Annalen, 60, 1905, 471-490. 1 10 History of the Theory of Numbers. [Chap, iv H. F. Baker" extended Sylvester's theorem to any modulus N: A' »-l N — TTli where the m, denote the integers <A^ and prime to A'", N'N=1 (mod r), and \k\ is the least positive residue modulo r of k. Lerch^ extended Mirimanofif's^^ formula to the case of a composite modulus m. Set m Let a belong to the exponent <f){m)/e. Then q{a, m)=e'Za/^ (mod m), where /? ranges over the residues of the incongruent powers of a, and wa+j3=0 (mod a), 0^a<a. As an extension of Sylvester's theorem, T T ' q(a, w)=2-= —2-^ (mod m), V V where v ranges over the integers < m and prime to m, while 7nr,-\-v=0, wr/ — ?'=0 (mod a), 0^r,<a, 0^r,'<a. For m = mi. . .rtik, where the rrij are relatively prime, k q{a, m) = 2 njn/(l){nj)q{a, nij) (mod m), where m = mjnj, n/n/=l (mod mj). H. Hertzer-'* verified that, for a<p<307, a^^ — 1 is di\'isible by p^ only for a = 68, p = 113; a = 3, 9, p = ll. He examined all the primes between 307 and 751, but only for a and p — a when a<y/p, finding only p = 113, a = 68. Removing the restriction a< Vp^ be found only the solutions p = ll,a = 3; p = 331, a = 18, 71; p = 353,a = 14; p = 487, a = 10, 175; p = 673, a = 22, together with the square of each a. A. Friedmann and J. Tamarkine^^ gave formulas connecting q^ with Bernoullian numbers and [u/p]. A. Wieferich^® proved that if x^+y^-\-z^ = Q is satisfied by integers X, y, z prime to p, where p is an odd prime, then 2""^ = 1 (mod p^). Shorter proofs were given by D. IMirimanoff-^ and G. Frobenius.'* D. A. Grave- ^ gave the residue of q^ for each prime p< 1000 and thought he could prove that 2^ — 2 is never divisible by p^ (error, Meissner^). A. Cunningham^" verified that 2^ — 2 is not divisible by p^ for any prime p< 1000, and^^ that 3^ - 3 is not divisible by p^ for a prime p = 2''3''+ 1< 100. W. H. L. Janssen van Raay^^ noted that 2^ — 2 is not divisible by p^ in general. »Proc. London Math. Soc, (2), 4, 1906, 131-5. "Comptes Rendus Paris, 142, 1906, 35-38. "Archiv Math. Phys., (3), 13, 1908, 107. «Jour. fur Math., 135, 1909, 146-156. »Jour. fiir Math., 136, 1909, 293-302. "L'enseignement math., 11, 1909, 455-9. "Sitzungsber. Ak. Wiss. Berlin, 1909, 1222-4; reprinted in Jour, fiir Math., 137, 1910, 314. **An elementary text on the theory of numbers (in Russian), I^ev, 1909, p. 315; Kiev Izv. Univ., 1909, Nos. 2-10. "Report British Assoc, for 1910, 530. L'interm^diaire des math., 18, 1911, 47; 19, 1912, 159. Proc. London Math. Soc, (2), 8, 1910, xiii. "L'interm^diaire des math., 18, 1911, 47. Cf., 20, 1913, 206. "Nieuw Archief voor Wiskunde, (2), 10, 1912, 172-7. Chap. IV] RESIDUE OF {U^^ — l)/p MODULO p. Ill ' L. Bastien^^" verified that (1) holds for p< 50 only for p= 43, a= 19, and for Jacobi's^ cases. He stated that, if p= 4p='= 1 is a prime, -1^2=1 + 1/3+1/5+... +1/(2/1-1) (mod p). W. Meissner^^ gave a table showing the least positive residue of (2' — l)/p modulo p for each prime p< 2000, where t is the exponent to which 2 belongs modulo p. In particular, 2^ — 2 is divisible by the square of the prime p = 1093, contrary to Proth^ and Grave,^^ but for no other p<2000. In the chapter on Fermat's last theorem will be given not only the con- dition q2—0 (mod p) of Wieferich^^ but also q^^O (mod p), etc., with cita- tions to D. Mirimanoff, Comptes Rendus Paris, 150, 1910, 204-6, and Jour, fiir Math., 139, 1911, 309-324; H. S. Vandiver, ibid., 144, 1914, 314-8; G. Frobenius, Sitzungsber. Ak. Wiss. Beriin, 1910, 200-8; 1914, 653-81. These papers give further properties of q^. P. Bachmann^^ employed the identity (a-\-h-\-cy-{a+b-cy+ia-h-cy-{a-h+cy = 2(^)cl(a+6r-^-(a-6r-n+2(|)c^l(a+6r-^-(a-6r-^f + ... for a = 6 = 1, c = 2 or 1 to get expressions for ^2 or q^, whence for an odd prime p. Comparing this with the value of (3^ — 3)/p obtained by expanding (2+1)^, we see that ?!ll2_2P-i_^i.2P-2_|_i.2P-3+ , . . +-^-2 (mod p). p p-l Again, 92^2- (^) Vssfirn.(s-0 {-ly+'+'s (mod p), summed for all sets of solutions of s^=f^+l (mod p). Finally, g2^s'|(r''-r-'')2(r2'"'-l)-i|, h=l[ " J where r is a primitive pth root of unity. *H. Brocard^^ commented on a^^=l (mod p""). *H. G. A. Verkaart^^ treated the divisibility of a^ — a by p. E. Fauquembergue^^ checked that 2^=2 (mod p2) for p = 1093. N. G. W. H. Beeger^^ tabulated all roots of a;^~^= 1 (mod p^) for each prime p<200. If w is a primitive root of p^, the absolutely least residue 32aSphinx-Oedipe, 7, 1912, 4-6. It is stated that G. Tarry had verified in 1911 that 2P-2 is not divisible by a prime p < 1013. "Sitzungsber. Ak. Wiss. Berlin, 1913, 663-7. "Jour, fiir Math., 142, 1913, 41-50. '^Revista de la Sociedad Mat. Espanola, 3, 1913-4, 113-4. '"Wiskundig Tijdschrift, vol. 2, 1906, 238-240. "L'interm6diaire des math., 1914, 33. 38Messenger Math., 43, 1913-4, 72-84. 112 History of the Theory of Numbers. [Chap, iv ±xi modulo p2 of CO" is a root, that (^xg) of xi^ is a second root, that ( ^Xg) of X1X2 is a third root, etc., until the root =tx, is reached, where s = (p-l)/2. The remaining roots are p^-x,(i = l,. . ., s). He proved that ixi...xy={-l)-^ (modp2). Hence Xi. . .x,= ±l if p = 4n+l. W. Meissner^^ WTote /i^ for the residue <p'" of /i^""' modulo p"*. When h varies from 1 to p-1, we get p-1 roots h^ of xP-^=1 (mod p"*). The product of the roots given by h = l,.. ., (p-l)/2, is =(-1)' or (-l)V (mod p'"), according as p = 4m-1 or 4n+l, where z is the number of pairs of integers <p/2 whose product is = -1 (mod p), and c is the smaller of the two roots of x^=-l (mod p). No number <p which belongs to one of the exponents 2, 3, 4, 6, modulo p, can be a root of x^^=l (mod p^). A root of the latter is given for each prime p< 300, and a root modulo p^ for each p<200; also the exponent to which each root belongs. N. Nielsen^^ noted that, if we select 2r distinct integers a„ &, (s = 1, . . . ,r) from 1, . . ., p-1, such that a,+b, = p, then "^ = i-mi-pA), A^ ^ = k^lg^-g^^'j (modp). Proof is given of various results by Lerch,-^ also of simple relations between Qa and BernouUian numbers, and of the final formula by Plana,^ here attrib- uted to Euler.^^ H. S. Vandiver^- proved that there are not fewer than [Vp] and not more than p — (l + \/2p— 5)/2 incongruent least positive residues of 1, 2^\..., (p-l)^\ modulo f. N. Nielsen^^ noted that, if a is not di\'isible by the odd prime p, a — I (p-3)/2 2 9a=-^+ S^ 2j^-ir'i5.(a^-'^-^-l) (modp), j gi+?2+...+gp-i=(-l)"-'5„+--l (modp2), n=ip-l)/2. V W. Meissner^ gave various expressions for ^2 and ^3. A. G^rardin^^ found all primes p<2000, including those of the form 2"—!, for which 52 is sjTnmetrical when written to the base 2. H. S. Vandiver^^ proved that 52—0 (mod p^) if and only if He gave various expressions for (n* — l)/m. "Sitzungsber. Berlin IMath. Gesell., 13, 1914, 96-107. "Ann. sc. I'^cole norm, sup., (3), 31, 1914, 171-9. "Euler, Institutiones Calculi Diff., 1755, 406. Proof, Math. Quest. Educ. Times, 48, 1888, 48. «BuU. AmQT. Math. Soc, 22, 1915, 61-7. «K)versigt Danske Vidensk. SeLsk. ForhandUnger, 1915, 518-9, 177-180; cf. Lerch's»» N. ♦^Mitt. Math. Gesell. Hamburg, 5, 1915, 172-6, 180. «Xouv. Ann. Math., (4), 17, 1917, 102-8. "Annals of Math., 18, 1917, 112. CHAPTER V. EULER'S (^-FUNCTION, GENERALIZATIONS. FAREY SERIES. Number <f){n) of Integers <n and Prime to n. L. Euler/ in connection with his generalization of Fermat's theorem, investigated the number <j){n) of positive integers not exceeding n which are relatively prime to n, without then using a functional notation for 0(n). He began with the theorem that, if the n terms a, a-\-d,. . ., a+(n — l)d in arithmetical progression are divided by n, the remainders are 0, 1,. . ., n — 1 in some order, provided d is prime to n; in fact, no two of the terms have the same remainder. If p is a prime, (^(p"") =p'"~^(p — 1), since p, 2p,. . ., p^~^-p are the only ones of the p"* positive integers ^ p^ not prime to p"*. To prove that (1) <t>{AB) =4>{A)<f>{B) {A, B relatively prime), let 1, a, . . ., CO be the integers <A and prime to A. Then the integers < AB and prime to A are 1 a . . . CO A + 1 A+a ... A+co 2A + 1 2A+a ... 2A+C0 (B-l)A+co. (5-l)A + l {B-l)A-\-a The terms in any column form an arithmetical progression whose difference A is prime to B, and hence include <^(J5) integers prime to B. The number of columns is (f>{A). Hence there are ({>{A)<f){B) positive integers <AB, prime to both A and B, and hence prime to AB. If p, . . . , s are distinct primes, the two theorems give (2) 0(p\ . .s«)=p^-np-l). . .s'-\s-l). Euler^ later used ttN to denote 4>{N) and gave a different proof of (2). First, let N = p'^q, where p, q are distinct primes. Among the N—1 integers <A^ there are p""—! multiples of q, and p'^'^q — l multiples of p, these sets having in common the p"~^ — 1 multiples of pq. Hence <^(iV)=iV-l-(p"-l)-(p'*-^g-l)+p"-i-l=p"-i(p-l)(g-l). A simpler proof is then given for the modified form of (2) : (3) iV(p-l)to-l)...(.-l), pq...s where p, q, r, . . . , s are the distinct primes dividing N. There are N/p multiples <N of p and hence N' = N{p — \)/p integers <A^ and prime to p. Of these, N' /q are divisible by q; excluding them, we have N" = N'{q — l)/q numbers < N and prime to both p and q. The rth part of these are said ^Novi Comm. Ac. Petrop., 8, 1760-1, 74; Comm. Arith., 1, 274, Opera postuma, I, 492-3. «Acta Ac. Petrop., 4 II (or 8), 1780 (1755), 18; Comm. Arith., 2, 127-133. He took 0(1)=O. 113 114 History of the Theory of Numbers. [Chap, v [cf. Poinsot^^] to be divisible by r; after excluding them we get N"{r — l)/r numbers; etc. Euler^ noted in a posthumous paper that, if p, q, r are distinct primes, there are r multiples ^pqr of pq, and qr multiples of p, and a single multiple of pqr, whence <t){pqr)=pqr-qr-pr-pq-^r+p-\-q-l = {p-l){q-l){r-l). In general, if M is any number not divisible by the prime p, and if fx denotes the number of integers ^M and prime to M, there are M—fj. integers ^M and not prime to M and hence p''{M—ii) integers ^Mp" and not prime to M and therefore not prime to Mp". Of the Mp^~^ multiples ^Mp^ of p, exclude the p"~^(ilf — /i) which are not prime to M; we obtain p^'V multiples of p which are prime to M. Hence <^(p"M)=p"M-p''(M-m)-p""V = P"~Hp-1)m- A. M. Legendre^ noted that, if ^, . . . , co are any odd primes not dividing A, the number of terms of the progression A+B, 2A+5, . . . , nA-\-B which are divisible by no one of the primes 0, . . . , co is approximately n(l — 1/0) . . . (1 — 1/co), and exactly that number if n is divisible by 0, . . . , co. C. F. Gauss^ introduced the symbol (i>{N). He expressed Euler's^ proof of (1) in a different form. Let a be any one of the 4>{A) integers <A and prime to A, while j8 is any one of the 4>{B) integers <B and prime to B. There is one and but one positive integer x<AB such that x=a (mod A), x=^ (mod B). Since this x is prime to A and to B, it is prime to AB. Making the agreement that <^(1) = 1, Gauss proved (4) 20(d) =N {d ranging over the divisors of N). For each d, multiply the integers ^d and prime to d by N/d; we obtain S0(d) integers ^N, proved to be distinct and to include 1, 2, . . ., iV. A. M. Legendre^ proved (3) as follows: First, let N = pM, where p is a prime which may or may not divide M; then Mp—M of the numbers 1,. . ., N are not divisible by p. Second, let N = pqM, where p and q are distinct primes. Then 1,. . ., N include M numbers divisible by both p and q; Mp — M numbers divisible by q and not by p; Mq — M numbers divisible by p and not by q. Hence there remain A^(l — l/p)(l — 1/q) num- bers divisible by neither p nor q. Third, a like argument is said to apply to N = pqrM, etc. Legendre (p. 412) proved that ii A,C are relatively prune and if 0,X,ju, . . . , CO are odd primes not dividing A, the number of terms kA — C{k = l,...,n), which are divisible by no one of 0, . . . , to, is *Tractatu3 de numerorum, Comm. Arith., 2, 515-8. Opera postuma, I, 1862, 16-17. *Es3ai siir la thdorie des nombres, 1798, p. 14. 'Disquisitionea Arithraeticse, 1801, Arts. 38, 39. •Th^orie des nombres, ed. 2, 1808, 7-8; German trans, of ed. 3 by Maser, 8-10. Chap. V] EulER's (^-FUNCTION. 115 where the summations extend over the combinations of 6,. . ., co taken 1, 2, . . ., at a time, while Aq is a positive integer <A for which ^Aq+C is divisible by A, and [x] is the greatest integer ^x. We thus derive the approximation stated by Legendre.^ Taking A = l, C = 0 (p. 420), we see that the number of integers ^n, which are divisible by no one of the dis- tinct primes 0, X, . . . , co is A. von Ettingshausen^ reproduced without reference Euler's^ proof of (3) and gave an obscurely expressed proof of (4) . Let A'' = p'g" . . . , where p, q,. . . are distinct primes. Consider first only the divisors d = p^q', where /i>0, v>0, so that d involves the primes p and q, but no others. By (3), ^(d)=d(i-^) (i-l), J^ |^py=(p+p'+. . .+r)(g+. . .+2"), S(^(pY) = (p"-i)(/-i). Similarly, S0(p'') =p"— 1. In this way we treat together the divisors of N which involve the same prime factors. Hence when d ranges over all the divisors of N, S<^(d) = l+S(p«-l) + S(p»-l)(g^-l)+ S (p»-l)(5''_l)(r-_l) + ... P P.Q P. 3. r =n]i+(p"-i)J=np"=iv, p where the summation indices range over the combinations of all the prime factors of N taken 1, 2, . . .at a time. [Cf, Sylvester .^^] A. L. Crelle^ considered the number Zj of integers, chosen from rii, . . . ,na, which are divisible by exactly j of the distinct primes Pi, . . ., Pm', and the number Sy of the integers, chosen from rii, . . . , n^, which are divisible by at least j of the primes Pi. Then Z1 + Z2+. . .+Zm = Si-S2 + Ss- . . .=tS^. Let V be the number of the integers rii, . . . , n^ which are divisible by no one of the primes Pi. Then a^'Ezi+v, p = a-Si+S2— . . .=PSrr,. In particular, take nj, ..., ria to be 1, 2,. .., iV, where N = p''q^r^ . . ., and take Pi,...,Pm to be p, g, r, . . . . Then N N ^ N N , N , P q pq pr ' pqr cf>{N)=N-s,-{-S2- . . . =n(i--^ 0"-) • • •• He proved (1) for 5 = 0", where a is a prime not dividing A (p. 40). By Euler's^ table there are B({)(A) integers <AB and prime to A. In Euler's ^Zeitschrif t fur Physik u. Math, (eds., Baumgartner and Ettmg8hauaen),Wien, 5, 1829, 287-292. *Abh. Akad. Wiss. Berlin (Math.), 1832, 37-50. 116 History of the Theory of Numbers. (Chap, v notation, a{kA + l), a{kA-\-a),. . ., a{kA-{-03) give all the numbers between kaA and {k-\-\)aA which are divisible by a and are prime to A. Taking A: = 0, 1,. . ., a°~^ — 1, we see that there are exactly a"~V(^) multiples of a which are <AB and prime to A. Hence 0(a"^) =aXA) -o^-VU) =(t>{a'')(f>{A). F. Minding^ proved Legendre's formula (5). The number of integers ^n, not divisible by the prime 0, is n — [n/d]. To make the general step by induction, let Pi, . . . , Pk be distinct primes, and denote by (5; pi, . . . , p^) the number of integers ^ 5 which are divisible by no one of the primes pi, . . . , Pk' Then, if p is a new prime, (B; pi, . . . , Pk, p) = {B;pu..., Pk) - ([B/p] ; Pi, . • • , Pk)- The truth of (4) for the special case N = p — 1, where p is a prime, follows (p. 41) from the fact that ((){d) numbers belong to the exponent d modulo p if d is any divisor of p — 1. N. Druckenmiiller^*^ evaluated (f>{b), first for the case in which 6 is a product cd. . .kl oi distinct primes. Set h=^l and denote by \f/{h) the num- ber of integers <b having a factor in common with 6. There are l\p{^) numbers < b which are divisible by one of the primes c, . . . , k, since there are \p{P) in each of the sets l,2,...,/3; ^+1,...,2^; ...; (i-l)/3+l,. . ., Z/3. Again, I, 21,..., pi are the integers <b with the factor I. Of these, 0(j3) are prime to jS, while the others have one of the factors c, . . . , k and occur among the above lxl/{^). Hence xl/{b)=l\l/i^)-\-<i>iP). But i/'O3)+0(/3)=/3. Hence </,(6) = a-l)(A(^) = (c-l)...(Z-l). Next, let 6 be a product of powers oi c, d,. . ., I, and set b = L^, ^ = cd. . .1. By considering L sets as before, we get E. Catalan^^ proved (4) by noting that 2(/,(py. . .)=n]i+(/)(p)+ . . . +<f>(p'')\ =np»=Ar, where there are as many factors in each product as there are distinct prime factors of N. A. Cauchy^^ gave without reference Gauss'^ proof of (1). E. Catalan^^ evaluated <t){N) by Euler's^ second method. C. F. Arndt^"* gave an obscure proof of (4), apparently intended for Catalan's. ^^ It was reproduced by Desmarest, Th^orie des nombres, 1852, p. 230. •Anfangsgriinde der Hoheren Arith., 1832, 13-15. »oTheorie der Kettenreihen . . .Trier, 1837, 21. "Jour, de Mathdmatiques, 4, 1839, 7-8. i»Compte8 Rendus Paris, 12, 1841, 819-821; Exercices d'analyse et de phys. math., Paris, 2, 1841, 9; Oeuvres, (2), 12. "Nouv. Ann. Math., 1, 1842, 466-7. "Archiv Math. Phys., 2, 1842, 6-7. Chap. V] EuLER's (^-FUNCTION. 117 J. A. Grunert^^ examined in a very elementary way the sets jk+1, jk-{-2,..., jk+k-l, {j+l)k (j = 0, l,...,p-l) and proved that 4){'pk)='p4){k) if the prime p divides k, while 4){pk) = (p — l)<j){k) if the prime p does not divide k. From these results, (2) is easily deduced [cf. Crelle^^ on (f){Z)]. L. Poinsot^® gave Catalan's^^ proof of (4) and proved the statements made by Euler^ in his proof of (3) . Thus to show that, of the N' = N{1- 1/p) integers < N and prime to p, exactly N'/q are divisible by q, note that the set 1,. . ., N contains N/q multiples of q and the set p, 2p, . . . contains {N/p)/q multiples of q, while the difference is N'/q. If P, Q, R,. . . are relatively prime in pairs, any number prime to N = PQR . . . can be expressed in the form pQR...+qPR...+rPQ... + ..., where p is prime to P, q to Q, etc. If also p<P, q<Q, etc., no two of these sums are equal. Thus there are 0(P)0(Q) . . . such sums [certain of which may exceed N]. To prove (4), take (pp. 70-71) a prime p of the form kN+l and any one of the N roots p of a;^= 1 (mod p). Then there is a least integer d, sl divisor of N, such that p'^= 1 (mod p). The latter has (f)(d) such roots. Also p is a primitive root of the last congruence and of no other such congruence whose degree is a divisor of N. A. L. Crelle^^ considered the product E = eie2. . .e„ of integers relatively prime in pairs, and set Ej = E/ej. When x ranges over the values 1, . . ., Ci, the least positive residue modulo E of EiXi-\- . . . +£'„a;„ takes each of the values 1, . . .,E once and but once. In case Xi is prime to ei for i = 1, . . . , n, the residue of SE'^Xi is prime to E and conversely. Let dn, di2, ... be any chosen divisors >1 of e^ which are relatively prime in pairs. Let \}/{ei) denote the number of integers ^e^ which are divisible by no one of the ^ti, di2,. . .. Let yl/{E) be the number of integers ^E which are divisible by no one of the dn, di2, c^2ij ■ ■ •> including now all the d's. Then \1/{E) = ^(ei) . . . i/'(en). In case dn, di2, . . . include all the prime divisors > 1 of e,-, ypie^ becomes ^(e^). Of the two proofs (pp. 69-73), one is based on the j&rst result quoted, while the other is like that by Gauss .^ As before, let ^{y) be the number of integers '^y which are divisible by no one of certain chosen relatively prime divisors di,...,dm of y. By con- sidering the xy numbers ny-\-r (0^n<x, I'^r^y), it is proved (p. 74) that, when X and y are relatively prime, ypixy) =x\p{y), \p2ixy) = {x-l)xl/{y), where \p2{^y) is the number of integers ^xy which are divisible neither by X nor by any one of the d's. These formulas lead (pp. 79-83) to the value of0(Z). Set Z = p/'...p/M, z = Pi...p^, n = Z/z, i^Archiv. Math. Phys., 3, 1843, 196-203. "Jour, de Math^matiques, 10, 1845, 37-43. "Encyklopadie der Zahlentheorie, Jour, fiir Math., 29, 1845, 58-95. 118 History of the Theory of Numbers. [Chap, v where Pi,. . ., p^ are distinct primes. For a prime p, not dividing y, we have (f){py) = {p- l)<t>iy) . Take y = Pi, p = P2', then <^(PiP2) = (Pi-l)(P2-l)- Next, take y = PiP2, P = P3} and use also the last result; thus <A(PlP2P3) = (Pl-l)(P2-l)(P3-l), and similarly for <t){z). When f ranges over the integers < z and prime to z, the numbers vz-\-^ {v = 0,l,. . .,n — l) give without repetition all the integers <Z and prime to Z. Hence (f>{Z)=n<i){z), which leads to (2). [Cf. Guil- min,2^ Steggall.'^^] The proofs of (4) by Gauss^ and Catalan^ ^ are reproduced without refer- ences (pp. 87-90). A third proof is given. Set N = a''h^c' . . ., where a, b, c, . . . are distinct primes. Consider any divisor e = b^"^' . . . of N such that e is not divisible by a. Then <t>i€a'')=a''-\a-l)(t>{e). Sum for /b = 0, 1, . . . , a; we get a°0(e). When k ranges over its values and /3i over the values 0, 1,. . ., j3, and 71 over the values 0, 1,. . ., 7, etc., ea* ranges over all the divisors d of iV. Hence 20 (d) =a''S0(e). Similarly, if Ci range over the divisors not divisible by a or b, S</)(€)=6^(^(ei),. . ., S<^(d)=a»6^ . . =N. E. Prouhet^^ proposed the name indicator and symbol i{N) for 0(iV). He gave Gauss' proof of (1) and Catalan's proof of (4). If 5 is the product of the distinct prime factors common to a and b, <j>{ab) =(i>{a)(}>ib)8/(f){8). As a generalization, let 5^ be the product of the distinct primes common to i of the numbers Oi, . . . , a„; then 2 § 2 2 n-l </)(ai. . .a„) =<j>{ai) . . .<f){an) ^ ^ Friderico Arndt^^ proved (1) by showing that, if x ranges over the integers <A and prime to A, while y ranges over the integers <B and prime to B, then Ay-{-Bx gives only incongruent residues modulo AB, each prime to AB, and they include every integer <AB and prime to AB. [Crelle's^^ first theorem for n = 2.] V. A. Lebesgue^° used Euler's^ argument to show that there are Nip-l){q-l)...{k-l) p-q. . .k integers < iV and prime top,q,...,k, the latter being certain prime divisors of A'' [Legendre,^ Minding^]. "Nouv. Ann. Math., 4, 1845, 75-80. "Jour, fur Math., 31, 1846, 246-8. "Nouv. Ann. Math., 8, 1849, 347. Chap. V] EuLER's 0-FuNCTION. 119 G. L. Dirichlet^^ added equations (4) for iV = n, . . ., 2, 1, noting that, if s^n, 4>(s) occurs in the new left member as often as there are mul- tiples ^n oi s. Hence i\-']<f>{s)=Un'+n). s=lLsJ The left member is proved equal to XxJ/in/s], where It is then shown that \p{n) —Zn^/ir^ is of an order 6i magnitude not exceed- ing that of n\ where 2>5>7>1,7 being such that 8=2 S^ P. L. Tchebychef^^ evaluated 0(n) by showing that, if p is a prime not dividing A, the ratio of the number of integers ^ pAN which are prime to A to the number which are prime to both A and p is p:p — l. A. Guilmin^^ gave Crelle's^^ argument leading to 0(Z). F. Landry ^^ proved (3). First, reject from 1, . . ., iV the N/p multiples of p; there remain A^(l — 1/p) numbers prime to p. Next, to find how many of the multiples q, 2q, . . . , N of q are prime to p, note that the coefficients 1, 2, . . ., N/q contain N/q-{l — l/p) integers prime to p by the first result, applied to the multiple N/q of p in place of N. Daniel Augusto da Silva^^ considered any set S of numbers and denoted by S{a) the subset possessing the property a, by S{ab) the subset with the properties a and b simultaneously, by {a)S the subset of numbers in S not having property a; etc. Then {a)S = S-S{a)=S\l-{a)\, symbolically. Hence (ha)S = {b)\(a)S\=S\l-{a)\\l-{h)\, {. . .cba)S^S\l-{a)\\l-{b)\ \l-{c)\ . . .. A proof of the latter symbohc formula was given by F. Horta.^^" With Silva, let *S be the set 1, 2, . . . , n, and let A, ^, . . . be the distinct prime factors of n. Let properties a, 6, ... be divisibility by A,B,. . .. Then there are n/A terms in >S(a), n/{AB) terms in S{ab), . . ., and <f){n) terms in ( . . .cba)S. Hence our symbolic formula gives *(«)=»(l-i)(l-|). "Abhand. Ak. Wiss. Berlin (Math.), 1849, 78-81; Werke, 2, 60-64. ^^Theorie der Congruenzen, 1889, §7; in Russian, 1849. «Nouv. Ann. Math., 10, 1851, 23. ^^Troisieme mlmoire siir la th^orie des nombres, 1854, 23-24. '^Proprietades geraes et resoluQao directa das Congruencias binomias, Lisbon, 1854. Report on same by C. Alasia, Rivista di Fisica, Mat. e Sc. Nat., Pavia, 4, 1903, i3-17; reprinted in Annaes Scientificos Acad. Polyt. do Porto, Coimbra, 4, 1909, 166-192. ""Annaes de Sciencias e Lettras, Lisbon, 1, 1857, 705. 120 History of the Theory of Numbers. [Chap, v E. Betti^® evaluated 0(w), where 7n is a product of powers of the distinct primes ai, a2> • •• • Consider the set Ci of the products of the a's taken i at a time and their multiples ^m. Thus Co is 1, . . . , w, while C2 is 0x02, 20x02, . . ., 0102; cii(h, 2ai03, . . ., diCis',- ■ •• aia2 ttitta Let X be an integer < w divisible by ai, . . . , a„. Then x occurs times in the sets Co, C2, C4, . . . ; and 2" ^ times in Ci, C3, . . .. Summing l-(l) + (2)-(3)+ =0 for each of the m—<t>{m) integers ^m having factors in common with m, we get m-0(m)-s(j)+s(2)-...=O. But ^i-i) is the niunber of integers having in common with m one of the factors ai, 02, . . ., and hence equals S— . Next, ^i^j ^^ *^® number of integers having in common with m one of the factors 0102, OiOa, . . . , and hence equals 2 { m/ (0102) } . Thus mm 4>{m) =m—L, — 1-2 .... R. Dedekind^^ gave a general theorem on the inversion of functions (to be explained in the chapter on that subject), which for the special case of </)(n) becomes a proof like Betti's. Cf. Chrystal's Algebra, II, 1889, 511; Mathews' Theory of Numbers, 1892, 5; Borel and Drach,^^ p. 27. J. B. Sturm^^ evaluated 4>{N) by a method which will be illustrated for the case N = \b. From 1,. . ., 15 delete the five multiples of 3. Among the remaining ten numbers there are as many multiples of 5 as there are multiples of 5 among the first ten numbers. Hence <^(15) = 10—2 = 8. The theorem involved is the following. From the three sets 1, 2, 3,* 4, 5; 6,* 7, 8, 9,* 10; 11, 12,* 13, 14, 15* delete (by marking with an asterisk) the multiples of 3. The numbers 11, 13, 14 which remain in the final set are congruent modulo 5 to the num- bers 6, 3, 9 deleted from the earUer sets. J. Liouville" proved by use of (4) that, for |x|<l, g <t>{m)x"' _ X (t>{m)x"'_ g <f>{m)x'^ _x{l-\-x^) m^ll-X"^ ~{l-xf l-X^^'m^ll+X"" ~{1-Xy ' "Bertrand's Alg^bre, Ital. transl. with notes by Betti, Firenze, 1856, note 5. Proof reproduced by Fontebasso'*, pp. 74-77. ^ "Jour, fvir Math., 54, 1857, 21. Dirichlet-Dedekind, Zahlentheohe, §138. ^ "Archiv Math. Phys., 29, 1857, 448-452. "Jour, de math6matiques, (2), 2, 1857, 433^40. Chap. V] EuLER's (^-FUNCTION. 121 where m in S' ranges only over the positive odd integers. The final fraction equals x-\-Sx^-{-5x^+ .... From the coefficient of x^ in the expansion of the third sum, we conclude that, if n is even, where d ranges over all the divisors of n. Let 5i range over the odd values of 5, and 82 over the even values of 5; then 0-0- the value n/2 following from (4). Another, purely arithmetical, proof is given. Finally, by use of (4), it is proved that, if s>2, n=l /fr n=l/t A. Cayley3° discussed the solution for N of <^(iV) = iV^ Set N = a^b^ ..., where a, 6, . . . are distinct primes. Multiply l + (a-l) \a\+a{a-l) \a^\ + ... +a''-\a-l) {a"} + . . . by the analogous series in 5, etc. ; the bracketed terms are to be multiplied together by enclosing their product in a bracket. The general term of the product is evidently Hence in the product first mentioned each of the bracketed numbers which are multiplied by the coefficient N' will be a solution N of <f){N)=N'. We need use only the primes a for which a — 1 divides N', and continue each series only so far as it gives a divisor of N' for the coefficient of a"~^(a — 1). V. A. Lebesgue^^ proved 4){Z)=n4>{z) as had Crelle^^ and then 4>{z) =n(pi — 1) by the usual method of excluding multiples of pi, . . . , p„ in turn. By the last method he proved (pp. 125-8) Legendre's (5), and the more general formula preceding (5). J. J. Sylvester^^ proved (4) by the method of Ettingshausen,' using (2) instead of (3) . By means of (4) he gave a simple proof of the first formula of Dirichlet;^^ call the left member u^', since [n/r] — [(n — l)/r] = l or 0, according as n is or is not divisible by r, v^^J^ w(n+l) The constant c is zero since Ui = l. He stated the generalization 2{*(i')(l-+2-+... + [?]'")}^ r+2'-+...+n'". He remarked that the theorem in its simplest form is "London Ed. and Dublin Phil. Mag., (4), 14, 1857, 539-540. "Exercicea d'analyse niim^rique, 1859, 43-45. "Quar. Jour. Math., 3, 1860, 186-190; CoU. Math. Papers, 2, 225-8. 122 History of the Theory of Numbers. [Chap, v the example given being r = 2, n = 4, whence the divisors of n are 1-1, 2-1, 4-1, 1-2, 2-2, 1-4 and the above terms are Ml, Ml, M-2, 2-M, 2M, 4-21, with the sum 4". [With this obscure result contrast that by Cantor/^] G. L. Dirichlet^^ completed by induction Euler's^ method of proving (3), obtaining at the same time the generalization that, if p, g, . . . , s are divisors, relatively prime in pairs, of N, the number of integers ^ N which are divi- sible by no one of p, . . . , s is H-;)04) 0--.) A proof (§13) of (4) follows from the fact that, if d is a divisor of N, there are exactly </)(d) integers ^N having with N the g. c. d. N/d. P. A. Fontebasso^^ repeated the last remark and gave Gauss' proof of (1). E. Laguerre^^ employed any real number k and integer m and wrote (m, m/k) for the number of integers ^m/k which are prime to m. By continuous variation of k he proved that i:{d,d/k) = [m/k], where d ranges over the divisors of m. For k = l, this reduces to (4). F. Mertens^^ obtained an asymptotic value for 0(1)+ . . . +(i>iG) for G large. He employed the function ju(n) [see Ch. XIX] and proved that I 0(m) = | S /.(n){r^lVr^l U^GHA TO=i n=i iLnJ Lnjj tt |A|<G(ilog,G+iC+f) + l, where C is Euler's constant 0.57721 .... This upper limit for A is more exact than that by Dirichlet.^^ T. Pepin^'' stated that, if n = a"6^. . . (a, 6, . . .distinct primes), n=0(n)+2a»-V(^) +2a-V-^(/,(^) + . . . -\-a-'b'-'. . .. Moret-Blanc^^ proved the latter by noting that the first sum is the num- ber of integers < n which are divisible by a single one of the primes a, 6, ... , the second sum is the number of integers < n divisible by two of the primes, . . ., while a''~^6^~\ . . is the number of integers <n divisible by all those primes. H. J. S. Smith^^ considered the m-rowed determinant A„, having as the element in the ith. row and jth column the g. c. d. {i, j) of i, j. Let li = m, "Zahlentheorie, §11, 1863; ed. 2, 1871; ed. 3, 1879; ed. 4, 1894. »*Saggio di una introd. all'arit. trascendente, Treviso, 1867, 23-26. «BuU. Soc. Math. France, 1, 1872-3, 77. »«Jour. fiir Math., 77, 1874, 289-91. »'Nouv. Ann. Math., (2), 14, 1875, 276. "/bid., p. 374. L. Gegenbauer, Monatsh. Math. Phys., 4, 1893, 184, gave a generahzation to primary complex numbers. »»Proc. London Math. Soc, 7, 1875-6, 208-212; CoU. Papers, 2, 161. Chap. V] EulER's </)-FuNCTION. 123 I2, I3,... be those divisors of m = p V • • • ^'^ which are given by the expansion of the product 0(m) = (p--p"-i). . .(r-r-i)=?i-Z2+?3-. . .-^v It is proved that cl>{m, k) = {k, k)-{k, k)+ . . . -{I, k) [called Smith's function by Lucas/^ p. 407] is zero if k<m, but equals <^(m) if ^ = m. Hence if to the mth column of A^ we add the columns with indices I3, I5,... and subtract the columns with indices I2, Z4, . . . , we obtain an equal determinant in which the elements of the mth column are zero with the exception of the element <^(m). Hence A^=A^_i0(m), so that (6) A^=<^(l)0(2)...0(m). If we replace the element 5 = {i, j) by any function /(5) of 5, we obtain a determinant equal to i^(l) . . .F{m), where nm)=/W-2/g)+S/g)-.,.. Particular cases are noted. For /(§) = S''', F{m) becomes Jordan's^"" func- tion Jkitn). Next, if /(5) is the sum of the kth. powers of the divisors of 5, then F{m)='m!'. Finally, if /(6) = 1^+2^+ . . . +5^, it is stated erroneous- ly that F{m) is the sum (i>k{'m) of the /bth powers of the integers ^m and prime to m. [Smith overlooked the factors o!', oJ^h^, ... in Thacker's^^° first expression for <l>k{n), which is otherwise of the desired form F{n). The determinant is not equal to 4>k{\) . . .^^kim), as the simple case k = l, w = 2, shows.] In the main theorem we may replace 1,. . ., m by any set of distinct numbers jui, . . . , jU;„ such that every divisor of each ju, is a number of the set; the determinant whose element in the ith. row and jth column is/(5), where 5 = (jUi, /xy), equals F()Ui) . . .F{fx^. Examples of sets of ^t's are the numbers in their natural order with the multiples of given primes rejected; the numbers composed of given primes; and the numbers without square factors. R. Dedekind^° proved that, if n be decomposed in every way into a product ad, and if e is the g. o,. &. oi a, d, then S|.#,(6) = nn(l+^), where a ranges over all divisors of n, and p over the prime divisors of n. P. Mansion^^ stated that Smith's relation (6) yields a true relation if we replace the elements 1,2,. . .of the determinant A^ by any symbols Xi,X2,. . ., and replace 0(m) by Xi^—Xi^-\-Xi — .... [But the latter is only another form of Smith's F{m) when we write x^ for Smith's /(5), so that the generali- zation is the same as Smith's.] "Jour, ftir Math., 83, 1877, 288. Cf. H. Weber, Elliptische Functionen, 1891, 244-5; ed. 2, 1908 (Algebra III), 234-5. ^Messenger Math., 7, 1877-8, 81-2. 124 History of the Theory of Numbers. [Chap, v P. Mansion^^ proved (6), showing that </>(m, k) equals <f){m) or 0, accord- ing as m is or is not a divisor of k. [Cf. Bachmann, Niedere Zahlentheorie, I, 1902, 97-8.] He repeated his*^ ''generalization." He stated that if a and b are relatively prime, the products of the 0(a) numbers <a and prime to a by the numbers <b and prime to b give the numbers <ab and prime to ab [false for a =4, 6 = 3; cf. Mansion^]. His proof of (4) should have been credited to Catalan." E. Catalan^^ gave a condensation and shght modification of Mansion's*' paper. C. Le Paige {ibid., pp. 176-8) proved ]\Iansion's^ theorem that every product equals a determinant formed from the factors. P. IMansion"" proved that the determinant |cy| of order n equals rriX2- • x^ if Cij=Xxp, where p ranges over the divisors of the g. c. d. of i, j. To obtain a "generahzation" of Smith's theorem, set Zi = Xi, Z2 = Xi-{-X2,. . ., Zi=J>Xi, where d ranges over all the divisors of i. Solving, we get where the Vs are defined above.^^ Thus each Cy is a z. For example, if n = 4, - 21 21 2l 2l Zi 22 2l 22 Zl 2l 23 2i 21 22 2l 24 Cii = For Zi = i, Xi becomes 4>{i) and we get (6). [As explained in connection with Mansion's*^ first paper, the generaUzation is due to Smith.] J. J. Sylvester^^ called (/)(7i) the totient T{n) of n, and defined the totitives of n to be the integers < n and prime to n. F. de Rocquigny^^ stated that, if ^"(A^) denotes <l)\(i>{N)\ , etc., if A^ is a prime and m>2, p>2. He stated incorrectly (ibid., 50, 1879, 604) that the number of integers ^ P which are prime to N = a^b^ ... is P(l — 1/a) (1-1/6).... A. Minine*^ noted that the last result is correct for the case in which P is divisible by each prime factor a, b,. . . oi N. He wrote symbolically nE— for [n/x], the greatest integer ^n/x. By deleting from 1, . . ., P the [P/a] numbers di\'isible by a, then the multiples of 6, etc., we obtain for the number of integers ^ P which are prime to N the expression [equivalent to (5)]. If N, N', N", ... are relatively prime by twos, ♦^Annalea de la Soc. Sc, Bru.\elles, 2, II, 1877-8, 211-224. Reprinted in Mansion's Sur la th^orie des nombres, Gand, 1878, §3, pp. 3-16. "Nouv. Corresp. Math., 4, 1878, 103-112. «Bull. Acad. R. Sc. de Belgique, (2), 46, 1878, 892-9. «Amer. Jour. Math., 2, 1879, 361, 378; Coll. Papers, 3, 321, 337. Nature, 37, 1888, 152-3. «Le8 Mondes, Revue Hebdom. des Sciences, 48, 1879, 327. "Ibid., 51, 1880, 333. Math. Soc. of Moscow, 1880. Jour, de math. 616in. et sp^c, 1880, 278. Chap. V] EulEr's 0-FunCTION. 125 cf>{N)p-(t>iN')prcf>{N")p.. . . =<f>(NN'Nr . .)p'P'P". . .. E. Lucas^^ stated and Radicke proved that a=l fc=2 0=1 k=2 if ^(a, n) is the number of integers > a, prime to a and ^ n. H. G. Cantor^^ proved by use of ^-functions that Svo^-Vr". . .vU24>M4>{vi) . . .0(^-i) ^n", summed for all distinct sets of positive integral solutions Vq,..., v^^i of Vq. . .Vp=n, and noted that this result can be derived from the special case (4). 0. H. MitchelP° defined the a-totient Taik) of k^a'b''. . . (where a,h,. .. are distinct primes) to be the number of integers <k which are divisible by a, but by no one of the remaining prime factors 6, c, ... of k. Similarly, the a6-totient Tabik) of k is the number of integers <k which are divisible by a and h, but not by c, . . . ; etc. If /c = a'6V, tM =a'-V(&V), Ta,{k)=a'-'h--'<t>{c^), TaUk)=a'-'b^-'c'-\ <f>ik) +2t,(A;) +2ra,(/c) +Tadk) = k. 3 3 If a contains the same primes as s, but with the same exponents as in k, so that o- = a' if s = a, it is stated (p. 302) that ■w=i*a- C. Crone" evaluated (^(n) by an argument valid only when n is a product of distinct primes Pi,...,Pq. The number of integers <n having a factor in common with n is then A=2(ii-l)-s(^-l) + ...+(-l).2(^^ 1). The sum of the second terms of each sum is -(0+a)- -(-^)'G^)=-i-(-i)"- Hence the number of integers <n and prime to n is n-\-A=n-l^—^^— — . . . -(-1)«S +(-!)« Pi V\V2 Pl-Pg-l provided n = pi. . .pg. [To modify the proof to make it vahd for any n, we need only add to A the term and hence replace (-1)*' by (-l)%/(pi. . .p^) in n-l-A.] *8Nouv. Corresp. Math., 6, 1880, 267-9. Also Lucas, '^ p. 403. "Gottingen Nachrichten, 1880, 161; Math. Ann., 16, 1880, 583-8. "Amer. Jour. Math., 3, 1880, 294. "Tidsskrift for Mathematik, (4), 4, 1880, 158-9. 126 History of the Theory of Numbers. [Chap. V Franz Walla^- considered the product P of the first n primes > 1. Let a*i, . . . , X, be the integers <P/2 and prime to P, so that v=4>{P)/2. Then, if n>2, half of the x's are =1 (mod 4) and the others are =3 (mod 4). Also, the absolute values of \P — 2Xj (j = 1, . . . , v) are the a:'s in some order. Half of the a:'s are <P/4. J. Perott^^ proved that the context showing that the summations extend over all the primes p< for which Kpi^N [Lucas"]. He proved that lim ^jN) _ 3 iV = «) m 7r2 and gave a table showing the approximation of SN^/tt^ to $(iV) for iV^ 100. The last formula, proved earlier by Dirichlet^^ and Mertens,^^ was proved by G. H. Halphen^^ by the use of integrals and f -functions. Sylvester^*" defined the frequency 5 of a divisor d of one or more given integers a, h, . . . , I to he the number of the latter which are divisible by d. By use of (4) he proved the generalization X8(f>{d)=a+h-\-...-\-l. d J. J. Sylvester^^ stated that the number of [irreducible proper] fractions whose numerator and denominator are ^j is T{j) = <f){l)+ . . . +<t>{j), and that 3 PoT •'" f-'/^] n^-\-i stU^s S(/,(t)=^, k=i L/CJ ^=1 i=i ^ whence T{j)/f approximates S/tt^ as j increases indefinitely. If u{x) denotes the sum of all the integers <x and prime to x, and if U(j)=u(l)+ . . .-{-u(j), then U{j) is the sum of the numerators in the above set of fractions, and* When j increases indefinitely, U{j)/f approximates I/tt^. For each integer n^ 1000 the values of (^(n), T{n), Srr/ir'^ are tabulated, Sylvester^^ stated the preceding results and noted that the first formula is equivalent to !I3^^^) l(/+i). "Archiv Math. Phys., 66, 1881, 353-7. "Bull, des Sc. Math, et Astr., (2), 5, I, 1881, 37-40. "Comptes Rendua Paris, 96, 1883, 634-7. "«Amer. Jour. Math., 5, 1882, 124; Coll. Math. Papers, 3, 611. "Phil. Mag., 15, 1883, 251-7; 16, 1883, 230-3; Coll. Math. Papers, 4, 101-9. Cf. Sylvester." "Comptes Rendus Paris, 96, 1883, 409-13, 463-5; Coll. Math. Papers, 4, 84-90. Proofs by F. Rogel and H. W. Curjel, Math. Quest. Educ. Times, 66, 1897, 62-4; 70, 1899, 56. *With denominator 3, but corrected to 6 by Sylvester," which accords with Ces&,ro." The editor of Sylvester's Papers stated in both places that the second member should be jij + l){2j+l)/12, evidently wrong for; =2. Chap. V] EulEE's (^-FUNCTION. 127 E. Ces^ro^^ proved that, if / is any function, X^i^Xx^Fin), F(n)^S/(d), n = ll X 71=1 where d ranges over the divisors of n. For / = 0, we have F(x)=x and obtain Liouville's^^ first formula. By the same specialization (p. 64) of another formula (given in Chapter X on sums of divisors^^), Cesaro derived the final formula of Liouville.^^ If (n, j) is the g. c. d. of n and j, then (p. 77, p. 80) Mn, j) =S#Q), X-^ = h:d<t>{d), X4>{n, j) =S(^(d)0(-). If (p. 94) p is one of the integers a, /3, . . . ^ n and prime to n, S^(a)F(a) =SG(a)/(a), /^(a:)^S/(d), G(p)^S^(pa), a where d ranges over the divisors of x. For g{x) = l, this gives S/(a)0(n, n/a)=SF(a), a where (p. 96) </)(n, x) is the number of integers ^x and prime to n. Cesaro (pp. 144-151, 302-3) discussed and modified Perott's^^ proof of his first formula, criticizing his replacement of [n/k] by n/k for n large. He gave (pp. 153-6) a simple proof that the mean^^ of <^(n) is Qn/w^ and reproduced the proofs by Dirichlet^^ and Mertens,^^ the last essentially the same as Perott's. For Kw) = l + l/2"^+l/3"+. . ., s4r(^>l), 2i 2-i-(m>l), 2-^ a"* ' '' a' a'"</)(a) ' '' 0(a) equal asymptotically (pp. 167-9) f(m)/r(m+l), (6 1og7i)/7r^ r(m+l), log n. As a corollary (p. 251) to Mansion's^^ generalization of Smith's theorem we have the result that the determinant of order n^, each element being 1 or 0 according as the g. c. d. of its two indices is or is not a perfect square, equals ( — 1)"+^+- , where pV- • • is the value of n\ expressed in terms of its prime factors. Ces^ro^* considered any function F{x, y) of the g. c. d. of x, y, and the determinant A„ of order n having the element F{Ui, u/) in the ith. row and ith column, where Ui,...,Un are integers in ascending order such that each divisor of every Ui is a u. Employing the function ix{n) [see Ch. XIX], he noted that i nO^)F(u„ud=f{u,) or 0, "M6m. Soc. R. Sc. de LiSge, (2), 10, 1883, No. 6, 74. "Atti Reale Accad. Lincei, (4), 1, 1884-5, 709-711. 128 History of the Theory of Numbers. [Chap, v according as u^ is or is not divisible by w„, while fix) =Kx)F{l) +M (2) F{2) +M (f) F{S) + . . . . Hence if we multiply the elements of the ith colmnn of A„ by fx{ujui) and add the products to the last column for 2 = 1, . . ., n — 1, the new elements of the last colunm are zero except the final element, which is/(w„). Thus A,=/(ii„)A„_i=/(ui)/(w2) . . .fM- [These results are due to Smith,^^ not merely the case Ui = i a.s stated.] Cesaro^^ noted that |wy|=/(l) . . ./(n) if «,= s/wft0ft.Q, where the function h has the property that the determinant with the general element h{i/j) is unity, and similarly for hi. Cesaro®° gave the last result for the case in which h{x)=hi{x) = l or 0 according as x is or is not an integer. P. Mansion (p. 250) stated that he** had employed a similar proof. Ces^ro®^ duphcated his paper^^ and transformed its final result into /(l)/(2) . . .fin) F[i,j] F\nl) where [i, j] = ij/{i, j) is the 1. c. m. of i, j, and F{x) is a function such that F{xy)=F{x)F{y). In particular, if F{x) = \/x, then J{x)=4){x)Tr{x)/x^, where 7r(n) is the product of the negatives of the distinct prime factors of n. Hence |Ki]|n=0(l)...0(n)7r(l)...7r(n). Ces^ro^^ investigated the r-rowed minors of the n-rowed determinant whose general element is F{b)=F{i, j), where 5 is the g. c. d. of i, j. It is shown that the {n — v)-Towed determinant whose general element is F{i-\-v, j-\-v) is equal to the sum of certain products of /(I), . . ., f{n) taken n — v&i a time, the case v = Q being Smith's theorem. Here /(x) =^J^F{j), Fix) =S/(d) (d divisor of x). Ces^ro^^ stated that the (n — l)-rowed determinant, whose general ele- ment Uij equals the number of divisors common to i+1 and j + 1, equals the number of integers ^ n deprived of square factors > 1 . "Atti. Reale Accad. Lincei, (4), 1, 1884-5, 711-5. •"Mathesis, 5, 1885, 248-9. "Giornale di Mat., 23, 1885, 182-197. "Annales de I'^cole normale sup., (3), 2, 1885, 425-435. "Nouv. Ann. Math., (3), 4, 1885, 56. Chap. V] EulEE's (^-FUNCTION. 129 Ces^ro" employed F(n)=S/(d), G(n)='Zg{d), where d ranges over the divisors of n, and proved that 0 G(l) G{2) ... G{n) G{\) F(l,l) F(l,2) ... F(l,n) -_j(i)..j(^)p'(^), G{n) F{n,l) F{n,2) Fin, n) In particular, if F{n) is the number of divisors of n and if G{n) is the number of prime divisors of n, the determinant, apart from signs, equals the number of primes ^ n. E. Cesaro^^ wrote (a, b) for the g. c. d. of a, h. If F{n) =S/(d), where d ranges over the divisors of n, then XF\(n,i)\=i:f(d)N/d. In particular, if /,(n) is the number of irreducible fractions ^e of denomi- nator n. IXn)=i:[j]n{d), S7,(d) = [ne]. The last formula, due to Laguerre,^^ follows by inversion (Ch. XIX), and directly from the fact that I^d) is the number of the first [ne] integers which with n have the g. c. d. n/d. The number of irreducible fractions ^ e of denominator ^n is $e(^) =-f «(!)+••• +-^.(^). We have 00 \n/j] O, ^M = SmO') S [ie], Imi $,(n)/n2 = -2 (€>0), j = l 1=1 n=oo T due to Sylvester^^ for € = 1. Let (^^g^(n) be the sum of the j'th powers of the numerators of the irreducible fractions < e of denominator n. Set Then $« in) = S </)(? ii) , sXn) = S ^^ 1=1 »-i i=l LzJ i.i which generalizes the two formulas of Sylvester .^^ Also, $^;^ (w) = — — — — — , asymptotically. TT V-\-l V-\-2 Ces^ro^^" factored determinants of the tj^e in his paper,^^ the function F now being such that Fixy)/ \Fix)Fiy)\ is a function of the g. c. d. oi x, y. L. Gegenbauer®^'' gave a complicated theorem involving several general functions, special cases of which give Sylvester's^^ two summation formulas. "Nouv. Ann. Math., (3), 5, 1886, 44-47. "Annali di Mat., (2), 14, 1886-7, 143-6. ""Giornale di Mat., 25, 1887, 18-19. «5fcSitzungsber. Ak. Wiss. Wien (Math.), 94, 1886, II, 757-762. 130 History of the Theory of Numbers. [Chap, v P. S. Poretzky^^ gave a formula for the function \l/{m) whose values are the 4>{7?i) integers <?fi and prime to m. For the case w = 2-3-5. . .p, where p is a prime, lpi-2 Pi J where K is an integer. Application is made to the finding of a prime exceeding a given number, and to a generalization of the sieve of Eras- tosthenes. E. Ces^ro^^ gave a very simple proof of the known fact that 2 2' n-00 n^ T which he expressed in words by saying that 0(n) is asymptotic to 6n/7r^ (not meaning that the limit of 4>{n)/n is G/tt"). On the distinction between asymptotic mean and median value, see Encyclop^die des sc. math., I, 17 (vol. 3), p. 347. Ces^ro^^ noted that if F{i, j) is a function of the g. c. d. of i, j, then Q=SF(i, j) XiXj {i, j = l,..., n) becomes q='Lf{i)yi^ by the substitution yk = Xk-{-X2k-\-Xsk-\- ■ ■ -, provided F{n) =2/(d), d ranging over the divisors of n. Since the determinant of the substitution is unity, the discriminants of Q and q are equal. Hence we have the theorem of Smith.^^ A gen- eralization is obtained by use of 2F(e„ e^XiXj, where the numbers ei, C2, . . . include the divisors of each €. E. Catalan^^ proved that, if d ranges over the divisors of iV = a"6^ . . . , E. Busche^° derived at once from Dirichlet's^^ formula the result S0(x))p(^)+p(^) + ...(=Snn', j=l \x/ \x / where p(a) =a — [a]. The case n = n' =n" = . . . leads to i:4>{x) = {v-\)n^, where x takes all values for which p{n/x)>p{vn/x). If we take n = l and add </)(!) = 1, we get (4) for N = v. Next, S0(a;) =rr'5", where x takes all values for which yJ±zi^,Q<yyi (y=i,...,.;,'=i,...,.'). r-\-r \xy r r 6«Math. phys. soc. Kasan, 6, 1888, 52-142 (in Russian). "Comptcs Rendus Paris, 106, 1888, 1651; 107, 1888, 81, 426; Annali di Mat., (2), 16, 1888-9, 178 (discussion with Jensen on terminology). •8Atti Rcale Accad. Lincei, Rendiconti, 2, 1888, II, 56-61. "M6m. Soc. Sc. Li^ge, (2), 15, 1888, No. 1, pp. 21-22; Melanges Math., Ill, No. 222, dated 1882. "Math. Annalen, 31, 1888, 70-74. Chap. V] EulEr's 0-FuNCTION. 131 For d = n,r' = l,r = v — l, this becomes the former result ; f or r = r ' = 1 , 5 = n, it becomes 20 (x) =n^, where x takes the values for which p(n/rc)^ 1/2. H. W. Lloyd Tanner^^ studied the group G of the totitives of n (the integers <n and prime to n), finding all its subgroups and the simple groups whose direct product is G. E. Lucas^^ proved that, in an arithmetical progression of n terms whose common difference is prime to n, there are (ji{d) terms having with n the g. c. d. n/d. If, when d ranges over the divisors of n, Xxpid) =n for every integer n, then (p. 401) \p{n)=(^{n), as proved by using n = l, a, a^,. . ., and n = ah, a^b, . . . , where a,h,. . . are distinct primes. He gave (pp. 500-1) a proof of Perott's^^ first formula by induction from N — 1