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) = (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 = - -,.. . — 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 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), = 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, = 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 = 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 = 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 = 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^ 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 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] = (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"'~^-\- . . . = (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 = 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 = or 2, r = w; if w = 1, r = 4iy if m>2. Hence r = 2" if m = 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 = 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 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 (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 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 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) = (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, - 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 = {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 (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) -2 i;--i) V 2 ; \ 3 ; ("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) = (mod p), S2„+i(p — 1)=0 (modp^). But if 2n is divisible by p — 1, S2n(p-1)=-1, S2„+i(p-l) = (modp), Sp(p - 1) = (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 — p — j,^^+.-l-+-i-+^+-^+-l-+... (p=10fc+7). p — 1 p—2 p — S p—4: p — 5 p — 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 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 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^- p roved 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 = (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 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 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 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 to N, communicated to him by J. Hammond. The name " indicateur "of n is given (preface, xv) to <f){n) [Prouhet^sj. C. Moreau (cf . Lucas,'^^ 501-3) considered the C{n) circular permutations of n objects of which a are alike, (3 alike, . . . , X alike. Thus, if a = 2, /3 = 4, the C(6) = 3 distinct circular permutations are aahbhb, ababbb, abbabb. In general, ^^^^=n^^^^^(a/d)!...(X/d)r where d ranges over the divisors of the g. c. d. of a, jS, . . . , X. In the example, d = 1 or 2, and the terms of the sum are 15 and 3. P. A. MacMahon^^ noted that C(n) = 1 if n = a, so that we have formula (4). His expression for the number of circular permutations of p things n at a time is quoted in Chapter III on Fermat's theorem. A. Berger^^" evaluated S^il k'^%{k). For a = 2 the result is 3nV7rH \n log n, where X is finite for all values of n. E. Jablonski'^^ considered rectilinear permutations of indices a, . . ., X, with the g. c. d. D. Set a = a'D,- • .,\ = \'D, a+ • . .+X = m = m'Z). Then the number of complete rectilinear permutations of indices a'n, . . . , \'n is P{n)=- ^'^'''^' {a'n)\...{\'n)\ The number of complete circular permutations is where d ranges over the divisors of D. If Q{D/d) is the number of rectilinear permutations of indices a, . . . , X which can be decomposed into d identical portions, ^Q(D/d)=P{D). Also 'iProc. London Math. Soc, 20, 1888-9, 63-83. "Theorie des nombres, 1891, 396-7. The first theorem was proved also by U. Concina, II Boll, di Matematica, 1913, 9. "Proc. London Math. Soc, 23, 1891-2, 305-313. '»«Nova Acta Regiae Soc. Sc. UpsaUensis, (3), 14, 1891, No. 2, 113. '♦Comptes Rendus Paris, 114, 1892, 904-7; Jour, de Math., (4), 8, 1892, 331-349. He proved Moreau's'* formula for C{n). 132 History of the Theory of Numbers. [Chap, v 2Q©d'=2pg)/,(d), where Jt{d) is Jordan's-"" function. S. Schatunowsky"* proved that 30 is the largest number such that all smaller numbers relatively prime to it are primes. He employed Tcheby- chef's^" theorem of Ch. XVIII that, if a> 1, there exists at least one prime between a and 2a. Cf. Wolfskehl,^^ Landau,^^. 113 Maillet,^^ Bonse/"« Remak.^^2 E. W. Da\ds''^ used points with integral coordinates ^0 to visualize and prove (1) and (4). K. Zsigmondy^^ wrote r, for the greatest integer ^ r/s and proved that, if a takes those positive integral values ^r which are di\asible by no one of the given positive integers rii, . . . , n^ which are relatively prime in pairs, r rn rnn' 2/(a) = S f{k) -S S f{kn) +22 f{knn') -..., k=l n k=l n, n' k = l n, n',. . . ranging over the combinations of rii,. . ., n^ taken 1, 2, . . . at a time. Taking /(A:) = 1, we obtain for the number (f>{r; rii, . . . , nj of integers ^r, which are divisible by no one of ni, . . . , n^, the expression (5) obtained by Legendre for the case in which the n's are all primes. By induction from p to p+1, we get + 2<^(r^;ni,. ..,n,)-..., p r=4>{r] ni, . . . , n,)+ 2 </)(r„<; rii, . . . , n,_i, n,+i, . . . , nj t=i + 2<^(r„„/; riiST^n, n') + r = 20(r,;ni,.. ., n,), where c ranges over all combinations of powers ^r of the n's. The last becomes (4) when ni,. . ., n^ are the different primes di\ading r. These formulas for r were deduced by him in 1896 as special cases of his inversion formula (see Ch. XIX). J. E. Steggair^ evaluated </)(n) by the second method of Crelle.^^ P. Bachmann^^ gave an exposition of the work of Dirichlet,^^ Mertens," Halphen^ and Sylvester^^ on the mean of <p{n), and (p. 319) a proof of (5). L. Goldschmidt^" gave an evaluation of <j){n) by successive steps which may be combined as follows. Let p be a prime not dividing k. Each of "Spaczinakis Bote (phys. math.), 14, 1893, No. 159, p. 65; 15, 1893, No. 180, pp. 276-8 (Russian). "Amer. Jour. Math., 15, 1893, 84. "Jour, fur Math., Ill, 1893, 344-6. "Proc. Edinburgh Math. Soc, 12, 1893-4, 23-24. "Die Anab-tische Zahlentheorie, 1894, 422^30, 481-4. ««Zeitschrift Math. Phys., 39, 1894, 203-4. Chap. V] EulER's 0-FuNCTION. 133 the <j)(k) integers ^k and prime to k occurs just once among the residues modulo k of the integers from Ik to {l-{-l)k; taking 1 = 0, 1,. . ., p — l, we obtain this residue p times. Hence there are p({>{k) numbers ^pk and prime to k. These include <j){k) multiples of p, whence 4){pk) = {p — l)(p{k). For, if r is one of the above residues, then r, r+k,. . ., r-{-{p — l)k form a complete set of residues modulo p and hence include a single multiple of p. Hence </)(a6c...) = (a-l)(6-l)(c-l)..., if a, b, c, . . . are distinct primes. Next, for n = a^h^ . . . , we use the sets of numbers from lab. . .to (l-]-l)ab. . ., for Z = 0, 1,. . ., a°-~^b^~^ . . . — 1. Borel and Drach^^ noted that the period of the least residues of 0, a, 2a,... modulo N, contains N/8 terms, if d is the g. c. d. of a, iV; conversely, if d is any divisor of N, there exist integers such that the period has d terms. Taking a = 0, 1,. . ., iV — 1, we get (4). H. Weber ^^ defined 0(n) to be the number of primitive nth roots of unity. If a is a primitive ath root of unity and /3 a primitive 6th root, and if a, b are relatively prime, a/3 is a primitive a6th root of unity and all of the latter are found in this way. Hence 0(a6) =</)(a)0(6). This is also proved for relatively prime divisors a, 6 of n — 1, where n is a prime, by use of integers a and jS belonging to the exponents a and b respectively, modulo n, whence a^ belongs to the exponent ab. K. Th. Vahlen^^ proved that, if la.^in) is the number of irreducible frac- tions between the limits a and /3, a>j8^0, with the denominator n, S/„.,(d) = [(a-^)n], ij~~^h,,{k)=i[{a-m], where d ranges over the divisors of n. For /3 = 0, the first was given by Laguerre.^^ Since /i,o(^)=<^(^), these formulas include (4) of Gauss and that by Dirichlet.2i J. J. Sylvester^ corrected his^^ first formula to read k^[k\ = 2Ui]H[i]t ^^U), r[n]=0(l)+. . .+c^([n]), and proved it. By the usual formula for reversion, A. P. Minin^^ solved ^(f){m)=R for m when R has certain values. The equation determines the number of regular star polygons of m sides. Fr. RogeP® gave the formula of Dirichlet.^^ *'Introd. thdorie des nombres, 1895, 23. «Lehrbuch der Algebra, I, 1895, 412, 429; ed. 2, 1898, 456, 470. "Zeitschrift Math. Phys., 40, 1895, 126-7. "Messenger Math., 27, 1897-8, 1-5; Coll. Math. Papers, 4, 738-742. "Report of Phys. Sec. Roy. Soc. of Friends of Nat. Sc, Anthropology, etc. (in Russian), Mos- cow, 9, 1897, 30-33. Cf. Hammond."^ «»Educat. Times, 66, 1897, 62. 134 History of the Theory of Numbers. [Chap, v RogeP^ considered the number of integers v<n such that v and n are not both di\'isible by the rth power of a prime. Also the number when each prime factor common to v and n occurs in them exactly to the rth power. I. T. Kaplan published at Odessa in 1897 a pamphlet in Russian on the distribution of the numbers relatively prime to a given number. M. Bauer^^ proved that, for x prime to in, kx-\-l represents \p{m) 4>{d^d2) <j) integers relatively prime to m and incongruent modulo m, where di is the g. c. d. {k, m) of k, m, and c?2= (/, m), {di, do) = 1, w^hile ^W=0Wn{i-^} is the number of incongruent integers prime to m = pi^ . . . p/* which are represented by kx+l when k, I, x are prime to 7n. Of those integers, \p{m)/\l/{pi. . .pr) are di\'isible only by the special prime factors Pi, . . ., Pr of m. J. de Vries^^" proved the first formula of Dirichlet's.^^ C. Moreau^^ evaluated 4){n) by the method of Grunert.^^ E. Landau^° proved that „=i<^(n) 27r^ \ *= pp2_p+iy where e is of the order of magnitude of x~^ log x, C is Euler's constant, and f is Riemann's ^-function. P. WolfskehP^ proved by Tchebychef's theorem that the 0(n) integers <n and prime to n are all primes only when n = 1, 2, 3, 4, 6, 8, 12, 18, 24, 30, [Schatunowsky.'°] E. Landau^^ gave a proof, without the use of Tchebychef's theorem, by finding a lower limit to the number of integers k ha\dng no square factor >1, where t^k>Dt/S. E. Maillet,^^ by use of Tchebychef's theorem, proved the same result and the generaUzation : Given any integer r, there exist only a finite number of integers N such that the <t>{N) integers <A^ and relatively prime to N contain at most r equal or distinct prime factors. Alois Pichler^'* noted that (}>(x)=n has no solution if n is odd and >1; while (i)(x) =2" has the solutions x = 2''bc. . . (a = 0, 1, . . ., n + 1) if «^Sitzungsber. Bohm. GeseU., Prag, 1897; 1900, No. 30. "Math. Natur. Berichte aua Ungam, 15, 1897, 41-6. ""K. Akad. Wetenschappen te Amsterdam. Verslagen, 5, 1897, 222. "Nouv. Ann. Math., (3), 17, 1898, 293-5. ' »»G6ttingen Nachrichten, 1900, 184. "L'interm^diaire des math., 7, 1900, 253-4; Math. Ann., 54, 1901, 503-4. "Archiv Math. Phys., (3), 1, 1901, 138-142. »»L'interm6diaire des math., 7, 1900, 254. "Ueber die Auflosimg der 01. <p{x) =n. . ., Jahres-Bericht Maximilians-Gymn. in Wien, 1900-1, 3-17. Chap. V] EulEK's 0-FuNCTION. 135 6 = 2^^+1, c = 22''+l,... are distinct primes and 2^+2"^+. . . =n or n — a+1 according as a = or a>0. When g- is a prime >3, <f){x) = 2q'' is impossible if p = 2q^-\-l is not prime; while if p is prime it has the two solutions p, 2p. If g = 3 and p is prime, it has the additional solutions 3"+\ 2-3''"^^ Next, 4>{x)=2''q is impossible if no one of p^ = 2''~''q-\-l{v = 0, 1,. . ., n — 1) is prime and q is not a prime of the form 2*+l, s = 2^^n; but if q is such a prime or if at least one p^ is prime, the equation has solutions of the respective forms bq^, where (/)(6) =2""*; ap„ where 0(o) =2". Finally, (f>{x)=2qr has no solution if p = 2gr+l is not prime and r9^2q-\-l. If p is a prime, but r9^2q-{-l, the two solutions are p, 2p. If p is not prime, but r = 2g+l, the two solutions are r^, 2r^. If p is prime and r = 2g-|-l, all four solutions occur. There is a table of the values n<200 for which (f){x)=n has solutions. L. Kronecker®^ considered two fractions with the denominator m as equivalent if their numerators are congruent modulo m. The number of non-equivalent reduced fractions with the denominator m is therefore 4){m). If m = m'm", where m' , m" are relatively prime, each reduced fraction r/m can be expressed in a single way as a sum of two reduced partial fractions r' /m', r' /m". Conversely, if the latter are reduced fractions, their sum r/m is reduced. Hence 0(m) =</)(m')</)(m"). The latter is also derived (pp. 245-6, added by Hensel) from (4), which is proved (pp. 243-4) by considering the g. c. d. of n with any integer ^n, and also (pp. 266-7) by use of infinite series and products. Proof is given (pp. 300-1) of (5). The Gaussian median value (p. 334) of (f>{n)/n is Q/w^ with an error whose order of magnitude is l/\/n, provided we take as the auxiliary number of values of 4>{n)/n a value of the order of magnitude ^yn log^ n. E. B. Elliott^^ considered monomials n = p'^q^. . . in the independent variables p,q,.... In the expansion of n(l — l/p)"'(l — l/g)"* . . . , the aggre- gate of those monomial terms whose exponents are all ^0 is denoted by Fm{n). Define iJi{p'q\ . .) to be zero if any one of r, s, . . . exceeds 1, but to be ( — 1)' if no one of them exceeds 1, and t of them equal 1. Then (7) F^_i(n) =Si^,,(d), F^^,{n) =Sm Q) F^((i), where d ranges over the monomials pV- • • with O^a^a, 0^/3^?),.... Henceforth, let p, q,... be distinct primes. Then Fi{n)=(j){n), while F_i(n) is the sum o-(n) of the divisors of n. In (7), d now ranges over all the divisors of n, and ai(/c) is Merten's function [Inversion]. For m = 0, (72) gives the usual expression for </)(n), while (7i) defines o-(n). For m = l, (7i) becomes (4). If T''^\n) ^T(n) is the number of divisors d of n, write r(2)(n)=ST(d),. . ., T(^>(n)=ST^^-i>(d). '^Vorlesungen iiber Zahlentheorie, I, 1901, 125-6. »«Proc. London Math. Soc, 34, 1901, 3-15. 136 History of the Theory of Numbers. [Chap, v Then Generalizing /x(s), let )u-*^(s) be zero if the expansion of the product n(l— p)*", extended over all primes p, does not contain a term equal to s, but let it equal the coefficient of s if s occurs in the expansion. Then i?,(n)=SdM'"Q) The 7i-rowed determinant in which the element in the rth row and sth column is F„_i(5), where 5 is the g. c. d. of r, s, is proved equal to F^(l) F^(2) . . .Fm{n), a generaUzation of Smith's^^ theorem. Finally, isf.,,Q)f_.(d)=siF,(d), the right member being T{n), 20(c?)/c?, lla{d)/d for r = 0, 1, —1. G. Landsberg^^" gave a simple proof of Moreau's"^ formula for the number of circular permutations. L. Carlini^^ proved Dirichlet's^^ formula by noting that (8) n(x''-l)=0 has unity as an n-f old root, while a root 7^ 1 of x'' — 1 is a root of [n/h] factors x"* — 1. Hence the 4){h) primitive roots of x^ = \ furnish <l>{h)[n/h] roots of (8). M. Lerch^^ found the number N of positive integers '^m which have no one of the divisors a, 6, . . . , k, I, the latter being relatively prime in pairs and ha\'ing m as their product. Let F{x) = 1 or 0, according as x is frac- tional or integral. Let L = ab. . .k. Then [Dirichlet^^] ^^m(Z-l)^ ,!/©-©-(-■) (-i) L E. Landau^^ proved that the inferior limit for a:= 00 of -<f>{x) log. log, X X is e~^, where C is Euler's constant. Hence <^(j) is comprised between this inferior limit and the maximum x — 1. R. Occhipinti^°° proved that, if aj is an nth root of unity, and if c?,i, • ■ • , dat are the divisors of i, n|s<^(di,)+a,S<^(d2i)+. . .+a/-4V(0| = i(-l)"-'n(n+l)n"-2. j-lU-l i-l i-l J ••"Archiv Math. Phys., (3), 3, 1902, 152-4. "Periodico di Mat., 17, 1902, 329. "Prag Sitzungsber., 1903, II. "Archiv Math. Phys., (3), 5, 1903, 86-91. "«Periodico di Mat., 19, 1904, 93. Handbuch,"' I, 217-9. Chap. V] Euler's 0-Function. 137 G. A. Miller^"^ proved (4) by noting that in a cyclic group G of order N there is a single cyclic subgroup of order d, a divisor of N, and it contains 0((i) operators of order d, while the order of any operator of G is a divisor of N. Thus (4) states merely that the order of G equals the sum of the numbers of the operators of the various possible orders. Next, (1) follows from an enumeration of the operators of highest period AB in a cyclic group of order AB, which is the direct product of its cyclic subgroups of orders A and B. Finally, if p is a prime, all the subgroups of a cyclic group of order p" are contained in its subgroup of order p"~\ whence <^(p") = p" — p"~^ G. A. Miller^°^ proved the last three theorems and the fact that 0(0 is even if Z>2 by means of the properties of the abelian group whose elements are the integers <m which have with m a g. c. d. equal to k. K. P. Nordlund^"^ proved 4){mn ...) = (m — l)(n — 1)..., where m, n,. . . are distinct primes, by writing down the multiples <mnp of m, the multi- ples of mn, etc., whence the number of integers Kmnp and not prime to it is mnp — l — {m — l){n — l){p — l), E. Busche^*^ treated geometrically systems (td) of four integers such that ad — hc>0, evaluated the number $(aS) of systems incongruent modulo S and prime to S, and generalized (4) to 2$(*S). L. Orlando^°^ showed that 0(m) is determined by (4) [Lucas''^]. H. Bonse^°^ proved Maillet's^^ theorem for r = l, 2, 3 without using •■Tcheby chef's theorem. His lemma was generalized by T. Suzuki. ^°^" J. Sommer^^^ gave without reference Crelle's^ final evaluation of (/)(n). R. D. CarmichaeP*'^ proved that if n is such that (l){x)=n is solvable there are at least two solutions x. He found solutions of </)(x) = 2" [in accord with Pichler^^] and proved that there are just n+2 solutions (a single one being odd) when n^31 and just 33 solutions when 32^ n^ 255. All the solutions of <^(x) = 4n — 2> 2 are of the form p", 2p", where p is a prime of the form 4s — 1 ; for example, if n = 5, the solutions are 19, 27 and their doubles. CarmichaeP°® gave a table showing every value of m for which 0(m) has any given value ^ 1000. A. Ranum^^^" would solve 4>{x) = n by resolving n in every possible way into factors no, ., n^, capable of being taken as the values of 0(2*"), 4>{pi'), . . ., 0(pA)) where 2, pi, . . ., p, are distinct primes. Then 2'^pi"'. . .p^^'r is a value of x. CarmichaeP^" gave a method of solving (j>(x)=a, based on the testing of the equation for each factor x of a definite function of a. M. Fekete^^^ considered the determinant pkn obtained by deleting the last row and last column of Sylvester's eliminant for a;''' — 1 = and a;** — 1 = "lAmer. Math. Monthly, 12, 1905, 41-43. "'Amer. Jour. Math., 27, 1905, 315. "'Nyt Tidsskrift for Mat., 16A, 1905, 15-29. iMJour. fiir Math., 131, 1906, 113-135. "»Periodico di Mat., 22, 1907, 134-6. iwArchiv Math. Phya., (3), 12, 1907, 292-5. i^^Tohoku Math. Jour., 3, 1913, 83-6. "'Vorlesungen liber Zahlentheorie, 1907, 5. "SBull. Amer. Math. Soc, 13, 1907, 241-3. "»Amer. Jour. Math., 30, 1908, 394-400. lo'^^Trans. Amer. Math. Soc, 9, 1908, 193-4. "«BuU. Amer. Math. Soc, 15, 1909, 223. "iMath. 6s Phys. Lapok (Math. Phys. Soc), Budapest, 18, 1909, 349-370. German transl., Math. Naturwiss. Berichte aus Ungam, 26, 1913 (1908), 196. 138 History of the Theory of Numbers. [Chap, v {k<n). Thus |p;tn| = 1 or according as k and n are relatively prime or not. Hence n n (t>{n) = 2 \pkn\, (t>i(n) = 2 fc|pt„|, A=l k=l where <i>i{n) is the sum of the integers ^n and prime to n. R. Remak^^- proved Maillet's^^ theorem without using Tehebychef's. E. Landau^^^ proved (5), Wolfskehl's^^ theorem and Maillet's^^ generali- zation. C. Orlandi^" proved that, if x ranges over all the positive integers for wliich [m/x] is odd, then 20(x) = (?w/2)'^ for 7fi even (Cesaro, p. 144 of this History), while 20 (x) = k^ for m = 2k — l. A. Axer^^° considered the system (P) of all integers relatively prime to the product P of a finite number of given primes and obtained formulas and asymptotic theorems concerning the number of integers ^x of (P) which are prime to x. Application is made to the probability that two numbers ^ n of (P) are relatively prime and to the asymptotic values of the number (i) of positive irreducible fractions with numerator and denominator in (P) and ^n and {ii) of regular continued fractions representing positive fractions m (P) with numerator and denominator S n. G. A. ]Miller^^^ defined the order of a modulo m to be the least positive integer h such that ab=0 (mod m). If p" is the highest power of a prime p dividing vi, the numbers ^7n whose orders are powers of p are km/p" (k = l, 2,. . ., p"). Hence l^kim/p-'i {ki = \,. . ., p-'i) form a complete set of residues modulo 7?i=Ilpi'i. If the orders of two integers are relatively prime, the order of their sum is congruent modulo 77i to the product of their orders. But the number of integers ^m whose orders equal m is (t>{7n). Hence (/)(np°) =n0(p°). Since all numbers ^m whose orders divide d, a di\'isor of 7n, are multiples of 7n/d, there are exactly d numbers ^m whose orders di\ide d, and (f){d) of them are of order d. Hence 7n = 'E4>{d). S. Composto^^^ employed distinct primes 7n, n, r, and the v=<t>{77in) integers P\,...,p^ prime to Tnn and ^ mn, and proved that Pi, Pi+7nn, pi+2mn, . . . , p,+(r- l)wn {i = l,. . .,v) include all and only the numbers rpi,. . ., rp, and the numbers not exceeding and prime to 7nnr. Hence 4>{vmr)=4>{m7i)-{r — \). A like theorem is proved for two primes and stated for any number of primes. [The proof is essentially Euler's^ proof of (1) for the case in which J5 is a prime not divid- ing a product A of distinct primes.] Next, if d is a prime factor of 7i, the integers not exceeding and prime to dn are the numbers ^ n and prime to n, together with the integers obtained by adding to each of them n, 2n, . . . , "2.\rchiv Math. Phys., (3), 15, 1909, 186-193. i^Handbuch. . .VerteUung der Primzahlen, I, 1909, 67-9, 229-234. "♦Periodico di Mat., 24, 1909, 17&-8. "'Monatshefte Math. Phys., 22, 1911, 3-25. "«.\mer. Math. Monthly, 18, 1911, 204-9. "'II Boll, di Matematica Gior. Sc.-Didat., 11, 1912, 12-33. Chap. V] EulEr's 0-FunCTION. 139 (d — l)n; whence 4>{dn) = d^{n) . Finally, let Pi, ..., Py be the v = (f){n) integers <n and prime to n. Then pi-\-kn (^ = l, . . .,v; k = 0, 1, . . .) give all integers prime to n; let Ph{n) denote the hth one of them arranged in order of magnitude. Then P,Xn)=kn-l (k^l), P,,+M=kn+pr {l^r^v-l, k^O). If h = kv-\-r, r<v, the sum of the first h numbers prime to n is where pi, . . . , p^ are the first r integers <n and prime to n. K. HenseP^^ evaluated <^(n) by the first remark of Crelle.^'^ J. G. van der Corput and J. C. Kuyver^^^ proved that the number /(a/4) of integers ^ a/4 and prime to a is N = \aJl{l — \/p) if a has a prime factor 4m+l, where p ranges over the distinct prime factors of a; but is N — 2^~^ if a is a product of powers of k prime factors all of the form 4m — 1. Also /(a/6) is evaluated. U. Scarpis^^'^ noted that 0(p" — 1) is divisible by n if p is a prime. Several writers^^^ discussed the solution of 4>{x)=4){y), where x, y are powers of primes. SeveraP^^ proved that (f){xy)>4>{x)4>{y) if x, y have a common factor. J. Hammond^^^ proved that there are ^^(n) — 1 regular star n-gons. H. Hancock^"^ denoted by ^{i, k) the number of triples {i, k, 1), {i, k, 2), . . . , {i, k, i) whose g. c. d. is unity. Let i = iid, k = kid, where ii, ki are relatively prime. Then ^{i, k)=ii(i>{d), $(/c, i)=ki(j}(d). A. Fleck^^^ considered the function, of m^Hp", <p,{m) = n|<^(p«) - (J)c/>(p"-^) +...+(- i)°(^y (p^-")}. Thus (f)o{'m) =4){'m), <^_i(m) = m, <^_2(w) is the sum of the divisors of m. Also S 4>k{d)=<i>k-i{m), (f>kimn)=(l)k{ni)(t)k{n), d:m if m, n are relatively prime. For f (s) =2m~*, ^ (f)k-i{m) ^ 4>k{m) 0.(p)=p-CI^), 0.(p^)=p^-(^l>+Ct')' ■•' </).(p'+'+o=p''(p-l)'^'. "sZahlentheorie, 1913, 97. "^Wiskundige Opgaven, 11, 1912-14, 483-8. i^'oPeriodico di Mat., 29, 1913, 138. i2iAmer. Math. Monthly, 20, 1913, 227-8 (incomplete); 309-10. li'^Math. Quest. Educat. Times, 24, 1913, 72, 106. ^^'Ibid., 25, 1914, 69-70. i^^Comptes Rendus Paris, 158, 1914, 469-470. "^Sitzungsber. Berlin Math. Gesell., 13, 1914, 161-9. 140 History of the Theory of Numbers. [Chap, v E. Cahen^^^ gave F. Arndt's^^ proof without reference. A. Cunningham^" tabulated all solutions N of 0(iV)=2' for r = 4, 6, 8, 9, 10, 11, 12, 16, each solution being a product of a power of 2 by distinct primes 22"+ 1. J. Hanmiond^-^ noted that, if 'Zf{k/n)=F{7i) or <l>(n), according as the summation extends over all positive integers k from 1 to n or only over such of them as are prime to n, then Z$(d)=F(n). This becomes (4) when /is constant. R. Ratat^29 ^oted that 0(n) = 0(n + l) for n= 1, 3, 15, 104. For n<125, 2n7^2, 4, 16, 104, he verified that (/)(2n=t l)>0(2n). R. Goormaghtigh^^o ^^^^^ ^j^j^^ 0(^^) = <^(^i_|_l) also for n= 164, 194, 255 and 495. He gave very special results on the solution of (f){x) = 2a. Formulas involving cf) are cited under Lipschitz,'''°' ^^ Cesaro,^^ Ham- mond,^" and Knopp^^^ of Ch. X, Hammond^ of Ch. XI, and RogeP« of Ch. XVIII. Cunningham^^ of Ch. VII gave the factors of (t>{f). Dede- kind^^ of Ch. VIII generalized ^ to a double modulus. Minin^^° of Ch. X solved 0(iV)=r(A^). Sum 0fc(n) of the A:th Powers of the Integers ^n and Prime to n. A. Cauchy^^^ noted that (piin) is divisible by n if n>2, since the integers <n and prime to n may be paired so that the sum of the two of any pair is n. A. L. Crelle^^ (p. 80, p. 84) noted that (^i(n) = |n<^(n). The proof follows from the remark by Cauchy. A. Thacker^^*^ defined (f)k{n) and noted that it reduces for k = to Euler's <i>{n). Set St(2) = l'"-|-2^+ . . .+2^n = a°6V. . ., where a, 6, . . . are distinct primes. By deleting the multiples of a, then the remaining multiples of b, etc., he proved that Mn)=sM -2a^s.(^) +2 aVs,(^) - S ^a*feVs,(^) + . . . , where the summation indices range over the combinations of a, 5, c, . . . one, two, ... at a time. In the second paper, he proved Bernoulli's^^"" formula where Bi, Bz,... are the Bernoullian numbers. Then, by substitution, ^^(^)=^n(l-i)+§(J)5,n^-^n(l-a)-i(3)53n*-^n(l-a^) i^TWorie des nombres, I, 1914, 393. i"Math. Quest. Educ. Times, 27, 1915, 103-6. 128/bid., 29, 1916, 53. i"L'interm(5diaire des math., 24, 1917, 101-2. "»/6ui., 25, 1918, 42-4. "'M6m. Ac. Sc. de I'Institut de France, 17, 1840, 565; Oeuvres, (1), 3, 272. "ojour. fur Math., 40, 1850, 89-92; Cambridge and Dublin Math. Jour., 5, 1850, 243. Repro- duced, with errors as to signs, by Zerr, Amer. Math. Monthly, 5, 1898, 93-5. Cf. E. Prouhet, Xouv. Ann. Math., 10, 1851, 324-330. """Jacques Bernoulli, Are conjectandi, 1713, 95-7. Chap. V] GENERALIZATIONS OF EuLER's 0-FuNCTION. 141 wheren(l-aO denotes {l-a^){l-h'). . .. J. Binet^" wrote Vif ■ ■> Vn for the integers <iV and prime to N^p^q". . .. Then, if B^, —B^, B^,... are the BernouUian numbers 1/6. 1/30, 1/42, . . ., andP,= (l-p^)(l-g'')..., for X sufficiently small to insure convergence. Expanding each member into negative powers of x and comparing coefficients, we get n =277/ = P_,N, 2Sr7i = P_,N\ SXrjf = P_,N^+SB,P,N, ^n,^ = P_,N^+QB,P,N^.. . the first being equivalent to the usual formula for 0(iV). The general law can be represented symbolically by givr'=^\{N+Bpy-h{N-Bpy\, where, after expanding the binomials, we are to replace N"/{BP) by P^iN" and any other term {BPY^~'^ by B2h-\P2h-\- It is easily shown that, if k is odd, Hit]'' is divisible by N. Silva^^ used his symbolic formula, taking S to be the sum of 1, . . ., n, whence S{a) is the sum §n(l+n/A) of the multiples ^n of A. Thus ^i(^) = 2^(^) • This proof of Crelle's result is thus like that by Brennecke.^" W. Brennecke^^^ proved Crelle's result by means of H-...+n-la(l+2+...+^)+6(l+...+^) + ...t + ]4+...+;J + . ..! + .... Set )Li = 0(n) , a = ahc .... He proved that <i>^{n)=^}xn'-^\aixn^-^n{\-a^){\-h^) . . ., the signs being + or — according as the number of the distinct prime factors a, 6, . . . of n is even or odd. •"Comptes Rendus Paris, 32, 1851, 918-921. "^Programm Realschule, Posen, 1855, §§5-6. 142 History of the Theory of Numbers. [Chap, v G. Oltramare^^ obtained for the sum, sum of squares, sum of cubes, and sum of biquadrates, of the integers <7na and relatively prime to a the respective values ^m~a<}>{a), JmV0(a) + (-l)»— a0(a,), ll o imV</>(«) + (-l)"^a2<A(ai), Ttl 111 6 z-O'O where a is the number and Oi the product of the distinct prime factors ju, I', . . . of a, while ^(aO = (ju^ — l)(j/^ — 1) . . .. The number of integers <n which are prime to a is 4>{a)n/a. J. Liou\'ille^^ stated that Gauss' proof of S0(d) =iV may be extended to the generalization 2QWc?) = l*+2*+...+iV*, where d ranges over the di\'isors of N. He remarked that Binet's^" results are readily proved in various ways. Also, e);3w={z>wf. N. V. Bougaief^^^ stated that, if ^(n) is the number of distinct prime factors of n>\, and ^i(n) is their product, also a result quoted below with Gegenbauer's^"° generalization. August BUnd^^^ reproduced without reference the formulas and proofs by Thacker,^^° and gave 0,(m)=w'</)o(7n)-^^^w'-Vi(/7O + ('2)^«''-'<A2W- • . . +(-l)'<^.(^0. E. Lucas^^^ indicated a proof that 7?<^„_i(x) is given symbolically by {x+QY-Q\ where, if n = a°6^ . ., 0, = 5,(l-a'-')(l-5'-0 - • •• Thus, if IT is the product of the negatives of the primes a, b, . . . , 2</)i(x) =x4>{x), 3<t>2{x) =(j>{x) (x~ + hA , 403Ct) =.T(/)(.T)(x2+7r). '"Mdmoires de I'lnstitut Nat. Gr^nevois, 4, 1856, 1-10. ""Comptea Rendus Paris, 44, 1857, 753-4; Jour, de Math., (2), 2, 1857, 393-6. i"Nouv. Ann. Math., (2), 13, 1874, 381-3; Bull. Sc. Math. Astr., 10, I, 1876, 18. '**Ueber die Potenzsummen der iinter einer Zahl ?« Uegenden und zu ihr relativ primen Zahlen, Diss., Bonn, 1876, 37 pp. i^^Nouv. Ann. Math., (2), 16, 1877, 159; Throne des nombres, 1891, 394. Chap. V] GENERALIZATIONS OF EuLER's 0-FuNCTION. 143 Several ^^^^ found expressions for 0„=<^„(iV) and proved that </>ox'»+n</)ia;"-i+i7i(n- 1) (/)2a;"-2+ . . . +<^„= (n odd) has the root —4>\/4>q, while the remaining roots can be paired so that the sum of the two of any pair is — 20i/(/)o. If n=3 the roots are in arith- metical progression. H. Postula^^^ proved Crelle's result by the long method of deleting multiples, used by Brennecke.^^^ Catalan {ibid., pp. 208-9) gave Crelle's short proof. Mennesson^^^ stated that, if q is any odd number, <^» = *0(^'+') (modg), and (Ex.366) that the sum of the products (/)(n) — 1 at a time of the integers ^n and prime to n is a multiple of n. E. Cesaro^^° proved the generalization: The sum rprn of the products m at a time of the integers a, ^,. . .^N and prime to N is divisible by iV if m is odd. For by replacing a by iV— a, /3 by A^"— j8, . . . and expanding. '^^-&^Ht-\y-Mt'-2y ¥2-... where 0=0(iV). Also (l>m{N) is divisible by iV if m is odd. F. de Rocquigny^^^ proved Crelle's result. Later, he"^ employed con- centric circles of radii 1, 2, 3, . . . and marked the numbers {m — l)N-}-l, (m — l)N-\-2, . . ., mN at points dividing the circle of radius m into A'' equal parts. The lines joining the center to the 0(iV) points on the unit circle, marked by the numbers <N and prime to N, meet the various circles in points marked by all the numbers prime to N. He stated that the sum of the 4>{N) numbers prime to N appearing on the circle of radius m is |(2m — l)0(iV^), and [the equivalent result] that the sum of the numbers prime to N from to mN is ^'m^(i>{N^). He later recurred to the subject {HUd., 54, 1881, 160). A. Minine^®^ noted that, if P>N> 1 and k is the remainder obtained by dividing P by N, the sum s{N, P) of the integers <P and prime to N may be computed by use of s{N, mN+k)=s{N, k)+^4>{N')+mN4>{N)„ where (Minine^O <i>{N)k is the number of integers ^k prime to N. *A. Minine^^^ considered the number and sum of all the integers < P which are prime to N [Legendre's (5) and Minine^*^^]. i"«Matli. Quest. Educ. Times, 28, 1878, 45-7, 103-5. i"Nouv. Corresp. Math., 4, 1878, 204-7. Likewise, R. A. Harris, Math. Mag., 2, 1904, 272. ^^lUd., p. 302. i6«76id., 5, 1879, 56-59. "iLes Mondes, Revue Hebdom. des Sciences, 51, 1880, 335-6. i62/6id., 52, 1880, 516-9. i"/6id., 53, 1880, 526-9. "^Nouveaux theoremes de la th^orie des nombres, Moscow, 1881. 144 History of the Theory of Numbers. [Chap, v A. Minine^^ investigated the numbers N which divide the sum of all the integers < N and prime to N. E. Cesaro^^^ proposed his theorems^®^ as exercises. Proofs, by associa- ting a with N — a, etc., were given by Moret-Blanc (3, 1884, 483-4). Ces^ro" (p. 82) proved the formula of Liouville.^" Writing (pp. 158-9) <f)„ for </),„(A0 and expanding 0„=2(iV— a)"*, where a, /3, . . . are the integers ^ A^ and prime to A'^, we get whence <^^ is di\'isible by N if m is odd, but not if m is even. This is e\'ident (p. 257) since aJ^ -{- {N — a)"" is di\isible by a+A'' — a if m is odd. The above formula gives A'" = (1 — A)"*, symboUcally, where " 4> AT-" is the arithmetic mean of the mth powers of a/N, ^/N, .... The mean value of <j)m{N) is 6A„A"'"+V'''"^- He reproduced (pp. 161-2) an earHer for- mula,^^° which shows that B"' = {l-B)'", symbolically, if B^ is the arith- metic mean of the products of a/N, ^/N, . . . taken m at a time. We have (p. 165) the approximation X x"*"^^ 6 2 <f)m(j) = 7 — rrr} — r^ * ~2> y=i (7M+l)(m-|-2) tT whence (p. 261) the mean of (t>^{N) is 6A''"+7(m+l)7r2. Proof is given (pp. 255-6) of Thacker's^^° formula *-«'" '"'*'::;"'*'"" -iiiij.cr)'-»-""'"' where UN)=^d'-'f^(d)=Il{l-u^-'), d ranging over the divisors of A^, and u over the prime divisors of N. Here nix) is Merten's function (Ch. XIX). It is proved (pp. 258-9) that 2d^-Vp(^ = 1, 2^V.(^ =2dV,-.(rf), the first characterizing the function \pp{N), and reducing to (4) for p = 0. If a ranges over the integers for which [2n/a] is odd, then (p. 293) exactly if 7?7 = 0, 1, 2, 3, approximately if m>3, where A^, is the excess of the sum of the inverses of 1,. . ., n over that of n + l, . . . , 2n. In particular, 20(a) =nl '"Math. Soc. Moscow (in Russian), 10, 1882-3, 87-101. »«Nouv. Ann. Math., (3), 2, 1883, 288. Chap. V] GENERALIZATIONS OF EuLER's </)-FuNCTION. 145 P. Nazimov^" (Nasimof) noted that, when x ranges over the integers ^m and prime to n, the sum of the values taken by any function /(x) equals \mld\ 7:ix{d)Xf{dx), d 1=1 where d ranges over all divisors of n. The case f{x) = 1 yields Legendre's formula (5) . The case/(a;) = xyields a result equivalent to that of Minine.^^^"'* A generalization was given by Zsigmondy'^'^ and Gegenbauer.^'^^ E. Cesaro^^^ noted that, if A^ is the arithmetic mean of the mth powers of the integers ^ N and prime to N, and B^ that of their products m at a time, we have the symbolic relations Cesaro^^^ proved Thacker's^^'^ formula expressed as the last being symboHc, where f^ is a function such that l^^,,{d)='n}~^, d ranging over the divisors of n. By inversion n(n)=2M©<i'-'=;^n (!-»-'), where u ranges over the distinct prime factors of n. L. Gegenbauer^'^° proved that, if j/= yln , x-l n=lL3; J ,-1 For the case /c = 0, p = 2, this becomes Bougaief 's^^^ formula ig2{x)=i\^^<i>{x), v = [Vn]. C. Leudesdorf^'^^ considered for fx odd the sum i/'^(iV) of the inverses of the juth powers of the integers < N and prime to N. Then ^|^,{N)=^kN'-hfiN^P,+,iN), where k is an integer. Thus, if N = p^q, where q is not divisible by the prime p>3, »/'^(A^) is divisible by p^' unless ju is prime to p, and )U+1 is divisible by p — 1; for example, \{/^{p) is divisible by p^. If p = 3, ^l/^iN) is divisible by p^' if fx is an odd multiple of 3. If p = 2, it is divisible by 2^'~^ except when q = l. Cesaro"^ inverted his" symbolic form of Thacker's formula for <l)m{N) in terms of xf/'s and obtained nB,rPp{n) = {<f>-nBy. i"Matem. Sbomik (Math. Soc. Moscow), 11, 1883-4, 603-10 (Russian). "'Mathesis, 5, 1885, 81. "'Giomale di Mat., 23, 1885, 172-4. ""Sitzungsber. Ak. Wiss. Wien (Math.), 95, II, 1887, 219-224. i"Proc. London Math. Soc, 20, 1889, 199-212. "Teriodico di Mat., 7, 1892, 3-6. See p. 144 of this history. s 146 History of the Theory of Numbers. [Chap, v Hence if a ranges over the integers ^ n and prime to n, Z(a — nBy = or a multiple of mpp according as p is odd or even. By this recursion formula, L. Gegenbauer"^ gave a formula including those of Nazimov^^^ and Zsigmondy." For any functions xid), Xiid) , f i^i, ■ ■ ■ , x,), m /^\ /^\1 [m/d] f{KXi,. . ., /cx,)2x(5)xiM =^x(d)xA^) ^ 2 ^ fidKX^,. . ., dKX,), where d ranges over all divisors of n which have some definite property P, while 5 ranges over those common divisors of n, Xi,..., x, which have property P. Various special choices are made for x> Xi> / and P. For instance, property P may be that d is an exact pth power, whence, if p = 1, d is any divisor of n. The special results obtained relate mainly to new number-theoretic functions without great interest and suggested apparently by the topic in hand. T. del Beccaro^'^ noted that (t>k{n) is divisible by n if A; is odd [Binet^^^]. When n is a power of 2, l^+2*+...4-(n-l)* = Oor0(n) (modn), according as k is odd or even. His proof of (1) is due to Euler. J. W. L. Glaisher^^^ proved that, if a, h,. . . are any divisors of x such that their product is also a divisor, the sum of the nth. powers of the integers < X and not divisible by a or 6, . . . , is where s is the number of the divisors a, 6, ... , and li a,h,. . . are all the prime factors of x, this result becomes Thacker's.^^" N. Nielsen^^^ proved by induction on y that the sum of the nth powers of the positive integers <mM and prime to M = pi^. . .ply is "'""'^'^W+(-i)-'f'^-^'"' C^+iV (™m)"--' n (P---1). n+l «=in+l \ zs y ,=i The case m= 1 gives Thacker's^^° result. That result shows {ihid., p. 179) that 02n(w) and <^2n+i(^) are divisible by m and m^ respectively, for l^n ^ (pi— 3)/2, where pi is the least prime factor of m, and also gives the resi- dues of the quotients modulo m. Corresponding theorems therefore hold for the sum of the products of the integers < m and prime to m, taken t at a. time. i"Sitzungsberichte Ak. Wiss. Wien (Math.), 102, 1893, Ila, 1265-94. "♦Atti R. Accad. Lined, Mem. CI. Fis. Mat., 1, 1894, 344-371. i«Messenger Math., 28, 1898-9, 39-41. "»Oversigt Danske Vidensk. Selsk. Forhandlinger, 1915, 509-12; cf. 178-9. Chap. V] GENERALIZATIONS OF EuLER's 0-FuNCTION. 147 Schemmel's Generalization of Euler's ^-Function. V. SchemmeP^° considered the $„(m) sets of n consecutive numbers each <m and relatively prime to m. If m = a'^}f . . ,, where a, h,. . . are distinct primes, and m, m' are relatively prime, he stated that $„(m) =a"~^(a — n)6^~-^(5— n) . . ., <^n{mm') =<l>„(m)$n(m'), Sn"-''V-^'. . .<l>„(5)=w, 6 = a«V. . ., a'^a, /S'^/S, . . ., the third formula being a generalization of Gauss' (4) . If ^ is a fixed integer prime to m, $„(w) is the number of sets of n integers <m and prime to m such that each term of a set exceeds by k the preceding term modulo m. Consider the productPof the Xth terms of the *J>„(m) sets. If n = 1, P= =t 1 (mod m) by Wilson's theorem. If n> 1, P"-i=)(-l)^-ir-i(X-l)!(n-X)!j*>) (modm). For the case A: = X = 1, n = 2, we see that the product of those integers < m and prime to m, which if increased by unity give integers prime to m, is = 1 (mod m) . E. Lucas^^^ gave a generalization of Schemmel's function, without men- tion of the latter. Let ei,..., e^ be any integers. Let ^(n) denote the number of those integers h, chosen from 0, 1, . . ., n — 1, such that h — ei, h — e2,. . ., h — Ck are prime to n. For k<n, ei = 0, 62= —I,. .., ei,— — {k — 1), we have k con- secutive integers h, h-j-1,. . ., h+k — l each prime to n, and the number of such sets is */c(n). Lucas noted that ^(p)'^(g) =^{pq) if p and q are rela- tively prime. Let n = a°-h^ . . ., where a, h,. . . are distinct primes. Let X be the number of distinct residues oi ei, . . . , e^ modulo a; fx the number of their distinct residues modulo h; etc. Then ^(n)=a»-i(a-X)&^-\6-M). . .. L. Goldschmidt^®^ proved the theorems stated by Schemmel, and himself stated the further generalization: Select any a — A positive integers <a, any h—B positive integers <b, etc.; there are exactly a''-\a-A)¥-\h-B)... integers <m which are congruent modulo a to one of the a — A numbers selected and congruent modulo b to one of the h — B numbers selected, etc. P. Bachmann^^^ proved the theorems due to Schemmel and Lucas. Jordan's Generalization of Euler's ^-Function. C. Jordan,^*^*^ in connection with his study of linear congruence groups, proved that the number of different sets of k (equal or distinct) positive integers ^n, whose g. c. d. is prime to n, is* ^^ ■/>.w=»'(i-^.)-..(i-^j "ojour. fur Math., 70, 1869, 191-2. "'Th^orie des nombres, 1891, p. 402. ""Zeitschrift Math. Phys., 39, 1894, 205-212. i^^Niedere Zahlentheorie, 1, 1902, 91-94, 174-5. ^iioTraitg des substitutions, Paris, 1870, 95-97. *He used the symbol [n, k]. Several of the writers mentioned later used the symbol (f>k(n), which, however, conflicts with that by Thacker.^^" 148 History of the Theory of Numbers. [Chap, v if Pi, . . ., Pg are the distinct prime factors of n. In fact, there are n*" sets of k integers ^n, while {n/piY of these sets have the common divisor pi, etc., whence k + .... «"'"•-©•-©■- -(^y Jordan noted the corollary: if n and n' are relatively prime, (11) J,{nn')=J,{n)J,{n'). A. Blind^^^ defined the function (10) also for negative values of k, proved (11), and the following generalization of (4): (12) 2Ji.(d) =n''' (d ranging over the di\'isors of 7i). W. E. Story^°^ employed the s>Tnbol r'^in) for Jk{n) and called it one of the two kinds of kth totients. The second kind is the number </)*(n) of sets of k integers ^?? and not all di\isible by any factor of n, such that we do not distinguish between two sets differing only by a permutation of their numbers. He stated that <t>\n) =|-,ir*(n)+f,V-nn)+t2V-2(n)+ . • • +<tiT(n)[ , where 1, fi*, W,. . . are the coefficients of the successive descending powers of X in the expansion of (x+l)(x+2). . .{x-\-k — \). Story-°- defined "the kih. totient of n to the condition k to be the num- ber of sets of k numbers ^ n which satisfy condition k. The number of sets of k numbers ^n, all containing some common di\'isor of n satisfying the condition k, but not all containing any one di\'isor of n satisfying the con- dition X is (if different permutations of k numbers count as different sets) '^\H''^-y~b,')y~b,'>) where 5, 5', . • • are the least divisors of n satisfj-ing condition /c, while 5i, 5/, . . . are the least di\'isors of n satisfying condition x- Here a set of least divisors is a set of divisors no one of which is a multiple of any other." E. Ces^ro" (p. 345) stated that, if $;.(a:) is the number of sets of k integers ^x whose g. c. d. is prime to x, then where J* is to be replaced by J/n), and d ranges over the di\'isors of n. J. W. L. Glaisher-^^ proved (12) by means of a symbolic expression for the infinite series 2/t(n)/(a:''). If ^t(n) is Merten's function, JM -2p,V,(^) +2pi V«/.(^^) - • • =M(n), where the summations relate to the distinct prime factors p, of n. Using "* Johns Hopkins University Circulars, 1, 1881, 132. ^^Ihid., p. 151. Cf. Amer. Jour. Math.. 3, 1880, 382-7. »<»London, Ed. Dublin Phil. Mag., (5), 18, 1884, 531, 537-8. Chap. V] Generalizations of Euler's (^-Function. 149 these formulas for n = l, 2,. each equal to { — lY~'^Jkin): 1^ 2^ 3^ 4^ 1111 10 1 10 n, we obtain two determinants of order n, 1 -1 -1 -1 1 ... 1 -2" -3^ -5^ 6^ ... 1' -2' -3* . . . 1* -2' ... L. Gegenbauer^*^ proved (12). For n = 'pi^ . . .p/'^, set 7r(n) = (-irpi...p„ X(n) = (-l^+••+^ where w{n) denotes the number of distinct prime factors of n. By means of the series f (s) =Sn~*, he proved that, when d ranges over the divisors of r, ZF(d)d2' = r^ SF(d)d2' = r'SdV,(d), ^F{d)Jk{d)d'' = 0, S(-l)(^+i)"'(^/V7rM^j =0, the last holding if r has no square factor and following from the third in view of (11), Mr) =2c^m(2) , i:F{d)d'tx{d) =r''fx{r), i:Md)Ud)J,(^^ =0 or J,,{Vr), according as r is or is not a square, S ( - 1) ^'+^)"'('"V^(m)/fe(m)/2fe(^)w'* = r''\{r)Jk{r) {mn^ = r) , TO, n ^JkM . . .JkinW-^'W-^'" .nti=r"', where rii, . . ., n^ range over all sets of solutions of nin2. . .n,+i = n, the case A; = 1 being due to H. G. Cantor .^^ E. Cesaro^^^ derived (10) from (12), writing ^i_k for Jk. E. Cesaro^"^ denoted J kin) by xf^'^in) and gave (12). L. Gegenbauer^^" gave the further generaUzation X{g,{x)f==i: i[f]u-), [^]. J. Hammond^°^ wrote \f/(n, d) for 2/(5), where / is an arbitrary function and 5 ranges over all multiples ^ n of the fixed divisor d of n. Then (13) mt)=^^P{n, l)-2)/^(n, pi)+2iA(n, p,P2)-..., ""Sitzungsber. Ak. Wiss. Wien (Math.), 89 II, 1884, 37-46. Cf. p. 841. See Gegenbauer" of Ch. X. "'Annali di Mat., (2), 14, 1886-7, 142-6; "'Messenger Math., 20, 1890-1, 182-190. 150 History of the Theory of Numbers. [Chap, v where i ranges over the integers ^n which are prime to n, while pi, P2> • • • denote the distinct prime factors of n. If f{t) = l, then \l/{n, d)=n/d and (13) becomes ^in)=n-i:^+X-^-...=n(l-^)(l-^).... Pi P1P2 \ Pi/ V P2/ Next, take f(t) = ao+ait-\-a2f+ .... Using hyperboUc functions, S/(0 = Jcoth(./2)=l+^-4+..., provided Z be replaced by nJXn)J_r{n), where /i(n) =/'(n) -a„ Mn) =f{n) -2a2, . . ., /_i(n) = ff{n)dn. Hence, since Ji(n) =(j){n), 2/(0 =^/-i(^) +^^-i(n)/i(n) -^V_3(n)/3(n) + . . . . In particular, for f{t)=t'', we get ^^-(n). In Prouhet's^^ first formula, 5 may be replaced by the g. c. d. A,,, b of a and h. The generalization J,{ah)=Ma)J,{h)j^^ is proved. From (12) we get by addition* (14) i\-]j,{j) = l'+2' y=iLjJ + . . . +n*. Taking n = l, 2,..., n, we obtain equations whose solution gives Jk(n) expressed as a determinant of order n in which the elements of the last coluimi are 1, 1+2*, 1+2^+3*, . . ., while for s<n the sth column consists of s — 1 zeros followed by s units, then s twos, etc. For s>0, the element in the (s + l)th row and rth column in Glaisher's^"^ first determinant is 1 or according as r/s is integral or fractional. J. Valyi^°^ used J2{n) ■^({>{n) in his enumeration of the n-fold perspective polygons of n sides inscribed in a cubic curve. H. Weber^os proved (10) for k = 2. L.Carlini209 gave without references (10), (11), (12), with</)(^) for J„(A;). E. Cesaro^io noted that (12) implies (10). For, if 2/(d) =F(n), we have by inversion (Ch. XIX), /(n) =Xii{d)F(n/d). The case f=Ji gives Jijn) _^IJLid) The latter is a case of G{n) ='2g{d) and hence, with (12) and W)QQ^2,Wf(^). ♦This work, Mess. Math., 20, 1890-1, p. 161, for k = l, is really due to Dirichlet." Formula (14) is the case p = 1 of Gegenbauer's, p. 217. "^Math. Nat. Berichte aus Ungarn, 9, 1890, 148; 10, 1891, 171. "'EUiptische Functioncn, 1891, 225; ed. 2, 1908 (Algebra III), 215. "»Periodico di Mat., 6, 1891, 119-122. "o/fcid., 7, 1892, 1-6. Chap. V] GENERALIZATIONS OF EuLER's </)-FuNCTION. 151 Ji+M=i:d'JMJi(^, which is next to the last formula of Gegenbauer's.^"^ Similarly, which is the case i = 1 of Gegenbauer's'^^ fifth formula in Ch. X, (Tk{n) being the sum of the A;th powers of the divisors of n. E. Weyr^^^ interpreted J2in) in connection with involutions on loci of genus 1. From the same standpoint, L. Gegenbauer^^^ proved (12) for k = 2 and noted that the value (10) of J-zin) then follows by the usual method of number-theoretic derivatives. L. Gegenbauer^^^" wrote cf)kim, n) for the number of sets of k positive integers ^ m whose g. c. d. is prime to n = pi°' . . . p/'' and proved a formula including [mf=Um, n)+i S {\„ . . . XT 4>k i -^ • , -^^ ) where (Xi, . . . , X,,) is the determinant derived from that with unity through- out the main diagonal and zeros elsewhere by replacing the 7th row by the X^th row for 7 = 1,. . ., c. The case m = n, k — l, is due to Pepin.^^ There is an analogous formula involving the sum of the /cth powers of the positive integers ^m and prime to n. E. Jablonski^^ used Jk{n) in connection with permutations. G. Arnoux^^^ proved (10) in connection with modular space. *J. J. Tschistiakow^^"* (or Cistiakov) treated the function /^(n). R. D. von Sterneck^^^ proved that J,{n) =SJ,(Xi)J,_,(X2) =S0(X,) . . 4{\), the X's ranging over all sets of integers S. n whose 1. c. m. is n. To generalize this, let Jk{n; mi, ... , rrik) be the number of sets of integers z'l, . . . , ik, whose g. c. d. is prime to n, while ij^n/mj for j = 1, . . . , k. Then Jk{n; Wi, . . ., m^)=SJ,(Xi; m\,. . ., 'm'r)Jk-r0^2] ^'r+i,- • •, rn'k) =2:Ji(Xi; mi). . . J'i(X^; m^), the X's ranging over all sets of integers ^n whose 1. c. m. is n, while m'l, . . . , m'k form any fixed permutation of mi, . . . , m^t, and J"i(n; m), designated <f)^"'\n) by the author, is the number of integers ^n/m which are prime to n. Also, "iSitzungsberichte Ak. Wiss. Wien (Math.), 101, Ila, 1892, 1729-1741. 2i2Monatshefte Math. Phys., 4, 1893, 330. 2i2aDenkschr. Ak. Wiss. Wien (Math.), 60, 1893, 25-47. 2i'Arithm6tique graphique; espaces arith. hypermagiques, 1894, 93. 2"Math. Soc. Moscow, 17, 1894, 530-7 (in Russian). "'Monatshefte Math. Phys., 5, 1894, 255-266. 152 History of the Theory of Numbers. [Chap, v SJ.(d;m„...,m.) = [ii]W...[iL], where d ranges over the divisors of n, the case A; = 1 being due to Laguerre.'* In the latter case, take n = 1, . . . , n and add. Thus k=i LfcJ -^^LmJ Ini m \ m/ j the last equality, in which (n, h) is the g. c. d. of n, h, following from expres- sions for (n, h) given by Hacks^^ of Ch. XL In the present paper the above double equation was proved geometrically. For m = l, we get Dirichlet's^^ formula. The g. c. d. of three numbers is expressed in terms of them and [x]. The initial formulas were proved geometrically, but were recognized to be special cases of a more general theorem. Let 2Md)=FM (1 = 1, ...,k), where d ranges over all divisors of n. Then the function ^(n) =S/i(Xi) . . .A(X,) (1. c. m. of Xj, . . . , X* is n) has the property S^(d)=Fi(n)...n(n). Hence in the terminology of Bougaief (Ch. XIX) the number-theoretic derivative ^{n) of Fi{n) . . .Fk{n) equals the sum of the products of the derivatives /» of the factors Fi, the arguments ranging over all sets of k numbers having n as their g. c. d. L. Gegenbauer^^^" proved easily that, if [n, . . , t] is the g. c. d. of n, . . . , f 2 F{[n,x^,...,x,]) = 'E F{d)jJ fj, where d ranges over all divisors of n, and F is any function. K. Zsigmondy^^^ considered any abelian (commutative) group G with the independent generators ^i,. . ., Qs of periods ni, . . ., n^, respectively. Any element g'l''' . . . gj"' of G is of period 5 if and only if 5 is the least positive value of X for which xhi,. . ., xhs are multiples of rii, . . ., n^, respectively. The number of elements of period 5 of G is thus the number of sets of posi- tive integers hi,. . ., hg {hi^rii,. . ., /ij^nj such that 5 is the least value of X for which xhi,. . . , xhs are divisible by ni, . . . , n^, respectively. The num- ber of sets is shown to be rPid;ni,...,n,)=lldjIl{l-l/q.'*), where 5_, is the g. c. d. of 5 and Uj] q\,...,qr are the distinct prime factors of 5; while U is the number of those integers nx, . . ., n^ which contain q^ at least as often as 5 contains it. If 5 and 5' are relatively prime, ypib] ni,. . ., n,)\pi8'; rii,. . ., n,)=i/'(55'; nj. . ., nj. "li^Sitzungsber. Akad. Wiss. Wien (Math.), 103, Ila, 1894, 115. "'Monatshefte Math. Phys., 7, 1896, 227-233. For his we write \p, as did Carmichael." I Chap. V] GENERALIZATIONS OF EuLER's 0-FuNCTION. 153 If d ranges over all divisors of the product ni . . . n^, Si//(5;ni, . . ., n,)=nin2. . .n,. d In case 5 divides each ni{i = \, . . ., s), 4/ becomes Jordan's Js(5). As a generalization (pp. 237-9) consider sets of positive integers ai, . . . , a„ where aj = l, 2, . . . , 7_, for j = 1, 2, . . . , s. Counting the sets not of the form n^ai, nf a2, . . . , n^f a, (i = l,. ..,r), we get the number n 7,-2 n \-%\ +s n f . J' ,,, ~\ - where (ni, n2, . . .) is the 1. c. m. of rii, n2, . . .. In particular, take n^P= . . . =n^f = ni (i = l,...,r), where ni, . . . , n^. are relatively prime in pairs, and let iV be a positive mul- tiple of ni, . . . , n^ such that Then the above expression equals J/(iV; mi,. . ., m,) = n T-l-S II [—1 + 2 H f-^^l - . • ., y=iLmyJ i j=\unjniA i,i' j=\unjnini>j which determines the number of sets tti,. . ., a, (ay = l, 2,. . ., — ;i=l,. . ., s) l_A/tyJ whose g. c. d. is divisible by no one of ni, ^2, . . . , n^. By inversion, S//g;^„...,m.) = n[|], where d ranges over the divisors of N which are products of powers of rii, . . . , Ur. When ni,...,ns are the distinct prime factors of N,J/{N; rrii, . . , , m,) becomes the function Js{N; mi, . . ., Wj) of von Sterneck.^^^ As in the case of the latter function, we have J/{N; Wi,. . ., m,)=SJi'(Xi; mi). . .//(X,; mj, the X's ranging over all sets whose 1. c. m. is N. L. Carhni^^^ proved that if a ranges over the integers for which [2n/a] = 2/c+l, then XJM = sg^ - 2s^'J, s^^ ^1'+... +m*. For k = l, this becomes 2(^(a) =n^ [E. Cesaro, p. 144 of this History]. D. N. Lehmer^^^ called Jmin) the m-fold totient of n or multiple totient of n of multiplicity m. He proved that, if A: = pi"'. . .pr°^ Jm{k'')=k^'''-''Jm{k), Jm(ky)=JM n \pr^-pr"^-'Xy, Pi) \ , 1=1 where \(y, pj =0 or 1 according as Pi is or is not a divisor of y. In the '"Periodico di Mat., 12, 1897, 137-9. "»Amer. Jour. Math., 22, 1900, 293-335. 154 History of the Theory of Numbers. [Chap, v second formula the product equals the similar function of y' if y and y' are congruent modulo pip^ ■ ■ Pr- Consider the function U/k] 1=1 where m, n, k are positive integers and x is a positive number. Then if S{x, k) denotes l*+2''-f . . . H-[x]*, it is proved that which for m = n = 1 becomes Sylvester's^^ formula. By inversion, where ai(i) is Merten's function. For k as above and k' = k/pr''^, ^, .(X, n, A;)=p,-(v-i)|(p^-i)ci,^(^^, n, A:')+$^(^, n, p,k') | = p,'"(«r-i)(p^_l)Sp^-^-i),^^/'^, n, A;'), where I is the least value of j for which [x/p^"'""^-'] = 0. Hence $^(x, n, k) can be expressed in terms of functions $,„(?/, n, 1). True relations are derived from the last four equations by replacing n by 1 — n and ^m{Xy 1 — n, A:) by \xlk\ n^{x,n,k)=i:j^{ik)'(ik)-'''". 1=1 Proof is given of the asymptotic formula „»n7i+l p ^'"(^' ^' ^)=:;;;:;rxT 7r^+^' hl^^x-- log x, wn + l D^+i where A is finite and independent of x, ??2, n, while « 1 *■ Pi — I Dm+i = 2 — q:Y> P^. fc = n a^_w „+i 7TJ P„,. 1 = 1. j = U i=lPi \Pi —i-j For m = n = fc = l, this result becomes that of Mertens^* (and Dirichlet'^). The asymptotic expressions found for ^^i^, n, k) are different for the cases n = l, n = 2, n>2. A set of m integers (not necessarily positive) having no common divisor > 1 is said to define a totient point. Let one coordinate, as x^, have a fixed integral value 5^0, while Xi,. . ., x^-i take integral values such that [xi/x^],. . ., [X;„_i/X;„] have prescribed values; we obtain a compartment in space of m dimensions which contains /m-i(^m) totient points. For example, if m = 3, X3 = 6, and the two prescribed values are zero, there are 24 totient points (xi, X2, 6) for which 0^Xi<6, 0^X2<6, while Xi and X2 have no common divisor dividing 6. For Xi = l or 5, Xo has 6 values; for Xi = 2 or 4, X2=l, 3 or 5; for Xi = 3, X2 = l, 2, 5; for Xi = 0, X2 = 0, 1, 5. Given a closed curve r=f{d), decomposable into a finite number of seg- ments for each of which f{d) is a single- valued, continuous function. Let I Chap. V] FarEY SeRIES. 155 K be the area of the region bounded by this curve, and N the number of points {x, y) within it or on its boundary such that a; is a multiple of k and is prime to y. Then ^ ^ lim— = -2Pi.fc, k=aa A TT where K increases by uniform stretching of the figure from the origin. In particular, consider the number A^ of irreducible fractions x/y^\ whose denominators are ^n. Since x^y, the area K of the triangular re- gion is n^/2. Hence N = {n^/2) (6/7r^) , approximately (Sylvester^^) . Again, the number of irreducible fractions whose numerators he between I and l-\-m, and denominators between V and I'+m', is Qmm'/ir^, approximately. There is a similar theorem in which the points are such that y is divisible by k', while three new constants obey conditions of relative primality to each other or to x, y, k, k'. Extensions are stated for m-dimensional space. E. Cahen^^^ called /^(n) the indicateur of /cth order of n. G. A. Miller^^° evaluated Jk{in) by noting that it is the number of operators of period m in the abeUan group with k independent generators of period m. G. A. Miller^^^ proved (10) and (11) by using the same abelian group. E. Busche^^^ indicated a proof of (10) and (12) by an extension to space oi k+1 dimensions of Kronecker's^^^ plane, in which every point whose rectangular coordinates x, y are integers is associated with the g. c. d. of x, y. A. P. Minin^^^ proved (14) and some results due to Gegenbauer.^"^ R. D. CarmichaeP^^ gave a simple proof of Zsigmondy's^^® formula for ^. G. Metrod^^^ stated that the number of incongruent sets of solutions of xy' — x'y = a (mod m) is 'EdmJ2{m/d), where d ranges over the common divisors of m and a. When a takes its m values, the total number of sets of solutions is vJ'Ay^ rt'A It is asked if like relations hold for Jk, k>2. Cordone^^ and Sanderson^^^ (of Ch. VIII) used Jordan's function in giving a generalization of Fermat's theorem to a double modulus. Farey Series. Flitcon^^ gave the number of irreducible fractions <1 with each denominator <100, stating in effect the value of Euler's (/)(n) when n is a product of four or fewer primes. "9Th6orie des nombres, 1900, p. 36; I, 1914, 396-400. «»Amer. Math. Monthly, 11, 1904, 129-130. 2"Amer. Jour. Math., 27, 1905, 321-2. 222Math. Annalen, 60, 1905, 292. ^^'Vorlesungen iiber Zahlentheorie, 1901, I, p. 242. 224Matem. Sbomik (Moscow Math. Soc), 27, 1910, 340-5. "^''Quart. Jour. Math., 44, 1913, 94-104. «2«L'interm6diaire des math., 20, 1913, 148. Proof, Sphinx-Oedipe, 9, 1914, 4. ^'Ladies' Diary, 1751. Reply to Question 281, 1747-8. T. Leybourn's Math. Quest, pro- posed in Ladies' Diary, 1, 1817, 397-400. 156 History of the Theory of Numbers. [Chap, v C. Haros-"*^ proved the results rediscovered by Farey-^° and Caiichy.^^- J. Farey-^" stated that if all the proper vulgar fractions in their lowest terms, having both numerator and denominator not exceeding a given number n, be arranged in order of magnitude, each fraction equals a frac- tion whose numerator and denominator equal respectively the sum of the numerators and sum of the denominators of the two fractions adjacent to it in the series. Thus, for n = 5, the series is 1112 13 2 3 4 Z' T' -JT' T' IT' T' 7' T' T' and 1_1 + 1 2_1 + 1. 4 5+3' 5 3+2* Henry Goodwyn mentioned this property on page 5 of the introduction to his "tabular series of decmial quotients" of 1818, published in 1816 for private circulation (see Goodwyn,^^' ^- Ch. VI), and is apparently to be credited with the theorem. It was ascribed to Goodwyn by C. W. Merri- field.2" A. L. Cauchy^^^ proved that, if a/b, a'/b', a"/b" are any three consecu- tive fractions of a Farey series, b and b' are relatively prime and a'b—ab' = 1 (so that a'/b'-a/b = l/bb'). Similarly, a"b'-a'b" = l, so that a+a": b+b" = a': b', as stated by Farey. StouveneP^^ proved that, in a Farey series of order n, if two fractions a/b and c/b are complementary (i. e., have the sum unity), the same is true of the fraction preceding a/b and that following c/b. The two fractions adjacent to 1/2 are complementary and their common denominator is the greatest odd integer ^n. Hence 1/2 is the middle term of the series and two fractions equidistant from 1/2 are complementary. To find the third of three consecutive fractions a/b, a'/b', x/y, we have a+x = a'z, b+y = b'z (Farey), and we easily see that z is the greatest integer ^ {n-\-b)/b', M. A. Stern-^^ studied the sets m, n, and m, m-\-n, n, and m, 2m-\-n, m-\-n, m-\-2n, n, etc., obtained by interpolating the sum of consecutive terms. G. Eisenstein^^" briefly considered such sets. *A. Brocot^^'' considered the sets obtained by mediation [Farey] from U/1, 1/0: oil. 01121. T' T' ITJ T' Y' 1' T' TJ">- • •• Herzer^^^ and Hrabak^^^ gave tables with the limits 57 and 50. G. H. Halphen^^^ considered a series of irreducible fractions, arranged in order of magnitude, chosen according to a law such that if any fraction / is excluded then also every fraction is excluded if its two terms are at least 2<9Jour. de I'dcole polyt., cah. 11, t. 4, 1802, 364-8. "oPhilos. Mag. and Journal, London, 47, 1816, 385-6; [48, 1816, 204]; Bull. Sc. Soc. Philomatique de Paris, (3), 3, 1816, 112. "'Math. Quest. Educat. Times, 9, 1868, 92-5. "*Bufl. Sc. Soc. Philomatique de Paris, (3), 3, 1816, 133-5. Reproduced in Exercices de Math., 1, 1826, 114-6; Oeuvres, (2), 6, 1887, 146-8. "»Jour. de mathdmatiques, 5, 1840, 265-275. »«Jour. fur Math., 55, 1858, 193-220. "laBericht Ak. Wiss. Berlin, 1850, 41^2. "'Calcul des rouages par approximation, Paris, 1862. Lucas.'" 2^«Tabellen, Basle, 1864. ""Tabellen-Werk, Leipzig, 1876. "'Bull. Soc. Math. France, 5, 1876-7, 170-5. Chap. V] FaREY SeRIES. 157 equal to the corresponding terms of /. Such a series has the properties noted by Farey and Cauchy for Farey series. E. Lucas^^^ considered series 1, 1 and 1, 2, 1, etc., formed as by Stern. For the nth series it is stated that the number of terms is 2"~-^ + l, their sum is 3""^ + l, the greatest two terms (of rank 2""^+l=±=2"~^) are (i+V5r+^-(i-\/5r+^ 2"+V5 Changing n to p, we obtain the value of certain other terms. J. W. L. Glaisher^^° gave some of the above facts on the history of Farey series. Glaisher^" treated the history more fully and proved (p. 328) that the properties noted by Farey and Cauchy hold also for the series of irre- ducible fractions of numerators ^ m and denominators ^ n. Edward Sang^^" proved that any fraction between A/ a and C/y is'of the form {'pA-[-qC)/{'pa-{-qy), where p and q are integers, and is irreducible if p, q are relatively prime. A. Minine^®^ considered the number S{a, N) of irreducible fractions a/h such that h-\-aa^N. Let 0(6)p denote the number of integers ^p which are prime to h. Then, for a > 0, >S(a,iV)= S0(6)p, P=L a J' since for each denominator h there are (/)(6)p integers prime to h for which h+aa-^N and hence that number of fractions. A. F. Pullich^^^ proved Farey's theorem by induction, using continued fractions. G. Airy^^^ gave the 3043 irreducible fractions with numerator and denom- inator ^ 100. J. J. Sylvester^^^ showed how to deduce the number of fractions in a Farey series by means of a functional equation. Sylvester,^^' ^^ Cesaro,^^ Vahlen,^^ Axer,^^^ and Lehmer^^^ investigated the number of fractions in a Farey series. Sylvester^^^" discussed the fractions x/y for which x<n^ y<n, x-\-y^n. M. d'Ocagne^^'^ prolonged Farey's series by adding 1/1 in the pth place, where p=<^(l)+ . . . +(j>{n). From the first p terms we obtain the next p by adding unity, then the next p by adding unity, etc. Consider a series S{a, N) of irreducible fractions Ui/hi in order of magnitude such that bi+atti^N, where a is any fixed integer called the characteristic. All the series S(_a, N) with a given base N may be derived from Farey's series ««Bull. Soc. Math. France, 6, 1877-8, 118-9. ^oProc. Cambr. Phil. Soc, 3, 1878, 194. MiLondon Ed. Dub. Phil. Mag., (5), 7, 1879, 321-336. '"Trans. Roy. Soc. Edinburgh, 28, 1879, 287. "'Jour, de math. el6m. et spec, 1880, 278. Math. Soc. Moscow, 1880. '"Mathesis, 1, 1881, 161-3. '"Trans. Inst. Civil Engineers; cf. Phil. Mag., 1881, 175. '"Johns Hopkins Univ. Circulars, 2, 1883, 44-5, 143; Coll. Math. Papers, 3, 672-6, 687-8. 266aAmer. Jour. Math., 5, 1882, 303-7, 327-330; Coll. Math. Papers, IV, 55-9, 78-81. '"Annales Soc. Sc. Bruxelles, 10, 1885-6, II, 90. Extract in Bull. Soc. Math. France, 14, 1885-6, 93-7. 158 History of the Theory of Numbers. [Chap, v 5(0, N) by use of a.(a, N) =a,(0, N), &.(a, iV) =6,(0, N) -aa,(0, N). Thus a,6._i — a,_i6i = l, so that the area of OA.A,_i is 1/2 if the point Ai has the coordinates o,, 6,. All points representing terms of the same rank in all the series of the same base he at equally spaced intervals on a parallel to the x-axis, and the distance between adjacent points is the num- ber of units between this parallel and the a:-axis. A. Hurwitz-^^ apphed Farey series to the approximation of numbers by rational fractions and to the reduction of binary quadratic forms. J. Hermes-^^ designated as numbers of Farey the numbers ri = l, 72 = 2, X3 = 7^ = 3, To = 4, tq = t7 = 5, t8 = 4, . . . with the recursion formula T„ = r„_2^+r2^+i-n+i, 2''<n^2'+\ and connected with the representation of numbers to base 2. The ratios of the r's give the Farey fractions. K. Th. Vahlen-^^'' noted that the formation of the convergents to a fraction w by Farey's series coincides with the development of w into a con- tinued fraction whose numerators are ±1, and made an application to the composition of linear fractional substitutions. H. Made-'° apphed Hurwitz's method to numbers a+hi. E. Busche"^ apphed geometrically the series of irreducible fractions of denominators ^a and numerators ^b, and noted that the properties of Farey series {a = h) hold [Glaisher-®^]. W. Sierpinski^^^ used consecutive fractions of Farey series of order m to show that, if x is irrational. ===« U=i^ ^ 2 2j Expositions of the theory of Farey series were given by E. Lucas,'" E. Cahen,-'^ Bachmann.^'^ An anonymous writer,^'^ starting with the irreducible fractions <1, arranged in order of magnitude, with the denominators ^ 10, inserted the fractions with denominator 11 by listing the pairs of fractions 0/1, 1/10; 1/6, 1/5; 1/4, 2/7;. . ., the sum of whose denominators is 11, and noting that between the two of each pair lies a fraction with denominator 11 and numerator equal the sum of their numerators. *«8Math. Annalen, 44, 1894, 417-436; 39, 1891, 279; 45, 1894, 85; Math. Papers of the Chicago Congress, 1896, 125. Cf. F. Klein, Ausgewahlte Kapitel der Zahlentheorie, I, 1896, 19^210. Cf. G. Humbert, Jour, de Math., (7), 2, 1916, 116-7. *«9Math. Annalen, 45, 1894, 371. Cf. L. von Schrutka, 71, 1912, 574, 583. ^s'ajour. fiir Math., 115, 1895, 221-233. ^'^Ueber Fareysche Doppelreihen, Diss. Giessen, Darmstadt, 1903. "'Math. Annalen, 60, 1905, 288. *"BuU. Inter. Acad. Sc. Cracovie, 1909, II, 725-7. »"Th6orie des nombres, 1891, 467-475, 508-9. *'^fil4ments de la theorie des nombres, 1900, 331-5. "'Niedere Zahlentheorie, 1, 1902, 121-150; 2, 1910, 55-96. i^'OZeitschrift Math. Naturw. Unterricht, 45, 1914, 559-562. CHAPTER VI. PERIODIC DECIMAL FRACTIONS; PERIODIC FRACTIONS: FACTORS OF I0"±1. Ibn-el-Banna^ (Albanna) in the thirteenth century factored 10" — 1 for small values of n. The Arab Sibt el-Maridini^" in the fifteenth century noted that in the sexagesimal division of 47° 50' by 1° 25' the quotient has a period of eight terms. G. W. Leibniz^ in 1677 noted that \/n gives rise to a purely periodic fraction to any base h, later adding the correction that n and h must be relatively prime. The length of the period of the decimal fraction for 1/n, where n is prime to 10, is a divisor of n — 1 [erroneous for n = 21 ; cf . Wallis^] . John Wallis^ noted that, if N has a prime factor other than 2 and 5, the reduced fraction M/N equals an unending decimal fraction with a repetend of at most A^ — 1 digits. If N is not divisible by 2 or 5, the period has two digits if N divides 99, but not 9; three digits if A^ divides 999, but not 99. The period of 1/21 has six digits and 6 is not a divisor of 21 — 1. The length of the period for the reciprocal of a product equals the 1. c. m. of the lengths of the periods of the reciprocals of the factors [cf. Bernoulli^]. Similar results hold for base 60 in place of 10. J. H. Lambert^ noted that all periodic decimal fractions arise from rational fractions; if the period p has n digits and is preceded by a decimal with m digits, we have lO'" ' lO'^lO" lO'^lO^" lO'^ClO^-l) John Robertson^ noted that a pure periodic decimal with a period P of k digits equals P/9 ... 9, where there are k digits 9. J. H. Lambert^ concluded from Fermat's theorem that, if a is a prime other than 2 and 5, the number of terms in the period of \/a is a divisor of a — 1. If S' is odd and \/g has a period oi g — 1 terms, then ^ is a prime. If \/g has a period of m terms, but ^ — 1 is not divisible by m, g is composite. Let 1/a have a period of 2m terms; if a is prime, A; = lO'^+l is divisible by a; if a is composite, k and a have a common factor; if k is divisible by a and if m is prime, each factor other than 2^5^ of a is of period 2m. Let a be a composite number not divisible by 2, 3 or 5. If 1/a has a period of m terms, where w is a prime, each factor of a produces a period 'Cf. E. Lucas, Arithm^tique amusante, 1895, 63-9; Brocard.'o^ i«Carra de Vaux, Bibliotheca Matb., (2), 13, 1899, 33-4. ^Manuscript in Bibliothek Hannover, vol. Ill, 24; XII, 2, Blatt 4; also. III, 25, Blatt 1, seq., 10, Jan., 1687. Cf. D. Mahnke, BibUotheca Math., (3), 13, 1912-3, 45-48. ^Treatise of Algebra both historical & practical, London, 1685, ch. 89, 326-8 (in manuscript, 1676). *Acta Helvetica, 3, 1758, 128-132. »Phil. Trans., London, 58, 1768, 207-213. "Nova Acta Eruditorum, Lipsiae, 1769, 107-128. 159 160 History of the Theory of Numbers. [Chap, vi of m terms. If \/a has a period of mn terms, where m and n are primes, while no factor has such a period, one factor of a divides 10'" — 1 and another di\'ides 10" — 1. If \/a has a period of mnp terms, where vi, n, p are primes, but no factor has such a period, any factor of a divides 10"*- 1,. . ., or 10"" — !. These theorems aid in factoring a. L. Euler^ gave numerical examples of the conversion of ordinary frac- tions into decimal fractions and the converse problem. Euler^'' noted that if 2p+l is a prime 40?2±1, ±3, ±9, ±13, it divides lO^-ljif 2p + l isaprime40;?±7, ±11, ±17, ±19, it divides 10^+1. Jean BernouUi^ gave a r^sum6 of the work by Wallis,^ Robertson,* Lambert^ and Euler,^ and gave a table showing the full period for 1/D for each odd prime D<200, and a like table when Z) is a product of two equal or distinct primes < 25. When the two primes are distinct, the table con- firms Wallis' assertion that the length of the period for 1/D is the 1. c. m. of the lengths of the periods for the reciprocals of the factors. But for l/D^, where D is a prime > 3, the length of the period equals D times that for 1/D. If the period for 1/D, where D is a prime, has D — 1 digits, the period for ?n/D has the same digits permuted cyclically to begin with m. He gave (p. 310) a device communicated to him by Lambert: to find the period for 1/D, where Z) = 181, we find the remainder 7 after obtaining the part p composed of the first 15 digits of the period; multiply l/D = p-\-7/D by 7 ; thus the next 15 digits of the period are given by 7p ; since 7^ = /)+ 162, the third set of 15 digits is found by adding unity to 7~p, etc.; since 7 belongs to the exponent 12 modulo D, the period for 1/D contains 15-12 digits. Jean Bernoulli^ made use of various theorems due to Euler which give the possible linear forms of the divisors of 10*±1, and obtained factors of (10*-l)/9 when A-^30, except for k = ll, 17, 19, 23, 29, with doubt as to the primality of the largest factor when A' = 13, 15 or ^19. He stated (p. 325) erroneously^^ that (10^^ + l)/ll-23 has no factor <3000. Also, 10''+1 = 7-1M3-211-9091-520S1. He gave part of the periods for the reciprocals of various primes ^601. L. Euler^^ wTote to Bernoulli concerning the latter's^ paper and stated criteria for the divisibility of 10^±1 by a prime 2p + l=4n±l. If both 2 and 5 or neither occur among the divisors of n, n=F2, n=F6, then 10'' — 1 is divisible by 2p-\-l. But if only one of 2 and 5 occurs, then 10^+1 is divisible by 2p+l [cf. Genocchi^^]. Henry Clarke^^ discussed the conversion of ordinary fractions into decimals without dealing with theoretical principles. 'Algebra, I, Ch. 12, 1770; French trans!., 1774. '"Opusc. anal., 1, 1773, 242; Comm. Arith. Coll., 2, p. 10, p. 25. 'Nouv. m6m. acad. roy. Berlin, ann^e 1771 (1773), 273-317. *Ibid., 318-337. "P. Seelhoff, Zeitschrift Math. Phys., 31, 1886, 63. Reprinted, Sphinx-Oedipc, 5, 1910, 77-8. "Nouv. m(Sm. acad. roy. Berlin, annde 1772 (1774), Histoire, pp. 35-36; Comm. Arith., 1, 584. ^^he rationale of circulating numbers, London, 1777, 1794. \ Chap. VI] PERIODIC DECIMAL FRACTIONS. 161 Anton FelkeP^ showed how to convert directly a periodic fraction written to one base into one to another base. He gave all primes < 1000 which can divide a period with a prime number of digits <30, as 29m +1 = 59,233,.... Oberreit^* extended Bernoulli's^ table of factors of 10*=*=!. C. F. Gauss^^ gave a table showing the period of the decimal fraction for Vp", p"<467, V a prime, and the period for 1/p", 467^ p"^ 997. W. F. Wucherer^® gave five places of the decimal fraction for n/d, d<1000, n<dford<50, n^ 10 for (^^ 50. Schroter published at Helmstadt in 1799 a table for converting ordinary fractions into decimal fractions. C. F. Gauss^'^ proved that, if a is not divisible by the prime p (^7^2, 5), the length of the period for a/p" is the exponent e to which 10 belongs modulo p^. If we set 0(p") =ef and choose a primitive root r of p^ such that the index of 10 is /, we can easily deduce from the periods for k/p^, where k = \, r, . . ., r^~\ the period for m/p", where m is any integer not divisible by p. For, if i be the index of m to the base r, and if i = af-\-^, where 0^/3</, we obtain the period for m/p" from that for rVp" by carrying the first a digits to the end. He computed^^ the necessary periods for each p"<1000, but published here the table only to 100. By using partial fractions, we may employ the table to obtain the period for a/b, where b is a product of powers of primes within the limits of the table. H. Goodwyn^^ noted that, if a<17, the period for a/17 is derived from the period for 1/17 by a cyclic permutation of the digits. Thus we may print in a double line the periods for 1/17, . . . , 16/17 by showing the period for 1/17 and, above each digit d of the latter, showing the value of a such that the period for a/ 17 begins with the digit d, while the rest of the period is to be read cyclically from that for 1/17. Goodwyn^^ noted that when 1/p is converted into a decimal fraction, p being prime, the sum of corresponding quotients in the two half periods is 9, and that for remainders is p, if p^7. J. C. Burckhardt^" gave the length of the period for 1/p for each prime p^2543 and for 22 higher primes. It follows that 10 is a primitive root of 148 of the 365 primes p, 5<p<2500. "Abhand. Bohmiachen Gesell. Wias., Prag, 1, 1785, 135-174. "J. H. Lambert's Deutscher Gelehrter Briefwechsel, pub. by J. Bernoulli, Leipzig, vol. 5, 1787, 480-1. The part (464-479) relating to periodic decimals is mainly from Ber- noulli's' paper. "Posthumous manuscript, dated Oct., 1795; Werke, 2, 1863, 412-434. ^'Beytrage zum allgemeinem Gebrauch der Decimal Brliche. . . ., Carlsruhe, 1796. "Disq. Arith., 1801, Arts. 312-8. A part was reproduced by Wertheim, Elemente der Zahlen- theorie, 1887, 153-6. I'Jour. Nat. Phil. Chem. Arts (ed., Nicholson), London, 4, 1801, 402-3. "76id., new series, 1, 1802, 314-6. Cf. R. Law, Ladies' Diary, 1824, 44-45, Quest. 1418. '"Tables des diviseurs pour tous les nombres du premier milMon, Paris, 1817, p. 114. For errata see Shanks," Kessler," Cimningham,^^! and G^rardin."^ 162 History of the Theory of Numbers. [Chap, vi H. Goodw-jTi-^ gave for each integer d^ 100 a table of the periods for n/d, for the various integers n<d and prime to d. Also, a table giving the first eight digits of the decimal equivalent to everj^ irreducible vulgar fraction < 1/2, whose numerator and denominator are both ^ 100, arranged in order of magnitude, up to 1/2. GoodwATi"' -^ was without doubt the author of two tables, which refer to the preceding ''short specimen" by the same author. The first gives the first eight digits of the decimal equivalent to every irreducible \'ulgar fraction, whose numerator and denominator are both ^ 1000, from 1/1000 to 99/991 arranged in order of magnitude. In the second volume, the "table of circles" occupies 107 pages and contains all the periods (circles) of ever}' denominator prime to 10 up to 1024; there is added a two-page table showing the quotient of each number ^ 1024 by its largest factor 2°5''. For example, the entry in the "tabular series" under -^^ is .08689024. The entry in the two-page table under 656 is 41. Of the various entries under 41 in the "table of circles," the one containing the digits 9024 gives the complete period 90243. Hence /^V = -086890243. Glaisher"^ gave a detailed account of Goodwyn's tables and checks on them. They are described in the British Assoc. Report, 1873, pp. 31-34, along -wdth tables showing seven figures of the reciprocals of numbers < 100000. F. T. Poselger'^ considered the quotients 0, a, h,. . . and the remainders 1, a, j8, . . . obtained by di\'iding 1, ^, A~, ... by the prime p; thus A a A' .,.B —=a-i — , — = aA+b-\ — ,.... P P P P Adding, we see that the sum lH-a-}-/3-|- ... of the remainders of the period is a multiple TTzp of p; also, w(A — 1) =a+6-f- . . .. Set M = k+...+hA'-'--\-aA*-\ where A belongs to the exponent t modulo p. Then — = -+MS, S = l+A'+ . . . +A^"-'\ P P "The first centenarj' of a series of concise and useful tables of all the complete decimal quotients which can arise from dividing a unit, or any whole number less than each divisor, by all integers from 1 to 1024. To which is now added a tabular series of complete decimal quotients for all the proper vulgar fractions of which, when in their lowest terms, neither the numerator nor the denominator is greater than 100; with the equivalent vulgar fractions prefixed. By Henry Goodwyn, London, 1818, pp. xiv + lS; vii+30. The first part was printed in 1816 for private circulation and cited by J. Farey in Philos. Mag. and Journal, London, 47, 1816, 385. "A tabular series of decimal quotients for all the proper vulgar fractions of which, when in their lowest terms, neither the numerator nor the denominator is greater than 1000, London, 1823, pp. v + 153. "A table of the circles arising from the division of a unit, or any other whole number, by all the integers from 1 to 1024; being all the pure decimal quotients that can arise from this source, London, 1823, pp. v + 118. "Abhand. Ak. Wiss. BerUn (Math.), 1827, 21-36. Chap. VI] PERIODIC DECIMAL FRACTIONS. 163 If M is divisible by p, we may take n = 1 and conclude that A^p^ differs from 1/p^ by an integer. If M is not divisible by p, S must be, so that n is divisible by p and the length of the period is pt. In general, for the denom- inator p^, we have n = l if M is divisible by p^~^, but in the contrary case n is a multiple of p'"'^. If the period for a prime p has an even number of digits, the sum of corresponding quotients in the two half periods is p. An anonymous writer^^ noted that, if we add the digits of the period of a circulating decimal, then add the digits of the new sum, etc., we finally get 9. From a number subtract that obtained by reversing its digits; add the digits of the difference; repeat for the sum, etc.; we get 9. Bredow^^ gave the periods for a/p, where p is a prime or power of a prime between 100 and 200. He gave certain factors of 10" — 1 for w = 6-10, 12-16, 18, 21, 22, 28, 33, 35, 41, 44, 46, 58, 60, 96. E. Midy" noted that, if a"", a"', . . . are the least powers of a which, diminished by unity, give remainders divisible by q^, qi''', . . . , respectively {q, qi,... being distinct primes), and if the quotients are not divisible by q, qi,. . ., respectively, and if t is the 1. c. m. of n, ni, . . ., then a belongs to the exponent t modulo p = q^qi^' . . . , and a' — 1 is divisible by q only h times. Let the period of the pure decimal fraction for a/h have 2n digits. If h is prime to 10" — 1, the sum of corresponding digits in the half periods is always 9, and the sum of corresponding remainders is h. Next, let 6 and 10" — 1 have d>l as their g. c. d. and set h' = h/d. Let a„ be the nth re- mainder in finding the decimal fraction. Then a+a„ = 6'A:, ai+a„+i = 6'/ci, etc. The sums q-\-qn, 5i+g„+i, ... of corresponding digits in the half periods equal {\{)k-k^)/d, il0k,-k2)/d,.. ., {10k,_r-k)/d. Similar results hold when the period of mn digits is divided into n parts of m digits each. For example, in the period 002481389578163771712158808933 for 1/403, the two halves are not complementary (10^^ — 1 being divisible by 31); for i = l, 2, 3, the sum of the digits of rank i, i-\-3, i+6, . . ., i+27 is always 45, while the corresponding sums of the remainders are 2015. N. Druckenmiiller^^" noted that any fraction can be expressed as a/x-\- ai/x^-l- .... J. Westerberg^^ gave in 1838 factors of 10"± 1 for nS 15. G. R. Perkins^^ considered the remainder r^ when N'^ is divided by P, and the quotient q in Nrj._i = Pqx-\-rj.. If Tk'^P—l, there are 2k terms in the period of remainders, and rk+x+r^ = P, qk+x+qx = N-l. [These results relate to 1/P written to the base N.] ^^Polytechnisches Journal (ed., J. G. Dingier), Stuttgart, 34, 1829, 68; extract from Mechanics' Magazine, N. 313, p. 411. ^*Von den Perioden der Decimalbriiche, Progr., Oels, 1834. '^'De quelques propriet^s des nombres et des fractions d^cimales p^riodiques, Nantes, 1836,21 pp. ""T.heorie der Kettenreihen . . ., Trier, 1837. 28See Chapter on Perfect Numbers."* 2»Amer. Jour. Sc. Arts, 40, 1841, 112-7. 164 History of the Theory of Numbers. [Chap, vi E. Catalan^" converted periodic decimals into ordinary fractions without using infinite progressions. When 1/13 is converted into a decimal, the period of remainders is 1, 10, 9, 12, 3, 4; repeat the period; starting in the series of 12 terms with any term (as 10), take the fourth term (4) after it, the fourth term (12) after that, etc.; then the sum 26 of the three is a multiple of 13. In general, if D is a prime and D — l=mn, the sum of n terms taken w by m in the period for N/D is a multiple of D [cf. Thibault'^]. If the sum of two terms of the period of remainders for N/D is D, the same is true of the terms following them. Hence the sum of corresponding terms of the two half periods is D. This happens if the number of terms of the period is <f){D). Thibault^^ denoted the numbers of digits in the periods for l/d and 1/d' by m and m'. If d' is divisible by d, m' is divisible by m. If d and d' have no common prime factor other than 2 or 5, the number of digits in the period for \/dd' is the 1. c. m. of m, m'. Hence it suffices to know the length of the period for 1/p", where p is a prime. If 1/p has a period of m digits and if 1/p" is the last one of the series 1/p, 1/p^, . . . which has a period of m digits, then the period for 1/p" for a >n has mp"'^ digits. For p = 3, we have w = 2; hence 1/3'^ for r^2 has a period of y~^ digits. For any prime p for which 7^p^ 101, we have n = 1, so that 1/p" has a period of mp°-~^ digits. Note that \/p and 1/p^ have periods of the same length to base h if and only if h^~^ = 1 (mod p^). Proof is given of Catalan's^" first theorem, which holds only when 10"' ^1 (mod D), i. e., when m is not a multiple of the number of digits in the period. For example, the sum of the /cth and (6+A;)th remainders for 1/13 is not a multiple of 13. E. Prouhet^^ proved Thibault's" theorem on the period for l/p". He^^" noted that multiples of 142857 have the same digits permuted. P. Lafitte^^ proved Midy's^^ theorem that, if p is a prime not dividing m and if the period for m/p has an even number of digits, the sum of the two halves of the period is 9 ... 9. J. Sornin^^ investigated the number m of digits in the period for 1/Z), where D is prime to 10. The period is a; = (10"* - l)/D. First, let D = lOA: + 1 . Then x = \Qy — \, where 10*"-^+ A; ,^ , , lO'-^-A:^ y = ^ = lOz+k, z = Finally, we reach v= \l — { — k)'^\/D, and x is an integer if and only if v is. Hence if we form the powers of the number k of tens in D, add 1 to the odd powers, but subtract 1 from the even powers of k, the first exponent giving a result divisible by D is the number m of digits in the period. »»Nouv. Ann. Math., 1, 1842, 464-5, 467-9. *nhid., 2, 1843, 80-89. "/bid., 5, 1846, 661. ^IhU., 3, 1844, 376; 1851, 147-152. 'Vbid., 397-9. Cf. Araer. Math. Monthly, 19, 1912, 130-2. w/Wd., 8, 1849, 50-57. Chap. VI] PeEIODIC DECIMAL FRACTIONS. 165 Next, if D = 10k — 1, we have a like rule to be applied only to the ^"^ — 1. If D = 10k=^3, 1/(3Z)) has a denominator lOZ^l, and the length of its period, found as above, is shown to be not less than that for 1/D. Th. Bertram^^ gave certain numbers p for which l/p has a given length k of period for k^ 100. Cf. Shanks.^^ J. R. Young^^ took a part of a periodic decimal, as .1428571 428 for 1/7, and marked off from the end a certain number (three) of digits. We can find a multipHer (as 6) such that the product, with the proper carrying (here 2) from the part marked off, has all the digits of the abridged number in the same cyclic order, except certain of the leading digits. In the special case the product is .8571428. W. Loof" gave the primes p for which the period for l/p has a given number n of digits, n^ 60, with no entry for n = 17, 19, 37-40, 47, 49, 57, 59, and with doubt as to the primality of large numbers entered for various other n's. E. Desmarest^^ gave the primes P< 10000 for which 10 belongs to the exponent {P — l)/t for successive values of t. The table thus gives the length of the period for 1/P. He stated (pp. 294-5) that if P is a prime < 1000, and if p is the length of the period for A/P, then except for P = 3 and P = 487 the length of the period for A/P^ is pP. A. Genocchi^^ proved Euler's^^ rule by use of the quadratic reciprocity law. Thus 5 is a quadratic residue or non-residue of N according as N = 5m=^l or 5m±3; for 4n+l = 5m='=l, n or n — 2 is divisible by 5; for 4n — l = 5m='=l, n or n-\-2 is divisible by 5. Also, 2 is a residue of 4n±l for n even, a non-residue for n odd. Hence 10 is a residue of A^ = 4n='= 1 for n even if n orn =f2 is divisible by 5, and for n odd if neither is. Thus Euler's inclusion of n=F6 is superfluous. By a similar proof, 10 is quadratic non- residue of A/' = 4n±l if both 2 and 5 occur among the divisors of n±2, n±6, or if neither occurs; a residue if a single one of them occurs. A. P. Reyer^^" noted that the period for a/3^ has 3^~^ digits and gave the length of the period for a/p for each prime p< 150. *F. van Henekeler^^^ treated decimal fractions. C. G. Reuschle^" gave for each prime p< 15000 the exponent e to which 10 belongs modulo p. Thus e is the length of the period for l/p. He gave all the prime factors of lO'^-l for n^l6, n = lS, 20, 21, 22, 24, 26, 28, 30, 32, 36, 42; those of 10"+1 for n^l8, n = 21; also cases up to n = 243 of the factors of the quotient obtained by excluding analytic factors. "Einige Satze aus der Zahlenlehxe, Progr. Coin, Berlin, 1849, 14-15. »«London, Ed. Dublin Phil. Mag., 36, 1850, 15-20. »^Archiv Math. Phys., 16, 1851, 54-57. French transl. in Nouv. Ann. Math., 14, 1855, 115-7. Quoted by Brocard, Mathesis, 4, 1884, 38. '^Th^orie des nombres, Paris, 1852, 308. For errata, see Shanks*^ and G^rardin.^'^ "Bull. Acad. Roy. Sc. Belgique, 20, II, 1853, 397-400. "<^Archiv Math. Phys., 25, 1855, 190-6. '^''Ueber die primitiven Wurzeln der Zahlen und ihre Anwendung auf Dezimalbriicbe, Leyden, 1855 (Dutch). "Math. Abhandlung.. .Tabellen, Progr. Stuttgart, 1856. Full title in Ch. I."* Errata, Bork,i''5 Hertzer,ii3 Cunningham. i" I 166 History of the Theory of Numbers. [Chap, vi W. Stammer^^ noted that n/p = 0.di . . . a^ implies -(10'-l)=ai...a,. V J. B. Sturm^^ used this result to explain the conversion of decimal into ordinary fractions without the use of series. M. Collins'^^ stated that, if we multiply any decimal fraction having m digits in its period by one with n digits, we obtain a product with Own digits in its period if vi is prime to n, but with 71(10™ — 1) digits if n is divisible by m. J. E. Oliver^^ proved the last theorem. If x'/x gives a periodic fraction to the base a with a period of ^ figures, then a^ = 1 (mod x) and conversely. The product of the periodic fractions for x'/x, . . . , z'/z with period lengths ^, . . . , f has the period length •M(^,...,f), M{x,...,z) where M{x, . . ., z) is the 1. c. m. of x, . . . , z. He examined the cases in which the first factor in the formula is expressible in terms of ^, . . . , f . Fr. Heime'*^ and M. Pokorny^^ gave expositions without novelty. Suffield*^ gave the more important rules for periodic decimals and indi- cated the close connection with the method of synthetic division. W. H. H. Hudson*^ called d a proper prime if the period for n/d has d — 1 digits. If the period for r/p has n = ip — l)/\ digits, there are X periods for p. The sum of the digits in the period for a proper prime p is 9{p — 1)/2. If 1/p has a period of 2n digits, the sum of corresponding digits in the two half periods is 9, and this holds also if p is composite but has no factor dividing 10" — 1 [Midy"]. If lOp+l is a proper prime, each digit 0, 1, . . . , 9 occurs p times in its period. If a, h are distinct primes with periods of a, /3 digits, the number of digits in the period for ab is the 1. c. m. of a, /8 [Bernoulli^]. Let p have a period of n digits and l/p = A-/(10" — 1). Let m be the least integer for which \ljp^-'^\2)p^-''^- ^\x-l) p is an integer; then 1/p^ has a period of mn digits. "Archiv Math. Phys., 27, 1856, 124. «/6id., 33, 1859, 94-95. «Math. Monthly (ed., Runkle), Cambridge, Mass., 1, 1859, 295. **Ibid., 345-9. ♦'Ueber relative Prim- und correspondirende Zahlen, primitive und sekundare Wurzeln und periodische Decimalbriiche, Progr., BerUn, 1860, 18 pp. "Ueber einige Eigenschaften periodischer Dezimalbriiche, Prag, 1864. ♦^Synthetic division in arithmetic, with some introductory remarks on the period of circulating decimals, 1863, pp. iv-|-19. *80xford, Cambridge and Dublin Messenger of Math., 2, 1864, 1-6. Glaisher" atrributed this useful anonymous paper to Hudson. Chap. VI] PERIODIC DECIMAL FRACTIONS. 167 V. A. Lebesgue^^ gave for iV^347 the periods for 1/iV, r/N,. . .[cf. Gauss^l. Sanio^° stated that, if m, n,. . . are distinct primes and 1/m, 1/n, . . . have periods of length q, q',. . ., then l/(wV. . .) has the period length ^a-i^b-i qq' ^ He gave the length of the period for l/p for each prime p^700, and the factors of 10" — 1, n^ 18. F. J. E. Lionnet^^ stated that, if the period for a/h has n digits, that for any irreducible fraction whose denominator is a multiple of h has a multiple of n digits. If the periods for the irreducible fractions a/6, a'/h', . . . have n, n', . • • digits, every irreducible fraction whose denominator is the 1. c. m. of b, h',. . . has a period whose length is the 1. c. m. of n,n',. . .. If the period for 1/p has n digits and if p" is the highest power of the prime p which divides 10" — 1, any irreducible fraction with the denominator p"'^^ has a period of np^ digits. C. A. Laisant and E. Beaujeux^^ proved that if g is a prime and the period for 1/q to the base B is P = ab. . .h, with q — 1 digits, then P-{a+h+...+h) = {B-l)a, ^{^+^-y) = B'-'-l and stated that a like result holds for a composite number q if we replace q—1 by/=</)(g). Their proof of the generaUzed Fermat theorem 5^=1 (mod q) is quoted under that topic. C. Sardi^^ noted that if 10 is a primitive root of a prime p = lOn+1, the period for 1/p contains each digit 0,..., 9 exactly n times [Hudson^^]. For p = 10n-f-3, this is true of the digits other than 3 and 6, which occur n+1 times. Analogous results are given for lOn+7, lOn+9. Ferdinand Meyer^^ proved an immediate generalization from 10 to any base k prime to 6, 6', ... of the statements by Lionnet.^^ Lehmann^" gave a clear exposition of the theory. C. A. Laisant and E. Beaujeux^^ considered the residues Vq, ri, . . . when A, AB, AB^,. . . are divided by A- Let ri_iB = QiZ)i+r,. When written to the base B, let Di = ap. . .02^1, and set Di = ap. . Mi. Then airi+ . . . +0prp = Z)i(ri-Q2A- • • • -QpDp). The further results are either evident or not novel. For G. Barillari^°" on the length of the period, see Ch. VII. *'M6m. soc. sc. phys. et nat. de Bordeaux, 3, 1864, 245. ^"Ueber die periodischen Decimalbrtiche, Progr., Memel, 1866. "Algebre 61em., ed. 3, 1868. Nouv. Ann. Math., (2), 7, 1868, 239. Proofs by Morel and Pellet, (2), 10, 1871, 39-42, 92-95. MNouv. Ann. Math., (2), 7, 1868, 289-304. "Giornale di Mat., 7, 1869, 24-27. "Archiv Math. Phys., 49, 1869, 168-178. ""Ueber Dezimalbriiche, welche aus gewohnUchen Briichen abgeleitet sind, Progr., Leipzig, 1869. 66N0UV. Ann. Math., (2), 9, 1870, 221-9, 271-281, 302-7, 354-360. 168 History of the Theory of Numbers. [Chap, vi *Th. Schroder^^ and J. Hartmann^^ treated periodic decimals. W. Shanks^^ gave Lambert's method (Bernoulli,^ end) for shortening the work of finding the length of the period for 1/A^. G. Salmon^^ remarked that the number 71 of digits in the period is known if we find two remainders which are powers of 2, since 10" = 2'' and 10'' = 2' imply 10"^"''^= 1; also if we find three remainders which are products of powers of 2 and 3. Muir''^ noted that it is here impUed that aq — bp equals n, whereas it is merely a multiple of n. J. W. L. Glaisher^" proved that, for any base r, 1 ir-iy .012...r-3r-l, a generalization of 1/81 = .012345679. W. Shanks^^ gave the length of the period for 1/p, when p is a prime < 30000, and a list of 69 errors or misprints in the table by Desmarest,^^ and 11 in that by Burckhardt.^° Shanks^- gave primes p for which the length n of the period for 1/p is a given number ^ 100, naturally incomplete. Shanks^^ gave additional entries p for n = 26, ?7 =99; noted corrections to his former table and stated that he had extended the table to 40000. Shanks^^ mentioned an extension in manuscript from 40000 to 60000. An extension to 120000 in manu- script was made by Shanks, 1875-1880. The manuscript, described by Cunningham, ^-^ who gave a list of errata, is in the Archives of the Royal Society of London. Shanks®^ stated that if a is the length of the period for 1/p, where p is a prime >5, that for 1/p'* is ap""'^ [^vithout the restriction by Thibault,^^ Muir^^]. G. de Coninck^® stated that, if the last digit (at the right) of A is 1 or 9, the last digit of the period for 1/A is 9 or 1 ; while, if A is a prime not ending in 1 or 9, its last digit is the same as the last in the period. Moret-Blanc^^ noted that the last property holds for any A not divisible by 2 or 5. For, if a is the integer defined by the period for 1/A, that for {A — l)/A is {A — l)a, whence a+ (A — 1)0 = 10" — 1, if n is the length of the periods. He noted corrections to the remaining nine laws stated by Coninck and implied that when corrected they become trivial or else known facts. "Progr. Ansbach, 1872, "Progr. Rinteln, 1872. "Messenger Math., 2, 1873, 41-43. "/bid., pp. 49-51, 80. ^^Ibid., p. 188. «iProc. Roy. Soc. London, 22, 1873-4, 200-10, 384-8. Corrections by Workman.*" "/bwi., pp. 381^. Cf. Bertram", Loof." *Hbid., 23, 1874-5, 260-1. ^Ibid., 24, 1875-6, 392. "Messenger Math., 3, 1874, 52-55. "Nouv. Ann. Math., (2), 13, 1874, 569-71; errata, 14, 1875, 191-2. *Ubid., (2), 14, 1875, 229-231. Chap. VI] PERIODIC DECIMAL FRACTIONS. 169 Karl Broda^^ considered a periodic decimal fraction F having an even number r of digits in the period and a number m of p digits preceding the period. Let x be the first half of the period, y the second half. Then 10"* 10"*+'" 10"'+^'" io"'+3^^- ■"10"' ' 10'"(102'"-1) _ 9(jp40'"+a;+p)+q 9-10"'(10^+l) ifx+2/ = o(10'' — l)/9 = a...a(tor terms). The first paper treated the case p = m = 0, and gave the generalization to base a in place of 10: £ ,^ , _£ , a+(a-l)x -r , _ a'-l a^^a"-^a'''^-- ' " (a-l)(a'-+l) " ^+2/-^ ^_j- The case a = a — l shows that a purely periodic fraction to the base a equals (a:+l)/(a'"+l) if the sum of the half periods has all its digits (to base a) equal to a— 1. Returning to the base 10, and taking A'= 9(10'"H-1), Z = 9x+a, where each digit of x is ^a, we see that Z/N equals a decimal fraction in which x is the first half of the period of r digits, while the second half is such that the sum of corresponding digits in it and x is a. If R is the remainder after r digits of the period have been obtained, R-\-Z = a (10''+ 1). C. G. Reuschle^^ gave tables which serve to find numbers belonging to a given exponent < 100 with respect to a given prime modulus < 1000. P. Mansion''" gave a detailed proof that, if n is prime to 2, 3, 5, and if the period for 1/n has n — 1 digits, the sum of corresponding digits in the half periods is 9. T. Muir^^ proved that, if p is a prime, either of N' = 1 (mod f) , iV^P" = 1 (mod p''+") follows from the other. If Xi is the least positive integer x for which the first holds and if p' is the highest power of p dividing N""' — !, then Xip" is the least positive integer y for which N^ = l (mod p'+"). Hence the known theorem: If N=JIpi'^, where Pi,P2,-- are distinct primes, and if the period for \/pi has m^ digits, and if Pi' is the highest power of Pi dividing 10"^ — 1, the number of digits in the period for \/N is the 1. c. m. of the niip^^i'^*. He asked if 6 = 1 when p>3, as affirmed by Shanks.^^ Mansion's proof {ibid., 5, 1876, 33) by use of periodic decimals of the generalized Fermat theorem is quoted under that topic. D. M. Sensenig'^ noted that a prime p?^2, 5, divides iV if it divides the sum of the digits of N taken in sets of as many figures each as there are digits in the period for l/p. «»Archiv Math. Phys., 56, 1874, 85-98; 57, 1875, 297-301. "Tafeln complexer Primzahlen, Berlin, 1875. Errata by Cunningham, Mess. Math., 46, 1916, 60-1. '"Nouv. Corresp. Math., 1, 1874-5, 8-12. "Messenger Math., 4, 1875, 1-5. "The Analyst, Des Moines, Iowa, 3, 1876, 25. i 170 History of the Theory of Numbers. [Chap, vi *A. J. M. Brogtrop^' treated periodic decimals. G. Bellavitis^* noted that the use of base 2 renders much more com- pact and convenient Gauss' ^^ table and hence constructed such a table. W. Shanks'^ found that the period for 1/p, where p = 487, is divisible by p, so that the period for 1/p^ has p — l digits. J. W. L. Glaisher^^ formed the period 05263. . . for 1/19 as follows: List 5; divide it by 2 and list the quotient 2; since the remainder is 1, divide 12 by 2 and list the quotient 6; divide it by 2 and list the quotient, etc. To get the period for 1/199, start with 50. To get the period, apart from the prefixed zero, for 1/49, start with 20 and divide always by 5; for 1/499, start wath 200. Glaisher^^ noted that, if we regard as the same periods those in which the digits and their cyclic order are the same, even if commencing at differ- ent places, a number q prime to 10 will have/ periods each of a digits, where af=4){q). This was used to check Goodwyn's table. ^^ If g = 39, there are four periods each of six digits, li q — 1 belongs to the period for 1/q, the two halves of every period are complementary; if not, the periods form pairs and the periods in each pair are complementary. For each prime N< 1000, except 3 and 487, the period for l/N" has nA^*"^ digits if that for 1/iV has n digits. Glaisher'^^ collected various known results on periodic decimals and gave an account of the tables relating thereto. If q is prime to 10 and if the period for 1/q has (/)(g) digits, the products of the period by the 4>{q) integers <q and prime to q have the same digits in the same cyclic order; for example, if g = 49. He gave (pp. 204-6) for each g<1024 and prime to 10 the number a of digits in the period for 1/q, the number n of periods of irreducible fractions p/q, not regarding as distinct two periods having the same digits in the same cychc order, and, finally Euler's (f>(,q). The values of a and n were obtained by mere counting from the entries in Good- wyn's^^ "table of circles"; in every case, an = <j){q). For the prime p = 487, he gave the full periods for 1/p and 1/p", each of 486 digits, thus verifying Desmarest's^^ statement of the exceptional character of this p [cf. Shanks'^]. Glaisher^^ again stated the chief rules for the lengths of periods. The problem was proposed^" to find a number whose products by 2, . . . , 6 have the same digits, but in a new order. Birger Hausted^^ solved this problem. Start with any number a of one digit, multiply it by any number p and let b be the digit in the units "Nieuw Archief voor VViskunde, Amsterdam, 3, 1877, 58-9. 7«Atti Accad. Lincei, Mem. Sc. Fis. Mat., (3), 1, 1877, 778-800. Transunti, 206. See 62a of Ch. VII. "Proc. Roy. Soc. London, 25, 1877, 551-3. ^^Messenger Math., 7, 1878, 190-1. Cf. Desmarest." "Report British Assoc, 1878, 471-3. "Proc. Cambridge Phil. Soc, 3, 1878, 185-206. "Solutions of the Cambridge Senate-House Problems and Riders for 1878, pp. 8-9. ""Tidsskrift for Math., Kjobenhavn, 2, 1878, 28. "/bid., pp. 180-3. Jornal de Sc. Math, e Ast., 2, 1878, 154-6. Chap. VI] PeEIODIC DECIMAL FRACTIONS. 171 place of the product ap, /S the digit in the tens place. Write the digit b to the left of digit a to form the last two digits of the required number P. The number c in the units place in 6p+j8 is written to the left of digit h in P. To cp add the digit in the tens place of bp and place the unit digit of the sum to the left of c in P. The process stops with the kth. digit t if the next digit would give a. Then P = t. . . cba and its products by k integers or fractions has the same k digits in the same cyclic order. For a = 2, p = 3, we get A; = 28 and see that P is the period of 2/27, and the k multipliers are m/2, m = l,. . ., 28. [To have an example simpler than the author's, take a = 7, p = 5; then P = 142857, the period of 1/7; the multipliers are 1, . . . , 6.] For proof, we have P = 10''-H-\- . . .-\-10h+10b+a, pP = 10''-'a+10''-H+ . . .+10c-\-h, pP = 10^-a+^, 10^ = 10^' so that P is the period with k digits for a/{lOp — l). E. LucasS2 gave the prime factors of lO'^^l, 10'^±1, lO^^^l, 10^^+1, 10^^+1, communicated to him by W. Loof, with the remark that (10^^ — 1)/9 has no prime factor < 3035479. Lucas gave the factors of 10^^+1. J. W. L. Glaisher^^ proved his^^ earlier statements, repeated his" earher remarks, and noted that, if g is a prime such that the period for 1/q has q — 1 digits, the products of the period for 1/q by 1, 2, . . ., g — 1 have the same digits in the same cyclic order. This property, well known for q = 7, holds also for g = 17, 19, 23, 29, 47, 59, 61, 97 and for q = 7\ 0. Schlomilch^^ stated that, to find every N for which the period for 1/N has 2k digits such that the sum of the sth and (fc+s)th digits is 9 for s = 1, . . . , A;, we must take an integer iV= (10*^+l)/r; then the first k digits of the period are the k digits of T — 1. C. A. Laisant^^ extended his investigations with Beaujeux^^'^^ and gave a summary of known properties of periodic fractions; also his^^ process to find the period of simple periodic fractions without making divisions. V. Bouniakowsky^^ noted that the property of the period of 1/N, observed by Schlomilch^^ for iV = 7, 11, 13, 77, 91, 143, holds also for the pe- riods of /c/iV, for A; = iV — 1 and (iV — 1)/2, with the same values of AT. Consider the decimal fraction Q.yiy2- ■ ■ with ym — ym-i+ym-2 (mod 9), replacing any residue zero by 9, and taking yi > 0, 1/2 > 0- The fraction is purely periodic and is either 0.9 or 0.33696639 or has the same digits permuted cyclically, or else has a period of 24 digits and begins with 1, 1 or 2, 2 or 4, 4, or has the same 24 digits permuted cyclically or by the interchange of the two halves s^Nouv. Corresp. Math., 5, 1879, 138-9. 8'Nature, 19, 1879, 208-9. s^Zeitschrift Math. Phys., 25, 1880, 416. 8*M6m. Soc. Sc. Phys. et Nat. de Bordeaux, (2), 3, 1880, 213-34. 86Le8 Mondes, 19, 1869, 331. "BuU. Acad. Sc. St. P^tersboiirg, 27, 1881, 362-9. 172 History of the Theory of Numbers. [Chap, vi of the period. The property of Schlomilch holds for these and the generali- zation to any base, as well as for those with the law xjm = '^ym-\-\-ym-2' But if ym = ^ym-i-2y^.2 (mod 9), ?/. = (2"'-'-l)(!/2-2/i)+2/i (mod 9), the fact that 2^ = 1 (mod 9) shows that the period has at most six digits. Those with six reduce by cychc permutation to nine periods : 167943, 235986, 278154, 197346, 265389, 218457, 764913, 329568, 751248. In the A-th of these the sum of corresponding digits in the two half periods is always =A- (mod 9). Karl Broda^^ examined for small values of r and certain primes p the solutions a: of x''= 1 (mod p) to obtain a base x for which the periodic frac- tion for 1/p has a period of r digits, and similariy the condition x^=—\ (mod p) for an even number of digits in the period (Broda^^). F. Kessler^^ factored 10"-1 forn = ll, 20, 22, 30. W. W. Johnson^° formed the period for 1/19 by placing 1 at the extreme right, next its double, etc., marking wdth a star a digit when there is 1 to carry: « * * * * * * «« 05263157894736842 1. ;^ To deduce the value of 1/19 written to the base 2, use 1 for each digit starred and for the others, reversing the order: .6 0001101011110010 i. If we apply the first process with the multipUer m, we get the period for the reciprocal of 10?7i — 1. E. Lucas^^ gave the prime factors of 10" — 1 for n odd, n^l7, 7i = 21, and certain factors forn = 19, . . . , 41 ; those of 10" + 1 for n^ 18 and n = 21. He stated that the majority of the results were given by Loof and pubUshed by Reuschle. In 1886, Le Lasseur gave 10^7-1 =3--2071723-5363222[3]57, said by Loof to have no divisor < 400,000 other than 3,9. On the omission of the digit 3, see Cunningham. ^-^ F. Kessler^- listed nine errors in Burckhardt's-" table and described his own manuscript of a table to p = 12553, i. e., for the first 1500 primes. Van den Broeck^^ stated that 10^" -1 is divisible by 3"+^ A. Lugli^^ proved that, if p is a prime 5^2, 5, the length of the period of 1/p is a divisor of p — 1. If the number of digits in the period of a/p is an even number 2t, the ^th remainder on dividing a by p is p — 1, and con- versely. Hence, if r^ is the hth remainder, rh+rh+i = p {h = l,. . ., t), and the sum of all the r's is tp. If the period of 1/p has s digits, s<p — 1, then ".\rchiv Math. Phys., 68, 1882, 85-99. «»Zeitschrift Math. Naturw. Unterricht, 15, 1884, 29. •"Messenger of Math., 14, 1884-5, 14-18. "Jour, de math. 6Um., (2), 10, 1886, 160. Cf . rinterm^diaire des math., 10, 1903, 183. Quoted by Brocard, Mathesis, 6, 1886, 153; 7, 1887, 73 (correction, 1889, 110). "Archiv Math. Phys., (2), 3, 1886, 99-102. "Mathesis, 6, 1886, 70. Proofs, 23.5-6, and Math. Quest. Educ. Times, 54, 1891, 117. "Periodico di Mat., 2, 1887, 161-174. Chap. VI] PERIODIC DECIMAL FRACTIONS. 173 p — l=sh and we have h sets of s fractions whose periods differ only by the cycHc permutation of the digits. If p is a product of distinct primes pi, P2 • • • and if the lengths of the periods of 1/p, 1/pi, l/p2, ■ ■ ■ are s, Si, S2, . . . , then s is the 1. c. m. of Si, S2,.... If p = Pi"P2^- • •> and s, Si, s' are the lengths of the periods of 1/p, 1/pi, l/pi", then s' is one of the numbers Si, Sip,. . ., SiPi°~^ and hence divides (pi — l)pi"~^; and s is a divisor of <f){p). Thus p divides lO^^^^-l. C. A. Laisant^^ used a lattice of points, whose abscissas are a+r,a-\-2r,..., a-\-pf and ordinates are their residues <p modulo p, to represent graphically periodic decimal fractions and to expand fractions into a difference of two series of ascending powers of fixed fractions. *A. Rieke^® noted that a periodic decimal with a period of 2m digits equals (i4. + l)/(10"*+l), where A is the first half of the period. He discussed the period length for any base. W. E. HeaP^ noted that, if B contains all the prime factors of N, the number of digits in the fraction to the base B for M/N is the greatest integer in (n+n' — l)/n', where n—n' is the greatest difference found by subtracting the exponent of each prime factor of N from the exponent of the same prime factor of B. If B contains no prime factor of N, the fraction for M/N is purely periodic, with a period of ^(A'') digits. If B contains some, but not all, of the prime factors of N, the number of digits preceding the period is the same as in the first theorem. The proofs are obscure. There is given the period for 1/p when p<100 and has 10 as a primitive root [the same p's as by Glaisher^^]. Likewise for base 12, with p<50. R. W. Genese^^ noted that, if we multiply the period for 1/81 [Glaisher^"] by m, where m<81 and prime to it, we get a period containing the digits 0, 1, . . ., 9 except 9n—m, where 9n is the multiple of 9 just exceeding m. Jos. Mayer^^ investigated the moduli with respect to which 10 belongs to a given exponent, and gave the factors of 10"— 1, n< 12. He discussed the determination of the exponent to which 10 belongs for a given modulus by use of the theory of indices and by the methods of quadratic, cubic, biquadratic,... residues. He used also the fact that there are (a — a') 08-/3') . . . divisors of Pi''p2^Vz ■ ■ • which divide no one of the fixed factors ViVVi ■ • ■, Pi>2W> • • • J where a<a,b<^,..., and pi, P2,-- are distinct primes. He gave the length of the period for 1/p, for each prime p^2543 and 22 higher primes [Burckhardt^^]. L. Contejean^°° proved that, in the conversion of an irreducible fraction a/h into a decimal fraction, if the remainders o^ and a^ are congruent modulo b, so that lO'a^lO^'a, then 10"'~''-1 is divisible by the quotient h' of b by the highest factor 2*5' of b. Thus the length of the period is "Assoc, fran?. avanc. sc, 16, 1887, II, 228-235. •'Versuch iiber die periodischen Bniche, Progr., Riga, 1887. •^Annals of Math., 3, 1887, 97-103. •^Report Britiah Assoc, 1888, 580-1. "Ueber die Grosse der Periode eines unendlichen Dezimalbruches, oder die Congruence lO^Sl (mod P). Progr. K. Studienanstalt Burghausen, Munchen, 1888, 52 pp. ""Bull. Boc. philomathique de Paris, (8), 4, 1891-2, 64-70. 174 History of the Theory of Numbers. [Chap, vi m—r, while r digits precede the period. The condition that the length of the period be the maximum 0(6') is that 10 be a primitive root of h', whence 5' = p", since 6'?^ 4 or 2^", p being an odd prime. P. Bachmann^°^ used a primitive root g of the prime p and set to the base g. We get the multiples Q, 2Q, . . . , (p — 1)Q by cyclic permuta- tion of the digits of Q. For p = 7, ^ = 10, Q = 142857. J. Kraus^"^ generaUzed the last result. When ri/n is converted into a periodic fraction to base g, prime to n, let ai, . . . , Ck be the quotients and ri, . . . , r^ the remainders. Then <7*-l rx = ax^^"^+ax+i/"^+- • -f«x-i (X = l,. • ■, k), n whence ^x(aiS'*"^+ • • • +(ik) =n(«x9'*"^+ • ■ • 4-ax-i). In particular, let n be such that it has a primitive root g, and take ri = 1. Then ft and if rx is prime to n, the product ry,Q has the same digits as Q permuted cyclically and beginning with a^. H. Brocard^"^ gave a tentative method of factoring 10" — 1. J. Mayer^°^ gave conditions under which the period of z/P to base a, where z and a are relatively prime to P, shall be complete, i. e., corresponding digits of the two halves of the period have the sum a — 1. Heinrich Bork^°^ gave an exposition, without use of the theory of num- bers, of kno^n results on decimal fractions. There is here first published (pp. 36-41) a table, computed by Friedrich Kessler, showing for each prime p< 100000 the value of q={p — l)/e, where e is the length of the period for 1/p. The cases in which ^ = 1 or 2 were omitted for brevity. He stated that there are many errors in the table to 15000 by Reuschle.'*" Cunningham^^^ listed errata in Kessler's table. L. E. Dickson^"^ proved, without the use of the concept of periodic fractions, that every integer of D digits written to the base N, which is such that its products by D distinct integers have the same D digits in the same cyclic order, is of the form A{N^ — 1)/P, where A and P are relatively prime. A number of this form is an integer only when P is prime "iZeitschrift Math. Phys., 36, 1891, 381-3; Die Elemente der Zahlentheorie, 1892, 95-97. Alike discussion occurs in l'interm(5diaire des math., 5, 1898, 57-8; 10, 1903, 91-3. "'Zeitschrift Math. Phys., 37, 1892, 190-1. "'El Progreso Matematico, 1892, 25-27, 89-93, 114-9. Cf. rinterm^diaire des math., 2, 1895, 323-4. '"Zeitschrift Math. Phys., 39, 1894, 376-382. losperiodische Dezimalbriiche, Progr. 67, Prinz Heinrichs-Gymn., Berlin, 1895, 41 pp. looQuart. Jour. Math., 27, 1895, 366-77. Chap. VI] PERIODIC DECIMAL FRACTIONS. 175 to N, and D is a multiple of the exponent d to which N belongs modulo P. The further discussion is limited to the case D = d, to exclude repetitions of the period of digits. Then the multipUers which cause a cyclic permuta- tion of the digits are the least residues of N, N^, . . . , A^^ modulo P. For A = 1, we have a solution for any N and any P prime to N. There are listed the 19 possible solutions with A>1, N^QS, and having the first digit >0. The only one with A^= 10 is 142857. General properties are noted. A like form is obtained (pp. 375-7) for an integer of D digits written to the base A^, such that its quotients by D distinct integers have the same D digits in the same cyclic order. The divisors are the least residues of N^, N^-\. . ., N modulo P. For example, if N = n, P = 7, A=4:, we get 4(11^ — 1)/7, or 631 to base 11, whose quotients by 2 and 4 are 316 and 163, to base 11. Another example is 512 to base 9. E. Lucas^ gave all the prime factors of 10"— 1 forn^ 18. F. W. Lawrence^"^ proved that the large factors of 10^^ — 1 and 10^^ — 1 are primes. C. E. Bickmore^'^^ gave the factors of 10" -1, n^ 100. Here (1023-l)/9 is marked prime on the authority of Loof , whereas the latter regarded its composition as unknown [Cunningham^^^]. There is a misprint for 43037 in 10^^-1. B. Bettini^"^ considered the number n of digits in the period of the deci- mal fraction for a/b, i. e., the exponent to which 10 belongs modulo h. If 10 is a quadratic non-residue of a prime b, n is even, but not conversely (p. 48). There is a table of values of n for each prime 6^277. V. Murer^^" considered the n = mq remainders obtained when a/b is converted into a decimal fraction with a period of length n, separated them into sets of m, starting with a given remainder, and proved that the sum of the sets is a multiple of 9 ... 9 (to m digits) . Further theorems are found when q = l, 2 or 3. J. Sachs ^^^^ tabulated all proper fractions with denominators <250 and their decimal equivalents. B. Reynolds^ ^^ repeated the rules given by Glaisher'^^' '^^ for the length of periods. He extended the rules by Sardi^^ and gave the number of times a given digit occurs in the various periods belonging to a denominator N, both for base 10 and other bases. Reynolds^^^ gave numerical results on periodic fractions for various bases the lengths of whose period is 3 or 6, and on the length of the period for 1/A^ for every base <N—1, when A^ is a prime. A. Cunningham^ ^^ applied to the question of the length of the period of a periodic fraction to any base the theory of binomial congruences [see i"Proc. London Math. Soc, 28, 1896-7, 465. Ci. Bickmore" of Ch. XVI. "'Nouv. Ann. Math., (3), 15, 1896, 222-7. "Teriodico di Mat., 12, 1897, 43-50. "o/bid., 142-150. uoaprogr. 632, Baden-Baden, Leipzig, 1898. ""Messenger Math., 27, 1897-8, 177-87. »"/feid., 28, 1898-9, 33-36, 88-91. "'/bid., 29, 1899-1900, 145-179. Errata.i" 176 History of the Theory of Numbers. [Chap.vi 201 of Ch. VII]. He gave extensive tables, and references to papers on higher residues and to tables relating to period lengths. O. Fujimaki^^* noted that if 10'" — 1 is exactly divisible by n, and the quotient is Qi. . .a^ of jn digits, the numbers obtained from the latter by cycHc permutations of the digits are all multiples of Ci . . . a^. J. Cullen, D. Biddle, and A. Cunningham^ ^^ proved that the large factor of 14 digits of (10-^+l)/(10Hl) is a prime. L. Kronecker^^^ treated periodic fractions to any base. W. P. Workman^^^ corrected three errors in Shanks'^^ table. D. Biddle^^^ concluded erroneously that (10^^ — 1)/9 is a prime. H. Hertzer"' extended Kessler's^"'^ table from 100000 to 112400, noted Reuschle's'*'^ error on the conditions that 10 be a biquadratic residue of a prime p and gave the conditions that 10 be a residue of an 8th power modulo p. For errata in the table, see Cunningham. ^^ P. Bachmann^'° proved the chief results on periodic fractions and cyclic numbers to any base g. A. Tagiuri^^^ proved theorems [F. Meyer ,^ Perkins^^] on purely periodic fractions to any base and on mixed fractions. E. B. Escott^^^ noted a misprint in Bickmore's^^^ table and two omissions in Lucas'^^ table, but described inaccurately the latter table, as noted by A. Cunningham. ^^^ A. Cunningham^^^ described various tables (cited above) which give the exponent to which 10 belongs, and listed many errata. J. R. Akerlund^^° gave the prime factors of 11 ... 1 (to n digits) for n^ 16, n = 18. K. P. Nordlund^^^ applied to periodic fractions the theorem that, if Til, . . ., rir are distinct odd primes, no one dividing a, then N = ni"''. . . Ur""^ di\'ides a^ — l, where A: = 0(iV)/2'""\ He gave the period of \/p for p a prime < 100 and of certain a/p. T. H. Miller, ^-^ generalizing the fact that the successive pairs of digits in the period for 1/7 are 14, 28, ... , investigated numbers n to the base r for which 1 _2n 4n 8n ~ — 2"~1 4 I 6 ~r • • • ) n r r r" »"Jour. of the Physics School in Tokio, 7, 1897, 16-21; Abh. Gesch. Math. Wiss., 28, 1910, 22. i"Math. Quest. Educat. Times, 72, 1900, 99-101. "'Vorlesungen iiber Zahlentheorie, I, 1901, 428-437. "'Messenger Math., 31, 1901-2, 115. "«7&id., p. 34; corrected, ibid., 33, 1903^, 126 (p. 95). "•Archiv Math. Phys., (3), 2, 1902, 249-252. ""Niedere Zahlentheorie, I, 1902, 351-363. "iPeriodico di Mat., 18, 1903, 43-58. »«Xouv. Ann. Math., (4), 3, 1903, 136; Messenger Math., 33, 1903-4, 49. '"Messenger Math., 33, 1903-4, 95-96. ^^Ibid., 14.5-155. '"Nj-t Tidsskrift for Mat., Kjobenhavn, 16 A, 1905, 97-103. '"Goteborgs Kungl. Vetenskaps-Handlingar, (4), VII-VIII, 1905. "'Proc. Edinburgh Math. Soc, 26, 1907-8, 95-6. Chap. VI] PERIODIC DECIMAL FRACTIONS. 177 whence r^ — 2n^ = 2. Besides the case r = 10, n = 7, he found r = 58, n = 41 , etc. A. Cunningham^^* noted two errors in his paper"^ and added 252^2 = ^j^o^ 9972>)^ 390112^ = 1 (mod 17«) and cases modulo p^, where p = 103, 487, attributed to Th. Gosset. A. Cunningham^^^ gave tables of the periods of \/N to the bases 2, 3, 5 for N^ 100. H. Hertzer^^" noted three errors in Bickmore's^°* table. A. Gerardin^^^ gave factors of 10" — 1, n<100, and a table of the expo- nents to which 10 belongs modulo p, a prime < 10000, with a list of errors in the tables by Burckhardt and Desmarest. A. Filippov^^^ gave two methods of determining the generating factor for the periodic fraction for 1/6 (cf. Lucas, Th^orie des nombres, p. 178). G. C. Cicioni^^^ treated the subject. E. R. Bennett^^^ proved the standard theorems by means of group theory. W. H. Jackson^^^ noted that, if a is prime to 10 and if h is chosen so that h< 10, a& = 10m — 1, the period for \/a may be written as 6]l + 10m+(10m)2+. . .+(10m)'-it -A;-10', where s is the exponent to which 10 belongs modulo a, and /c is a positive integer. Thus for a = 39, 6 = 1, we have m = 4, s = 6, and the period is 1+40+. . . + (40)^-A:-10^ ^ = .025641. G. Mignosi^^^ discussed the logic underlying the identification of an unending decimal with its generator y/q. A. Cunningham^^^ treated periodic decimals with multiples having the same digits permuted cyclically. F. Schuh^^^ considered the length g^ of the period for 1/p" for the base g, where p is a prime. He proved that qa is of the form qip% where 0^ c^ a — 2 when p = 2, a>2, while O^c^a — 1 in all other cases. For a>2, ?a-l = giP""\ • • • , qa-c+1 = qiVj Qa-c = • • • = ?2 = ?, where q = qi txcept when p = 2, gr = 4m — 1, and then g = 2. Equality of periods for moduli p" and p'' can occur for an odd prime p only when this period is gi, and for p = 2 only when it is 1 or 2. It is shown how to find the numbers g which give equal periods for p" and p, and the odd numbers g which give the period 2 for 2". "8Math. Gazette, 4, 1907-8, 209-210. Sphinx-Oedipe, 8, 1913, 131. "9Math. Gazette, 4, 1907-8, 259-267; 6, 1911-12, 63-7, 108-116. "OArchiv Math. Phys., (3), 13, 1908, 107. "^Sphinx-Oedipe, Nancy, 1908-9, 101-112. "'Spaczinskis Bote, 1908, pp. 252-263, 321-2 (Russian). "'La divisibiht^ dei numeri e la teoria delle decimaU periodiche, Perugia, 1908, 150 pp. "«Amer. Math. Monthly, 16, 1909, 79-82. "'Annals of Math., (2), 11, 1909-10, 166-8. "'II Boll. Matematica Gior. Sc.-Didat., 9, 1910, 128-138. "^Math. Quest. Educat. Times, (2), 18, 1910, 25-26. "'Nieuw Archief Wiskunde, (2), 9, 1911, 408-439. Cf. Schuh,"'"*, Ch. VII. 178 History of the Theory of Numbers. [Chap, vi T. Ghezzi^^^ considered a proper irreducible fraction m/p with p prime to the base b of numeration. Let h belong to the exponent n modulo p. In 7nb = pqi-\-ri, rih = pq2+r2,. . ., 0<ri<p, 0<r2<p,. . ., fi,. . ., r„ are distinct and r„ = w. Multiply the respective equations by 6""^, 6""^,. . . and add; we see that p hr-i A similar proof shows that m/p equals a fraction ^xith. the denominator b'{b'* — l) when 6 = aia2a3, p^piCL^a^a^^ the a's being primes and pi rela- tively prime to b, while 6' is the least power of b having the di\isor ai^a^a^, and n is the exponent to which b belongs modulo pi. F. Stasi^*° gave a long proof showing that the length of the period for b/a does not exceed that for 1/a. If the period A for 1/p has m digits and n = p5 is prime to 10, the length of the period for \/n is m if A is divisible by q; is mi if A is prune to q and if the least A(10'"^*"^^+ . . . +1) divisible by q has m = i; and is mj if A=A'a, q = aq', with A', q' relatively prime, while the least A' (10"'^*'-^^+ ... +1) divisible by q' has k=j. For a prime p5^2, 5, let 1 A h i p'' 10"* -1' and let A^ be the first of the periods of successive powers of 1/p not divisible by p; then the period for l/p''+^' has wp^' digits. If p, is a prime 9^2, 5, and Ti is the length of the period for l/p„ and if l/pj^< is the highest power of 1/pi with a period of Ti digits, the length of the period for l/p,"* is T-' = r{pi''i~^i and that for l/II Pi"< is a multiple of the 1. c. m. of the r/. If n is prime to 10 and if ri, . . . , r;„ = 1 are the successive remainders on reducing \/n to a decimal, then r^=r2i (mod n). Hence if 1/n has a period of 2i digits, r^ = \ (mod n) and conversely. But if it has a period of 2i+l digits, r-+i = 10 and conversely. *K. W. Lichtenecker^^^ gave the length of the period for 1/p, when p is a prime ^307, and the factors of 10^ — 1, r?^ 10. L. Pasternak^^^ noted that, after multiplying the terms of a fraction by 9, 3 or 7, we may assume the denominator iV' = 10m — 1. To convert Rq/N into a decimal, we have 10Rk-i = Nyk+Rk (^ = 1, 2, . . .). Set 7?^. = lOz^+e^t, ek^ 9. Since Vk'^ 9, e^ = Vk and Rk-i = mCk+Zk. Hence the successive digits of the period are the unit digits of the successive remainders. E. Maillet^^^ defined a unique development Oo+ai/n+ 02/11^+ ... of an arbitrary number, where the Oi are integers satisfjdng certain conditions. He studied the conditions that the development be limited or periodic. "»I1 Boll. Matematica Gior. Sc.-Didat., 9, 1910, 263-9. ""/bid., 11, 1912, 226-246. i"Zeit3chr. fur das Realschulwesen, 37, 1912, 338-349. i^L'enseignement math., 14, 1912, 285-9. »"L'interm6diaire des math., 20, 1913, 202-6. Chap. VI] PERIODIC DECIMAL FRACTIONS. 179 Welsch^** discussed briefly the length of the period of a decimal fraction. B. Howarth^^^ noted that D^ is not a factor of (10^"-1)/(10'*-1) if D is a prime and n is not a multiple of the length of the period for 1/D. Again/^^ (^IQmnp^- _i)/9 is not divisible by (lO'"^-!) (10"^-1)/81. A. Cunningham^^^ factored 10^^ ± 1. Known factors of lO"^ 1 are given. Cunningham^^^ gave factors of 10"*^'* — !. A. Leman^^^ gave an elementary exposition and inserted proofs of Fer- mat's theorem and related facts, with the aim to afford a concrete introduc- tion to the more elementary facts of the theory of numbers. S. Weixer^^'' would compute the period P for 1/p by multiplication, beginning at the right. Let c be the final digit of P, whence pc = 10z — l. Then c is the first digit of the period P^ for z/p. The units digit Ci of cz = 10zi-\-Ci is the tens digit of P and the units digit of P^. In CiZ-\-Zi = 1022+^2, C2 is the hundreds digit of P and the tens digit of P\ etc. A. Leman^^^ discussed the preceding paper. Problems^^^ on decimal fractions may be cited here. O. Hoppe^^^ proved that (10^^ — 1)/9 is a prime. M. Jenkins^^^ noted that if iV= a^6^. . . , where a,h,. . .are distinct primes 9^2, 5, the period for 1/N is complementary (sum of corresponding digits of the half periods is 9) if and only if the lengths of the periods for 1/a, 1/b,. . . contain the same power of 2. Kraitchik^^^ of Ch. VII and Levanen^^ of Ch. XII gave tables of ex- ponents to which 10 belongs. Bickmore and Cullen^^^ of Ch. XIV factored 10^^+1. Further Papers Involving No Theory of Numbers. J. L. Lagrange, Legons 61em. k I'^cole normale en 1795, Oeuvres 7, 200. James Adams, Annals Phil., Mag. Chem. (Thompson), (2), 2, 1821, 16-18. C. R. Telosius and S. Morck, Disquisitio. . . . Acad. Carolina, Lundae, 1838 (in Meditationum Math. . . . PubHce Defendant C. J. D. Hill, 1831, Pt. II). J. A. Arndt, Archiv Math. Phys., 1, 1841, 101-4. J. Dienger, ibid., 11, 1848, 232; Jour, fur Math., 39, 1850, 67. Wm. Wiley, Math. Magazine, 1, 1882, 7-8. A. V. Filippov, Kagans Bote, 1910, 214-221 (pedagogic). i*^L'intermediaire des math., 21, 1914, 10. "sMath. Quest. Educat. Times, 28, 1915, 101-4. "»76id., 27, 1915, 33-4. ^"Ibid., 29, 1916, 76, 88-9. i"Math. Quest, and Solutions, 3, 1917, 59. "'Vom Periodischen Dezimalbruch zur Zahlentheorie, Leipzig, 1916, 59 pp. ""Zeitschrift Math. Naturw. Unterricht, 47, 1916, 228-230. i"/6id., 230-1. i"Zeitschrift Math. Naturw. Unterricht, 12, 1881, 431; 20, 188; 23, 584. i^Proc. London Math. Soc, Records of Meeting, Dec. 6, 1917, and Feb. 14, 1918, for a revised proof. i"Math. Quest. Educ. Times, 7, 1867, 31-2. Minor results, 32, 1880, 69; 34, 1881, 97-8: 37, 1882, 44; 41, 1884, 113-4; 58, 1893, 108-9; 60, 1894, 128; 63, 1895, 34; 72, 1900, 75-6; 74, 1901, 35; (2), 2, 1902, 65-6, 84-5; 4, 1903, 29, 65-7, 95; 7, 1905, 97, 106, 109-10; 8, 1905, 57; 9, 1906, 73. Math. Quest, and Solutions, 3, 1917, 72 (table); 4, 1917, 22. I 4 I CHAPTER VII. PRIMITIVE ROOTS. BINOMIAL CONGRUENCES. Primitive Roots, Exponents, Indices. J. H. Lambert^ stated without proof that there exists a primitive root g of any given prime p, so that g^ — \ is divisible by p for e = p — 1, but not for 0<e<p — 1. L. Euler^ gave a proof which is defective. He introduced the term primitive root and proved (art. 28) that at most n integers a;<p make x" — 1 divisible by p, the proof applying equally well to any polynomial of degree n with integral coefficients. He stated (art. 29) that, for n<p, x" — 1 has all n solutions "real" if and only if n is a divisor of p — 1; in par- ticular, x^~^ — l has p — 1 solutions (referring to arts. 22, 23, where he repeated his earlier proof of Fermat's theorem). Very likely Euler had in mind the algebraic identity a;^"^ — l = (x" — 1)Q, from which he was in a position to conclude that Q has at most n— p+1 solutions, and hence x" — 1 exactly n. By an incomplete induction (arts. 32-34), he inferred that there are exactly </)(n) integers x<p for which x" — 1 is divisible by p, but x^ — \ not divisible by p for 0<Z<n, n being a divisor of p — 1 (as the context indicates). In particular, there exist <f>{p — l) primitive roots of p (art. 46). He listed all the primitive roots of each prime ^ 37. J. L. Lagrange^ proved that, if p is an odd prime and a:P-i-l=Z^+pF, where X, ^, F are polynomials in x with integral coefficients, and if x"" and x" are the highest powers of a; in X and ^ with coefficients not divisible by p, there are m integral values, numerically <p/2, of x which make X a mul- tiple of p, and fi values making ^ a multiple of p. For, by Fermat's theorem, the left member is a multiple of p for a; = ± 1, ± 2, . . . , =*= (p — 1)/2, while at most m of these values make X a multiple of p and at most fx make ^ a multiple of p. L. Euler^ stated that he knew no rule for finding a primitive root and gave a table of all the primitive roots of each prime ^41. Euler^ investigated the least exponent x (when it exists) for which fa'+g is divisible by N. Find X such that —g^\N is a multiple, say aV, of a. Then fa'-^-r is divisible by N. Set r=FX'iV = a^s, ^^1. Then j'gx-a-ff_^ is divisible by N; etc. If the problem is possible, we finally get / as the residue of /a"^""" • • • "^, whence x = a+. . . +f . For example, to find the least x for which 2"" — 1 is divisible by iV = 23, we have 1+23 = 2^3, 3-23= -2^5, -5-23= -2^7, -7-^23 = 2^, whence a^ = 3+2+2+4 = ll. ^Nova Acta Eruditomm, Leipzig, 1769, p. 127. ''Novi Comm. Acad. Petrop., 18, 1773, 85; Comm. Arith., 1, 516-537. ^Nouv. M6m. Ac. Roy. Berlin, ann^e 1775 (1777), p. 339; Oeuvres 3, 777. <0pu8c. Anal., 1, 1783 (1772),. 121; Comm. Arith., 1, 506. ^Opusc. Anal., 1, 1783 (1773), 242; Comm. .\i-ith., 2, p. 1; Opera postuma, I, 172-4. 181 r 182 History of the Theory of Numbers. [Chap, vii A. M. Legendre^ started with Lagrange's' result that, if p is a prime and n is a divisor of p — 1, (1) a:" = l(modp) has n incongruent integral roots. Let n = v^v'^ . . ., where v, v',. . . are dis- tinct primes. A root a of (1) belongs* to the exponent n if no one of a"/', a"^', . . . is congruent to unity modulo p. For, if a* = l, 0<d<n, let a be the g. c. d. of 6, n, so that a = ny—dz for integers y, z; then contrary to hypothesis. Next, of the n roots of (1), n/v satisfy x''^'' = l (mod p), and n{l — l/v) do not. Likewise, n{l — l/v') do not satisfy 2.n/«'' = 1 . qIq i^ jg gaj(j ^Q follow that there are *(„)=„ (i-i)(i-i) numbers belonging to the exponent n modulo p. If ^•^^1, ^-^-^Kmodp), /3 belongs to the exponent v"". If j8' belongs to the exponent v'^, etc., the product /3]S' ... is stated to belong to the exponent n. C. F. Gauss^ gave two proofs of the existence of primitive roots of a prime p. If d is a divisor of p — 1, and a'' is the lowest power of a congruent to unity modulo p, a is said to belong to the exponent d modulo p. Let ypid) of the integers 1, 2, . . . , p — 1 belong to the exponent d, a given divisor of p — 1 . Gauss showed that i/' (c?) = or (d) , 2 1/' (c^) = p — 1 = 2 <^ (d) , whence i^(d) —(t>{d). In his second proof, Gauss set p — 1 =a"6'^. . ., where a, 5, . . . are distinct primes, proved the existence of numbers A, B,. . . belonging to the respective exponents a", h^,. . ., and showed that AB . . . belongs to the exponent p — 1 and hence is a primitive root of p. Let a be a primitive root of p, h any integer not divisible by p, and e the integer, uniquely determined modulo p — 1, for w^hich o* = 6 (mod p). Gauss (arts. 57-59) called e the index of h for the modulus p relative to the base a, and wrote e = ind h. Thus a'^^^'^b (mod p), ind 66'=ind 6+ind h' (mod p-1). Gauss (arts. 69-72) discussed the relations between indices for different bases and the choice of the most convenient base. In articles 73-74, he gave a convenient tentative method for finding a primitive root of p. Form the period of 2 (the distinct least positive resi- dues of the successive powers of 2); if 2 belongs to an exponent ^<p — 1, select a number 6<p not in the period of 2, and form the period of 6; etc. If a belongs to the exponent t modulo p, the product of the terms in the period of a is = ( — 1)'+^ (mod p), while the sum of the terms is =0 unless a=l (arts. 75, 79). •M6m. Ac. R. Sc, Paris, 1785, 471-3. Thdorie des nombres, 1798, 413-4; ed. 3, 1830, Nos. 341-2; German transl. by Maser, 2, pp. 17-18. *This term was introduced later by Gauss.'' ^Disquisitiones Arith., 1801, arts. 52-55. Chap. VII] PRIMITIVE RoOTS, EXPONENTS, INDICES. 183 The product of all the primitive roots of a prime p^3 is =1 (mod p) ; the sum of the primitive roots of p is =0 if p — 1 is divisible by a square, but is =( — 1)" if p — 1 is the product of n distinct primes (arts. 80, 81). If p is an odd prime and e is the g. c. d. of ^(p")=p''~^(p — 1) and t, then x' = l (mod p") has exactly e incongruent roots. It follows that there exist primitive roots of p", i. e., numbers belonging to the exponent 0(p") (arts. 85-89). For n>2, every odd number belongs modulo 2" to an exponent which divides 2""^, so that primitive roots of 2" are lacking; however, a modified method of employing indices to the base 5 may be used (arts. 90, 91). If w = A"5^.., where A, B,... are distinct primes, and a=0(A"), ^=4>{B^), . . ., and if ii is the 1. c. m. of a, jS, . . ., then ^'' = 1 (mod m) for z prime to m. Now fiKa-^. . . =4>{m) except when m = 2", p" or 2p", where p is an odd prime. Thus there exist primitive roots of m only when m = 2, 4, p"or2p" (art. 92). Table I, at the end of Disq. Arith., gives on one page the indices of each prime <p for each prime and power of prime modulus < 100. Gauss gave no direct table to determine the number corresponding to a given index, but indicated (end of art. 316) how his Table III for the conversion of ordi- nary into decimal fractions leads to the number having a given index (cf. Gauss,i^'i^Ch. VI). S. F. Lacroix^ reproduced Gauss' second proof of the existence of primi- tive roots of a prune, without a reference. L. Poinsot^ argued that the primitive roots of a prime p may be obtained from the algebraic expressions for the imaginary (p — l)th roots of unity by increasing the numbers under the radical signs by such multiples of p that the radicals become integral. The (/)(p — 1) primitive roots of p may be obtained by excluding from 1, . . . , p — 1 the residues of the powers whose exponents are the distinct prime factors of p — 1; while symmetrical, this method is unpractical for large p. Fregier^° proved that the 2"th power of any odd number has the remainder unity when divided by 2""*"^, if n>0. Poinsot^^ developed the first point of his preceding paper. The equa- tion for the primitive 18th roots of unity is x^—x^-\-l=0. The roots are : = a^^Kl + ^^'=^ (a' = l). But \/^= ±4, -¥^=4:, ^-11 = 2 (mod 19). Thus the six primitive roots of 19 are x= —4, 2, —9, —5, —6, 3. In general, the algebraic expres- sions for the nth roots of unity represent the different integral roots of a;" = l (mod p), where p is a prime kn-\-\, after suitable integers are added to the numbers under the radical signs. Since unity is the only (integral) sCompldment des ^l^mens d'alg^bre, Paris, ed. 3, 1804, 303-7; ed. 4, 1817, 317-321. «M6m. Sc. Math, et Phys. de I'Institut de France, 14, 1813-5, 381-392. "Annales de Math, (ed., Gergonne), 9, 1818-9, 285-8. "M6m. Ac. Sc. de I'Institut de France, 4, 1819-20, 99-183. 184 History of the Theory op Numbers. [Chap, vii root of x^=l (mod p), if p is a prime >2, he concluded (p. 165) that p is a factor of the numbers under the radical signs in the formula for a primitive pth root of unity. Cf. Smith^^^ of Ch. VIII. Poinsot^^" again treated the same subject. J. Ivory^^ stated that a primitive root of a prime p satisfies x^^~^^^^^ — 1, but no one of the congruences x'=—l (mod p), <=(p — l)/(2a), where a ranges over the odd prime factors of p — 1 ; while a number not a primitive root satisfies at least one of the a;' = — 1 . Hence if each a' ^ — 1 and ^(p-i)/2^ — 1, then a is a primitive root. V. A. Lebesgue^^ stated that prior to 1829 he had given in the Bulletin du Nord, Moscow, the congruence X = of Cauchy^^ for the integers belonging to the exponent n modulo p. A. Cauchy^^ proved the existence of primitive roots of a prime p, essen- tially as in Gauss' second proof. If p — 1 is divisible by n = a"6V . . . , where a, b, c,. . . are distinct primes, he proved that the integers belonging to the exponent n modulo p coincide with the roots of Y (a^"-l)(a:"^°^-l)(a;"/°--l)... _^ , , , ^=(x"/"-l)(a:''/^-l)...(a:"/-^-l)...=^ ^^^^ P^' The roots of the equation X = are the primitive nth roots of unity. For the above divisor n of p — 1, the sum of the Zth powers of the primitive roots of a;" = 1 (mod p) is divisible by p if Z is divisible by no one of the numbers n, n/a, n/h, . . . , n/ab, . . . , n/ahc, .... But if several of these are divisors of I, and if we replace n, a, b,. . . by <t>{n), 1—a, 1 — 6, . . . in the largest of these divisors in fractional form, we get a fraction congruent to the sum of the Ith. powers. In case x'" = l (mod p) has m distinct integral roots, the sum of the lib. powers of all the roots is congruent modulo p to m or 0, according as I is or is not a multiple of m. M. A. Stern^^ proved that the product of all the numbers belonging to an exponent d is = 1 (mod p) , while their sum is divisible by p if d is divisible by a square, but is = ( — 1)" if d is a product of n distinct primes (generaliza- tions of Gauss, D. A., arts. 80, 81). If p = 2n+l and a belongs to the expo- nent n, the product of two numbers, which do not occur in the period of a, occurs in the period of a. To find a primitive root of p when p — 1 = 2ab . . . , where a,b,. . . are distinct odd primes, raise any number as 2 to the powers (p — l)/a, (p — 1)/6, . . . ; if no one of the residues modulo p is 1, the negative of the product of these residues is a primitive root of p; in case one of the residues is 1, use 3 or 5 in place of 2. If p = 2g+l and q are odd primes, 2 or — 2 is a primitive root of p according as p = 8n + 3 or 8n + 7 . If p = 4^^ -f-1 "''Jour, de I'dcole polytechnique, cah. 18, t. 11, 1820, 345-410. "Supplement to Encyclopaedia Britannica, 4, 1824, 698. "Jour, de Math., 2, 1837, 258. "Exercices de Math., 1829, 231; Oeuvrea, (2), 9, 266, 278-90. "Jour, fiir Math., 6, 1830, 147-153. Chap. VII] PRIMITIVE RoOTS, EXPONENTS, INDICES. 185 and q are primes, 2 and —2 are primitive roots of p. If p = 4g+l and g = 3n+l are primes, 3 and —3 are primitive roots of p. F. Minding^^ gave without reference Gauss' second proof of the exist- ence of primitive roots of a prime. F. J. Richelot^^ proved that, if p = 2"'+l is a prime, every quadratic non-residue (in particular, 3) is a primitive root of p. A. L. Crelle^^ gave a table showing all prime numbers ^ 101 having a given primitive root; also a table of the residues of the powers of the natural numbers when divided by the primes 3, . . ., 101. His device^^ for finding the residues modulo p of the powers of a will be clear from the example p = 7, a = 3. Write under the natural numbers <7 the residues of the successive multiples of 3 formed by successive additions of 3 ; we get 12 3 4 5 6 3 6 2 5 14. Then the residues 3, 2, 6, . . . of 3, 3^, 3^, . . . modulo 7 are found as follows: after 3 comes the number 2 below 3 in the above table; after 2 comes the number 6 below 2 in the table; etc. Crelle^° proved that, if p is a prime and X is prime to p — 1 and <p — 1, the residues modulo p of z^ range with z over the integers 1, 2,. . ., p — 1. His proof that there exist (^(n) numbers belonging to the exponent n modulo p, if n divides p — 1, is like that by Legendre.^ G. L. Dirichlet^^ employed 0(A:) systems of indices for a modulus /j = 2^p'p"'. . ., where p, p', . . . are distinct primes, and X^3. Given any integer n prime to k, and primitive roots c, c', . • • of P'> v" \ • • • > we can determine indices a, /3, 7, 7', . . . such that n=(-l)''5^ (mod 2^^), n = c^ (mod p'), n = c"'' (mod p"'),- • •• Michel Ostrogradsky^^ gave for each prime p<200 all the primitive roots of p and companion tables of the indices and corresponding numbers. (See Jacobins and Tchebychef .3^) C. G. J. Jacobi^^ gave for each prime and power of a prime < 1000 two companion tables showing the numbers with given indices and the index of each given number. In the introduction, he reproduced the table by Burckhardt, 1817, of the length of the period of the decimal fraction for 1/p, for each prime p^2543, and 22 higher primes. Of the 365 primes <2500, we therefore have 148 having 10 as a primitive root, and 73 of the form 4w+3 having —10 as a primitive root. Use is made also of the primes for which 10 or — 10 is the square or cube of a primitive root. "Anfangsgrlinde der hoheren Arith., 1832, 36-37. "Jour, ftir Math., 9, 1832, p. 5. ^Hhid., 27-53. "Also, ihid., 28, 1844, 166. "Abh. Ak. Wiss. BerUn, 1832, Math., p. 57, p. 65. ^HhU., 1837, Math., 45; Werke, 1, 1889, 333. "Lectures on alg. and transc. analysis, I-II, St. P^tersbourg, 1837; M6m. Ac. Sc. St. P6tera- bourg, s6r. 6, sc. math, et phys., 1, 1838, 359-85. 2'Canon Arithmeticus, Berlin, 1839, xl+248 pp. Errata, Cunningham.""-"" 186 History of the Theory of Numbers. [Chap. VII To find a primitive root g of p, select any convenient integer a and form the residues of a, or, a^,. . . [as by Crelle^®]. Let n be the exponent to which a belongs. Set nn' = p — \. If n<p — 1, select an integer h not in the period a, . . . , a". The residue of 6" is in this period of a. If y is the least power of 6 whose residue is in the period of a, then / divides n', say w'=i/' (P- xxiii). Since a=g'^', h^=a\ we have y^gf^''^g"''^''"'\ h = g^'^'+'^^ (mod p), for some value 0, 1, . . . , /—I of k. But A: must be chosen so that i+nk is prime to /. For, if i-\-nk = du, where d is a divisor of /, we would have 5^'' = a". The nf residues of a'b' (r = 0,. . ., n-1; s = 0,. . ., /-I) are dis- tinct ; their indices to base g are/', 2f', . . . , nff in some order and are known. If nf'<p — l, we employ an integer not in the set a^b' and proceed similarly. Ultimately we obtain a primitive root and at the same time the index of everj" number. This method was used for the primes between 200 and 1000. For primes < 200, the tables by Ostrogradsky^^ w^ere reprinted with the same errors (noted at the end of the Canon). Jacobi proved that, if n is an odd prime, any primitive root of n^ is a primitive root of any higher power of n (p. xxxv). For the modulus 2", 4^iu^9, the final tables give the index / of any positive odd number to base 3, where (_l)(Ar-l)W-3)/8^ = 37 (jjjQ^ 2"). Robert ]\Iurphy-^ stated the empirical theorem that every prime anr+p has a as a primitive root if p>a/2, p is a prime <a, and if a is a primitive root of p. For example, a prime 10nr-{-7 has 10 as a primitive root. H. G. Erlerus'^ considered two odd primes p and p' and a number m such that m=a (mod p), m = a' (mod p'). Let a belong to the exponent e modulo p, and a' to the exponent e' modulo p\ If 8 is the g. c. d. of e and e', then m belongs to the exponent ee'/8 modulo pp'. He discussed at length the number of integers belonging to the exponent n for a com- posite modulus. A. Cauchy^^ called the least positive integer i for which m' = 1 (mod n) the indicator relative (or corresponding) to the base m and modulus n, which are assumed relatively prime. If the base m is constant, and ii, 12 are the indicators corresponding to moduli nj, 112, and if n = nin2 is prime to 772, then the 1. c. m. of I'l and {2 is the indicator corresponding to modulus n. If the modulus n is constant, and ii, io are the indicators corresponding to bases Wi, ^2, and if I'l, 1*2 are relatively prime, then 1*112 is the indicator corresponding to the base 7^17^2. Let I'l, io be the indicators corresponding to the bases mi, 7722 and same modulus n. The g. c. d. 0; of I'l, 2*2 can be expressed (often in several ways) as a product uv such that ii/u, io/v are relatively prime. For, if co = a/3. . . , "Phil. Mag., (3), 19, 1841, 369. "Elementa Doctrinse Numerorum, Diss., Halis, 1841, 18-43. »«Comptes Rendus Paris, 12, 1841, 824-845; Oeuvres, (1), 6, 124-146. Chap. VII] PRIMITIVE RoOTS, EXPONENTS, INDICES. 187 where a, j8, . . . are powers of distinct primes, use a as a factor in forming u in case a is prime to ii/a, but as a factor of y in case a is prime to zVa, and as a factor of either u or v indifferently in case a is prime to both ii/a and 12/0. Since ii/u and i2/v are relatively prime indicators corresponding to bases mi" and m2^ it follows from the preceding theorem that the indicator corresponding to base mi"-W2'' and modulus n is ii ^2 iii2 1 r • • = — = 1. c. m. of ii, i2. U V £0 Hence, given several bases mi, m2, . . . and a single modulus n, we can find a new base relative to which the indicator is the 1. c. m. of the indicators corresponding to mi, m2, .... If the latter bases include all the integers <n and prime to n, the corresponding indicators give all indicators which can correspond to modulus n, so that all of them divide a certain maximum indicator I. Then for every integer m relatively prime to n,m^ = l (mod n) . If n = v°', where v is an odd prime, or if n = 2 or 4, l=^{n). If n = 2^ k>2, I=(f}{n)/2. If Ij is the maximum indicator corresponding to a power Uj of a prime, and if n = llnj, then I is the 1. c. m. of /i, /2, • • •• The equation mx — ny = l has the solution x = 7n^~^ (mod n). Cauchy^^ republished the preceding paper, but with an extension from the limit n = 100 to the limit n = 1000 for his table of the maximum indicator I. C. F. Arndt^^ gave (without reference) Gauss' second proof of the exist- ence of a primitive root of an odd prime p, and proved the existence of the <^(p") primitive roots of p'* or 2p'', and that there are no primitive roots for moduli other than these and 4. If i is a divisor of 2""^, n>2, exactly t numbers belong to the exponent t modulo 2'' (p. 18). If, for a modulus p", 2p", a belongs to the exponent t, then a-a^ . . .a' is congruent to ( — 1)'+^ (pp. 26-27), while the product of the numbers belonging to the exponent t is congruent to +1 if ^?^ 2 (pp. 37-38). He proved also Stern's^^ theorem on the sum of these numbers. He gave the same two theorems also in a later paper. ^^ L. Poinsot^" used the method of Legendre^ to prove the existence of 4>{n) integers belonging to the exponent n, sl divisor of p — 1, where p is a prime. He gave (pp. 71-75) essentially Gauss' first proof, and gave his own^ method of finding primitive roots of a prime. The existence of primitive roots of p", 2p", 4, but of no further moduli, is established by use of the number of roots of binomial congruences (pp. 87-101). C. F. Arndt^^ noted that if a belongs to an even exponent t modulo 2", then ±a, ±a^, ..., ±a'~^ give the t incongruent numbers belonging to the exponent t, and are congruent to A; • 2"" =f 1 (A; = 1 , 3, 5, . . . ) . The product of the numbers belonging to the exponent t modulo 2", n>2, is = +1. • "Exercices d' Analyse et de Phys. Math., 2, 1841, 1-40; Oeuvres, (2), 12. "Archiv Math. Phys., 2, 1842, 9, 15-16. "Jour, flir Math., 31, 1846, 326-8. 3«Jour. de Math^matiques, (1), 10, 1845, 65-70, 72. "Archiv Math. Phys., 6, 1845, 395, 399. 188 History of the Theory of Numbers. [Chap. VII E. Prouhet^" gave, without reference, Crelle's^' method of forming the residues of the powers of a number. The object of the paper is to give a uniform method of proof of theorems, given in various places in Legendre's text, relating to the residues of the first n powers of an integer belonging to the exponent n modulo P, especially when P is a prime or a power of a prime, and the existence of primitive roots. He gave (p. 658) the usual proof that =•= 2 is a primitive root of a prime 2^+ 1 if 5 is a prime 4/c± 1 (with a misprint). C. F. Arndt^^ proved that if ^ is a primitive root of the odd prime p and if p^ (SKn) is the highest power of p dividing G = g^~^ — 1, then g belongs to the exponent p'*~''(p — 1) modulo p". Conversely, if the last is true of a primitive root g of p, then G is divisible by p^ and not by p^"'"^ The first result with X = 1 shows that any primitive root of p^ is a primitive root of p", n>2. Let g he a, primitive root of p; if G is not divisible by p^, g is a. primitive root of p^; but if G is divisible by p^, and h is not divisible by p, then g+hp is a primitive root of p^. Any odd primitive root of p** is a primitive root of 2p". If gr is a primitive root of p'* or 2p'*, and t is a divisor of p"~^(p — 1), then if a ranges over the integers <t and prime to t, the <f>{t) integers belonging to the exponent t modulo p" or 2p" are g% where e = p"~^(p — l)a/<. The numbers belonging to the exponent 2"""* modulo 2" are found more simply than by Gauss'^ and Jacobi^^ (p. 37). P. L. Tchebychef^^ proved that if a, /3, . . . are the distinct prime factors of p — 1, where p is a prime, then a is a primitive root of p if and only if no one of the congruences x'' = a, xP = a,. . . (mod p) has an integral root. This furnishes a method (usually impracticable) of finding all primitive roots of p. A second method uses a number a belonging to the exponent n, and a number h not congruent to a power of a, and deduces a number belonging to an exponent >n. In the second supplement, he proved that 3 is a primitive root of any prime 2^"+ 1 ; that =*= 2 is a primitive root of any prime 2a +1 such that a is a prime 4A:± 1 ; 3 is a primitive root of 4iV2"'4-l if w>0 and iV is a prime >9^ /(4-2'"); 2 is a primitive root of any prime 4iVH-l such that A^ is an odd prune. The last result was later proposed'^ as a question for solution (with reference to this text) . There is given the table of primitive roots and indices for primes < 200, due to Ostrogradsky^^. Schapira (p. 314) noted that in the list of errata in Jacobi's^^ Canon (p. 222) there is omitted the error 8 for 6 in ind 14 for p = 25. V. A. Lebesgue^*^ remarked that Cauchy's^^ congruence X=0 shows the existence of 0(n) integers belonging to the exponent n modulo p, a prime. »«Nouv. Ann. Math., 5, 1846, 175-87, 659-62, 675-83. »3Jour. fiir Math., 31, 1846, 259-68. "Theory of Congruences (m Russian), 1849. German translation by Schapira, Berlin, 1889, p. 192. Italian translation by Mile. Massarini, Rome, 1895, with an extension of the tables of indices to 353. »Nouv. Ann. Math., 15, 1856, 353. Solved by use of Euler's criterion by P. H. Rochette, and., 16, 1857, 159. Also proved by Desmarest,*^ p. 278. »*Nouv. Ann. Math., 8, 1849, 352; 11, 1852, 420. Chap. VII] PRIMITIVE RoOTS, EXPONENTS, INDICES. 189 E. Desmarest" devoted the last 86 pages of his book to primitive roots; the 70 pages claimed to be new might well have been reduced to five by the omission of trivial matters and the use of standard notations. To find (pp. 267-8) a primitive root of the prime P = 6g+l, where q is an odd prime, seek an odd solution of ^^^+3 = (mod P) and set w = 2/2 — 1; then R^=—l and R belongs to the exponent 6; thus we know the solutions of x^ = l\ let a be any integer prime to P and not such a solution; if a^=±l, then =ta belongs to the exponent q, and ±ai2 is a primitive root of P; but, if a^« 7^ 1 , then a^^= =f 1 (mod P) , and =*= a is a primitive root of P. If P = 8Q + 1 and Q are primes, then P=5 (mod 12) and 3 is a quadratic non-residue and hence a primitive root of P. Let P be a prime of the form 5q^2. Then u^= 5 (mod P) is not solvable. Thus, if a is a primitive root of P, 5 = a% where e is odd. Thus if e is prime to P — 1, 5 is a primitive root of P. It is recommended that 5 be the first number used in seeking by trial a primitive root. And yet he announced the theorem (p. 283) that 5 is in general a primitive root. If P is a prime 5g±2 also of the form 2"Q+1, where Q is an odd prime including 1, then (pp. 284-6) 5 is a primitive root of P provided P is not a factor of 5^ —1. He gave the factors of the latter and of 10^" — 1 for n = 1, . . . , 5. ' Results, corresponding to those just quoted for 5, are stated for p = 7, — 7, 10, 17. What is really given is a Hst of the linear forms of the primes P for which p is a quadratic non-residue. If, in addition, P = 2''Q + 1, where Q is an odd prime, then p is a primitive root, provided p^^^^l (mbd P). The last condition is ignored in his statement of his results and again in his collection (pp. 297-8) of "principles which give primitive roots" entered in his table (pp. 298-300) giving a primitive root of each prime < 10000. V. A. Lebesgue^^ proved that, if a and p = 2'a+l are primes, any quad- ratic non-residue x of p is a primitive root of p if a;2*-'+1^0(modp). J. P. Kulik^^ gave for each prime p between 103 and 353 the indices and all the primitive roots of p. His manuscript extended to 1000. There is an initial table giving the least primitive root of the primes from 103 to 1009. G. 01tramare^° called x a root of order or index m of a prime piix belongs to the exponent {p — l)/m modulo p. Let Xm{x) = (mod p) be the con- gruence whose roots are exclusively the roots of order moi p. By changing X to x^^"", we obtain Xmn=<l>{^) ^0. li rii, n2, . . . , n are the divisors > 1 of w. Am — ■ Y Y '^Th^orie des nombres. Traits de I'analyse ind6terminee du second degr6 k deux inconnues suivi de I'application de cette analyse k la recherche des racines primitives avec une table de ces racines pour tous les nombres premiers compris entre 1 et 10000, Paris, 1852, 308 pp. For errata, see Cunningham, Mess. Math., 33, 1903, 145. 58Nouv. Ann. Math., 11, 1852, 422-4. '9 Jour, fur Math., 45, 1853, 55-81. "7Wd., 303-9. 190 History of the Theory of Numbers. [Chap, vii V. A. Lebesgue^^ noted that, given a primitive root g (g<p) of the prime p, we can find at once the primitive roots of p". Let g' be the positive residue <p~ when g^ is divided by p^ and set h = {g' — g)/p. Then g+px+p'^y {y = 0,. . ., p""^-!; x = 0,. . ., p-1; x^h) give p"~^(p — 1) primitive roots. Replacing g by g\ where i is less than and prime to p — 1, we obtain ]0(p") } primitive roots of p". In particular, a primitive root of p~ is a primitive root of p" (Jacobi^). But, if h = 0, g is not a primitive root of p". Since ginda+e^p_^ (mod p") , e = ip"-np-l), we can reduce by half the size of Jacobi's Canon. D. A. da Silva^^ gave two proofs that x'^ = l (mod p) has (f)(d) primitive roots, if d divides p — 1, and perfected the method of Poinsot^'^" for finding the primitive roots of a prime. F. Landry^^" was led to the same conclusion as Ivory.^^ In particular, if p = 2* + l, or if p = 2n+l (n an odd prime) and a7^p — l, any quadratic non-residue a of p is a primitive root. For each prime p< 10000, at least one prime ^ 19 is a quadratic non-residue of p. Cauchy's^* congruence for the primitive roots is derived and proved. G. Oltramare*^ proved that — 3°2^'' is a primitive root of the prime p = 2a/3 + l, if a^3, /3f^3, S'^^l, 22^^1 (mod p); that, if p = 3-2"'-M=g2+3r2, qx-]-ry = l, { — l+qy — 3rx)5i^/2 is a primitive root of p; and analogous theorems. If a and 2a-^l are primes, 2 or a is a primitive root of 2a -fl, according as a is of the form 4n-[-l or 4n+3. If a is a prime 9^3 and if p = 2a4-l is a prime and m> 1, then 3 is a primitive root of p unless 3^'"~^-|-l=0 (mod p). [Cf. Smith.''^] P. Buttel^ attributed to Scheffler (Die unbestimmte Analytik, 1854, §142) the method of Crelle^^ for finding the residues of powers. C. G. Reuschle's^^ table C gives the Haupt-exponent {i. e., exponent to which the number belongs) (a) of 10, 2, 3, 5, 6, 7 with respect to all primes p< 1000, and the least primitive root of p; (b) of 10 and 2 for 1000< p< 5000 and a convenient primitive root; (c) of 10 for 5000<p< 15000 (no primitive root given). Numerous errata have been listed by Cunningham."" Allegret^^ stated that if n is odd, n is not a primitive root of a prime 2^^n-f-l, X>0; proof can be made as in Lebesgue.^^ "Comptes Rendus Paris, 39, 1854, 1069-71; same in Jour, de Math., 19, 1854, 334-6. **Proprietades geraes et resolu^ao directa das Congruencias binomias, Lisbon, 1854. Report by C. Alasia, Rivista di Fisica, Mat. e Sc. Nat., Pavia, 4, 1903, 25, 27-28; and Annaes Scientificos Acad. Polyt. do Porto, Coimbra, 4, 1909, 166-192. **'Troi8i&me m6moire sur la thdorie des nombres, Paris, 1854, 24 pp. "Jour, fiir Math., 49, 1855, 161-86. "Archiv Math. Phys., 26, 1856, 247. "Math. Abhandlung. . .Tabellen, Prog. Stuttgart, 1856; full title in the chapter on perfect numbers. i''^ "Nouv. Ann. Math., 16, 1857, 309-310. Chap. VII] PRIMITIVE RoOTS, EXPONENTS, INDICES. 191 H. J. S. Smith^^ stated that some of Oltramare's^^ general results are erroneous at least in expression, and gave a simple proof that 0^"^= 1 (mod p**) has exactly d roots if d divides 0(p"). V. A. Lebesgue^^ proved that, if p is an odd prime and a, b belong to exponents a, (3, there exist numbers belonging to the 1. c. m. m of a, (3, as exponent. Hence if neither a nor /3 is a multiple of the other, w exceeds a and /3. If d<p — l is the greatest of the exponents to which 1, . . ., p — 1 belong, the latter do not all belong to exponents dividing d, since otherwise they would give more than d roots of x'^=l (mod p). Hence there exist primitive roots of p. If a is odd, ±l+2°a belongs to the exponent 2™~" modulo 2"" (p. 87). If h belongs to the exponent k modulo p, a prime, then h+Pz belongs modulo p" to an exponent which divides A;p"~^ (p. 101). If / is a primitive root of p, and f^~^ — l=pz, then / is a primitive root of p™ if and only if z is not divisible by p (p. 102). G. L. Dirichlet^^ proved the last theorem and explained his^^ system of indices for a composite modulus. V. A. Lebesgue^° published tables, constructed by J. Hoiiel,^^ of indices and corresponding numbers for each prime and power of prime modulus < 200, which differ from Jacobi's^^ only in the choice of the least primitive root. There is an auxiliary table of the indices of x\ for prime moduli <200. V. A. Lebesgue^^ stated that, if g'<p is a primitive root of the prime p and if g'=g^~^ (mod p), then g' is a primitive root of p; at least one of g and g' is a primitive root of p" for n arbitrary. V. Bouniakowsky^^ proved in a new way the theorems of Tchebychef^* that 2 is a primitive root of p = 8n+3 if p and 4n+l are primes, and of p = 4nH-l if p and n are primes. He gave a method to find the exponent to which 2 or 10 belongs modulo p. A. Cayley^^ gave a specimen table showing the indices a, j3,. . . for every number M = a"6^. . .(modiV), where ilf<iV and prime to iV, for iV = l,. . ., 50. There is no apparent way of forming another single table for all A^'s analo- gous to Jacobi's tables (one for each N) of numbers corresponding to given indices. F. W. A. Heime^^ gave the least primitive root of each prime < 1000. His other results are not new. A secondary root of a prime p is one belong- ing to an exponent < p — 1 modulo p. "British Assoc. Report, 1859, 228; 1860, 120, §73; Coll. Math. Papers, 1, 50, 158 (Report on theory of numbers). **Introd. th^orie des nombres, 1862, 94-96. "Zahlentheorie, §§128-131, 1863; ed. 2, 1871; ed. 3, 1879; ed. 4, 1894. soM^m. soc. sc. phys. et nat. de Bordeaux, 3, cah. 2, 1864-5, 231-274. "Formiiles et tables numer., Paris, 1866. For moduli ^ 347. ^Comptes Rendus Paris, 64, 1867, 1268-9. "Bull. Ac. Sc. St. Petersbourg, 11, 1867, 97-123. "Quart. Jour. Math., 9, 1868, 95-96. "Untersuchungen, besonders in Bezug auf relative Primzahlen, primitive u. secundare Wurzeln, quadratische Reste u. Nichtreste; nebst Berechnung der kleinsten primitiven Wurzeln vorf alien Primzahlen zwischen 1 und 1000. BerUn, 1868; ed. 2, 1869. 192 History of the Theory of Numbers. [Chap, vii C. J. D. Hill^^ noted that his tables of indices for the moduU 2" and 5" (n^5) give the residues of numbers modulo 10", i. e., the last n digits. Using also tables for the moduli 9091 and 9901, as well as a table of loga- rithms, we are able to determine the last 22 digits. B. M. Goldberg^^ gave the least primitive root of each prime < 10160. V. Bouniakowsky^^ proved that 3 is a primitive root of p if p = 24n+5 and (p — 1)/4 are primes; —3 is a primitive root of p if p = 12n+ll and (p — 1)/2 are primes; if p is a primitive root of the prime p = 4n+l, one (or both) of p, p—p is a primitive root of p"' and of 2p"'; 5 is a primitive root of p = 20?i+3 or 20n4-7 if p and {p — 1)/2 are primes, and of p = 40n + 13 or 4071-1-37 if p and (p — 1)/4 are primes; 6 is a primitive root of a prime 24n-|-ll and —6 of 24n-|-23 if (p — 1)/2 is a prime; 10 is a primitive root of p = 40n+7, 19, 23, and -10 of p = 40n+3, 27, 39, if (p-l)/2 is a prime; 10 is a primitive root of a prime 80n-(-73, n>0, or 80n+57, n>l, if (p — 1)/8 is a prime. If p = 8an-h2a — 1 or 8an+a— 2 and (p — 1)/4 are primes, and if a^-f-1 is not divisible by p, a is a primitive root of p. V. A. Lebesgue^^ proved certain theorems due to Jacobi^^ and the following theorem which gives a method different from Jacobi's for forming a table of indices for a prime modulus p: If a belongs to the exponent n, and if 6 is not in the period of a, and if / is the least positive exponent for which h^=a\ then x^=a has the root a'6", where ft-\-iu — l=nv; the root belongs to the exponent nf if and only if u is prime to /. Consider the congruence x*" = a (mod p) , where a belongs to the exponent n = (p — l)/n', and m is a divisor of n'. Every root r has a period of mn terms if no one of the residues of r, r^,. . ., r*""^ is in the period of a. If all the prime divisors of m divide n, the m roots have a period of mn terms; but if m has prime divisors g, r, . . . , not dividing n, there are only -(^X^)- roots having a period of mn terms. The existence of primitive roots follows; this is already the case if m = n'. Mention is made of companion tables in manuscript giving indices of numbers, and numbers corresponding to indices, constructed by J. Ch. Dupain in full for p<200, but from 200 to 1500 with reduction to one-half in view of ind p — a=ind a=t(p — 1)/2 modulo p — 1. L. Kronecker^^ proved the existence of two series of positive integers Qj, m, {j=l,. . ., p) such that the least positive residues modulo A:>2 of ^1 V2*' • • ■ ^p*" give all the (f){k) positive integers <A: and prime to k, if ii=0, 1,. . ., mi — 1; i2 = 0, 1,. . ., m2 — 1; etc. [cf. Mertens^^]. G. Barillari^"" proved that, if a is prime to h and belongs to the exponent "Jour, fur Math., 70, 1869, 282-8; Acta Univ. Lundensis, Lund, 1, 1864 (Math.), No. 6, 18 pp. "Rest- und Quotient-Rechnung, Hamburg, 1869, 97-138. "BuU. Ac. Sc. St. P6tersbourg, 14, 1869, 375-81. »«Compte8 Rendua Paris, 70, 1870, 1243-1251. "Monatsber. Ak. BerUn, 1870, 881. Cf. Traub, Archiv Math. Phys., 37, 1861, 278-94. •KJiomale di Mat., 9, 1871, 125-135. Chap. VII] PrBIITIVE RoOTS, EXPONENTS, INDICES. 193 m modulo h, and if h^ is the highest power of h which divides a"* — 1, and if n^/i, then 6" divides a* — 1 where e = m6"~\ Further, if 6 is a prime, a belongs to the exponent e modulo 6". For a new prime 6', let m', n', h' have the corresponding properties. Then the exponent to which a belongs modulo B = })%'"' ... is the 1. c. m. L of m6''-\ m'b'"'-''', .... For a = 10, we see that L is the length of the period for the irreducible fraction N/B. L. Sancery^^ proved that if p is a prime and a<p belongs to the exponent 6 modulo p, there exists an infinitude of numbers a-\-px = A such that A^—1 is divisible by p^, but not by p'''^^, where k is any assigned positive integer. If A belongs to the exponent 6 modulo p>2, A will belong to the exponent 6 modulo p" if the highest power of p which divides .A^ — 1 is ^p"; but if it be p"'^, A belongs to the exponent dp^ modulo p" [Barillari^°"]. Hence A is a primitive root of p" if a primitive root of p and if A^~^ — 1 is not divisible by p^, and there are ^j^CpOj primitive roots of p" or 2p\ [Generalization of Arndt.^^j C. A. Laisant®^ noted that if a belongs to the exponent 3 modulo p, a prime, then a + 1 belongs to the exponent 6, and conversely. If a belongs to the exponent 6, a+1 will not belong to the exponent 3 unless p = 7, a = 3. Hence if p is a prime 6m +1, there are two numbers a, h belonging to the exponent 3, and two numbers a + 1, 6+1 belonging to the exponent 6; also, a+6 = p — 1. If (p. 399) p+5 is an odd prime and p is even, then pV— 9> p^qP = p (mod p+g). G. Bella vitis^^'' gave, for each power p'^383 of a prime p, the periodic fraction for 1/p' to the base 2 and showed how to deduce the indices of numbers for the modulus p\ Let ? = p'~^(p — 1) and let 2 belong to the exponent q/r modulo p\ A root b of 6'"= 2 (mod p') is the base of the system of indices. G. Frattini^^ proved by the theory of roots of unity that, if p is a prime, the number of interchanges necessary to pass from 1, 2, . . ., p — 2 to ind 2, ind 3, . . . , ind (p — 1) and to ind 1— ind 2, ind 2 — ind 3, . . ., ind (p — 2)— ind (p — 1) are both even or both odd. Fritz Hofmann^^ used rotations of regular polygons to prove theorems on the sum of the primitive roots of a prime (Gauss^). A. R. Forsyth^^ found the sum of the cth powers of the primitive roots of a prime p. The sum is divisible by p if p — 1 contains the square of a prime not dividing c or if it contains a prime dividing c but with an exponent exceeding by at least 2 its exponent in c. If neither of these conditions is satisfied, the result is not so simple. "BuU. Soc. Math, de France, 4, 1875-6, 23-29. fi^M^m. Soc. So. Phys. et Nat. de Bordeaux, (2), 1, 1876, 400-2. "^lAtti Accad. Lincei, Mem. Sc. Fis. Mat., (3), 1, 1876-7, 778-800. "Giornale di Mat., 18, 1880, 369-76. "Math. Annalen, 20, 1882, 471-86. ^'Messenger of Math., 13, 1883-4, 180-5. 194 History of the Theory of Numbers. [Chap, vii J. Perott^^ gave a simple proof that x^ = l (mod p") has p'' roots. Thus there exists an integer b belonging to the exponent p""^ modulo p"*. Assum- ing the existence of a primitive root of p, we employ a power of it and obtain a number a belonging to the exponent p — 1 modulo p". Hence ab is a primitive root of p". Schwartz^" stated, and Hacken proved, the final theorem of Cauchy.^* L. Gegenbauer^^ stated 19 theorems of which a specimen is the follow- mg: If p = 8a(8/3+l) + 24/3+5 and (p-l)/4 are primes and if 64a2+48a + 10 is relatively prime to p, then 8a+3 is a primitive root of p. G. Wertheim^^ gave the least primitive root of each prime < 1000 and companion tables of indices and numbers for primes < 100. He reproduced (pp. 125-130) arts. 80-81 of Gauss^ and stated the generalization by Stern.i^ H. Keferstein''' would obtain all primitive roots of a prime p by excluding all residues of powers with exponents dividing p — 1 [Poinsot^]. IM. F. Daniels"^ gave a proof like Legendre's^ that there are <f>{n) num- bers belonging to the exponent n modulo p, a prime, if n divides p — 1. *K. Szily^- discussed the "comparative number" of primitive roots. E. Lucas"^ gave the name reduced indicator of n to Cauchy's^^ maximum indicator of n, and noted that it is a divisor <4){n) of 0(n) except when n = 2, 4, p* or 2p^', where p is an odd prime, and then equals (f>{n). The exponent to which a belongs modulo m is called the "gaussien" of a modulo m (preface, xv, and p. 440). H. Scheffler"'* gave, without reference, the theorem due to' Richelot^'^ and the final one by Prouhet.^- To test if a proposed number a is a primitive root of a prime p, note whether p is of one of the linear forms of primes for which a is a quadratic non-residue, and, if so, raise a to the powders whose exponents divide (p — 1)/2. L. Contejean^^ noted that the argument in Serret's Algebre, 2, No. 318, leads to the following result [for the case a = 10]: If p is an odd prime and a belongs to the exponent e = {p — l)/q modulo p, it belongs to the exponent p-'^e modulo p" when (a*— l)/p is not di\dsible by p, but to a smaller exponent if it is divisible by p [Sancery®^]. P. Bachmann^^ proved the existence of a primitive root of a prime p by use of the group of the residues 1, . . . , p — 1 under multiplication. **Bull. des Sc. Math., 9, I, 1885, 21-24. For k = n — l the theorem is contained imphcitly in a posthumous fragment by Gauss, Werke, 2, 266. "Mathesis, 6, 1886, 280; 7, 1887, 124-5. «8Sitzungsber. Ak. Wiss. Wien (Math.), 95, II, 1887, 843-5. "Elemente der Zahlentheorie, 1887, 116, 375-381. '"Mitt. Math. Gesell. Hamburg, 1, 1889, 256. ^'Lineaire Congruenties, Diss., Amsterdam, 1890, 92-99. "Math, ^s termes ^rtesito (Memou^ Hungarian Ac. Sc), 9, 1891, 264; 10, 1892, 19. Magyar Tudom. Ak. Ertesitoje (Report of Hungarian Ac. Sc), 2, 1891, 478. "Th^orie des nombres, 1891, 429. '*Beitrage zur Zahlentheorie, 1891, 135-143. "Bull. Soc. Philomathique de Paris, (8), 4, 1891-2, 66-70. "Die Elemente der Zahlentheorie, 1892, 89. Chap. VII] PeIMITIVE RoOTS, EXPONENTS, INDICES. 195 G. B. Mathews'^^ reproduced art. 81 of Gauss''' and gave a second proof by use of Cauchy's^^ congruence X=0 for n = p — 1. K. Zsigmondy^^ treated the problem to find all integers K, relatively prime to given integers a and 6, such that a''=b'' (mod K) holds for the given integral value <T = y, but for no smaller value. For 6 = 1, it is a question of the moduli K with respect to which a belongs to the exponent y. Set y=Ilqi*, where the q's are distinct primes and qi the greatest. Then all the primes K for which a''=b'' (mod K) holds for (T = 7, but for no smaller a, coincide with the prime factors of y y (a^-6^)n(a««'-6««'). . . A = n(a^/«-6^/«)... in which the products extend over the combinations of qi,q2,--- one, two, . . . at a time, provided that, if a''=h'' (mod qi) for (J=^ylqi\ but for no smaller (T, we do not include among the K's the prime q^, which then occurs in A to the first power only. If the prime p is a K and if p^ is the highest power of p dividing A, then p* is the highest power of p giving a K. The com- posite i^'s are now easily found. If a and 6 are not both numerically equal to unity, it is shown that there is at least one prime K except in the following cases: 7 = 1, a-6 = l; 7 = 2, 0+6 = ^2" (/x^l); 7 = 3, a = ±2, 6==f1; 7 = 6, a = ='=2, 6 = ±l. The case h = \ shows that, apart from the corre- sponding exceptions, there exists a prime with respect to which the given integer aj^^^X belongs to the given exponent 7. As a corollary, every arithmetical progression of the type mT+I ()" = 1? 2, . . .) contains an infini- tude of primes. Zsigmondy^^ considered the function A^(a) obtained from the above A by setting 6 = 1. If a is a primitive root of the prime p = l+7, the main theorem of the last paper shows that p divides A^(a). Conversely, I+7 is a prime if it divides A. Thus, if all the primes of a set of integers possess the same primitive root a, any integer p of the set is a prime if and only if Ap_i(a) is divisible by p. Hence theorems due to Tchebychef^^ imply criteria for primes. For example, a prime 2^"+l has the primitive root 3 implies that 2^"+l is a prime if and only if it divides 3^ + 1, where k = 2^'' . Since ±2 is a primitive root of any prime 2q-\rX such that g- is a prime 4/c± 1, we infer that, if g is a prime 4/c± 1, then 2g+l is a prime if and only if it divides (2^±1)/(2±1). Since 2 is a primitive root of a prime 4A''+1 such that N is an odd prime, we infer that, if N is an odd prime, 4A^+1 is a prime if and only if it divides (2^^-|-l)/5. G. F. Bennett^° proved (pp. 196-7) the first theorem of Cauchy,^^ and (pp. 199-201) the results of Sancery.^^ If a and a' belong to exponents t and t' which contain no prime factor raised to the same power in each, then the exponent to which aa' belongs is the 1. c. m. of t and t' (p. 194). "Theory of Numbers, 1892, 23-25. "Monatshefte Math. Phys., 3, 1892, 265-284. 'Hhid., 4, 1893, 79-80. 8»Phil. Trans. R. Sec. London, 184 A, 1893, 189-245. 196 History of the Theory of Numbers. [Chap, vii If 2*'^^ is the highest power of 2 dividing a^ — 1, where a is odd, the exponent to which a belongs modulo 2^ is 2^~' if X>s, but, if X^s, is 1 if a=l, 2 if o^=l, a^l (mod 2^^); the result of Lebesgue'*^ (p. 87) now follows (pp. 202-6) . In case a is not prime to the modulus, there is an evident theorem on the earliest power of a congruent to a higher power (p. 209). If e is a given divisor of 0(w), there is determined the number of integers belonging to exponent e modulo m [cf. Erlerus^^]. If a, a',. . . belong to the exponents t, t',... and if no two of the «' . . . numbers a'a"' . . . {0^r<t,0^r'<t',. . .) are congruent modulo m, then a, a', . . . are called independent generators of the 4>{m) integers <m and prime to m (p. 195); a particular set of generators is given and the most general set is investigated (pp. 220-241) [a special problem on abelian groups]. J. Perott^^ found a number belonging to an exponent which is the 1. c. m. of the exponents to which given numbers belong. If, for a prime modulus p, a belongs to an exponent t>l, and b to an exponent which divides t, then b is congruent to a power of a (proof by use of Newton's relations between the sums of like powers of a, . . . , a' and their elementary symmetric func- tions). Hence there exists a primitive root of p. M. Frolo . ^- noted that all the quadratic non-residues of a prime modulus m are primitive roots of m if m = 2^''4-l, m = 2n+l or 4n + l with n an odd prime [Tchebychef^^]. To find primitive roots of m "without any trial," separate the m — 1 integers <m into sets of fours a, b, —a, —b, where a6=l (mod m): Begin with one such set, say 1, 1, —1, —1. Either a or m — a is even; divide the even one by 2 and multiply the corresponding =t 6 by 2 ; we get another set of four. Repeat the process. If the resulting series of sets contains all m — 1 integers <m, 2 and —2 are primitive roots if w = 4/i+l, and one of them is a primitive root if m = 4/i — 1. If the sets just obtained do not include all m — 1 integers <m, further theorems are proved. G. Wertheim^^ gave the least primitive root of each prime p <3000. L. Gegenbauer^^" gave two expressions for the sum Sk of those terms of a complete set of residues modulo p which belong to the exponent k, and evaluated l>Sk/t fit) with t ranging over the divisors of k. G. Wertheim^^ proved that any prime 2'*" + l has the primitive root 7. If p = 2"g-|-l is a prime and ^ is a prime >2, any quadratic non-residue m of p is a primitive root of p if m"" — 1 is not divisible by p. As corollaries, we get primes q of certain linear forms for which 2, 5, 7 are primitive roots of a prime 2^-1-1 or 4g-f-l; also, 3 is a primitive root of all primes 8g-|-l or 16g+l except 41; and cases when 5 or 7 is a primitive root of primes 8^+1, lQq+1. There is given a table showing the least primitive root of each prime between 3000 and 3500. "BuU. des Sc. Math., (2), 17, I, 1893, 66-83. ""BuU. Soc. Math, de France, 21, 1893, 113-128; 22, 1894, 241-5. "Acta Mathematica, 17, 1893, 315-20; correction, 22, 1899, 200 (10 for p = 1021). 8"Denkschr. Ak. Wiss. Wien (Math.), 60, 1893, 48-60. "Zeitschrift Math. Naturw. Unterricht, 25, 1894, 81-97. Chap. VII] PRIMITIVE RoOTS, EXPONENTS, INDICES. 197 J. Perott^^ employed the sum Sk of the ^th powers of 1, 2, . . ., p — 1, and gave a new proof that Si=0, . . ., Sp_2=0, Sp_i= —1 (mod p). If m is the 1. c. m. of the exponents to which 1,2,. . ., p — 1 belong, evidently Sm=p — 1, whence m>p — 2. If A belongs to the exponent m, then A, A^, . . ., A"" are incongruent, whence mSp — 1- Thus A is a primitive root. N. Amici^^ proved that, if j'>2, a number belongs to the exponent 2""^ modulo 2" if and only if it is of the form 8/i±3, and called such numbers quasi primitive roots of 2\ For a base Sh=^S, numbers of the two forms 8A:+1 or 8^=*=3, and no others, have indices. The product of two numbers having indices has an index which is congruent modulo 2""^ to the sum of the indices of the factors. The product of two numbers 6i and 62? neither with an index, has an index congruent modulo 2""^ to the sum of the indices of —61 and —62- The product of a number with an index by one without an index has no index. K. Zsigmondy^^ proved by use of abelian groups that, if 8 = qi^\ . .Qr'"', m = pi'^K . .ps^s, where Qi,. . ., Qr are distinct primes, and Pi,..., Ps are dis- tinct primes, the number of incongruent integers belonging to the exponent 5 modulo m is 5i...5,n(l-l/g/0, 1=1 where d, is the g. c. d. of 5 and tj=(f>{pp), while li is the number of the integers ti,...,ts which contain the factor ql'K E. de Jonquieres^^ proved that the product of an even number of primi- tive roots of a prime p is never a primitive root, while the product of an odd number of them is either a primitive root or belongs to an exponent not dividing {p — l)/2. Similar results hold for products of numbers belonging to like exponents. Certain of the n integers r, for which f is a given num- ber belonging to the exponent e = {p — \)/n, belong to the exponent ne, while the others (if any are left) belong to an exponent ke, where k divides n. He conjectured that 2 is not a primitive root of a prime p=l, 7, 17 or 23 (mod 24); 3 not of p=l, 11, 13 or 23 (mod 24); 5 not of p=\, 11, 19, or 29 (mod 30). These results and analogous ones for 7 and 11 were shown by him and T. Pepin^^ to follow from the quadratic reciprocity law and Gauss' theorems on the divisors oi x^^A. G. Wertheim^°. added to his^* corollaries cases when 6, 10, 11, 13 are primitive roots of primes 2^+1, 4^+1; also, 10 is a primitive root of all primes 8g+l?^137 for which g- is a prime 10A;+7 or lOyc+9, and of primes IGg+l for which g is a prime 10/c+l or lO/c+7. Wertheim^^ gave the least primitive root of each prime between 3000 and 5000 and of certain higher primes. He noted errata in his^^ table to 3000. 85BuU. des Sc. Math6matiques, 18, I, 1894, 64-66. 8«Rendiconti Circolo Mat. di Palermo, 8, 1894, 187-201. "Monatshefte Math. Phys., 7, 1896, 271-2. 88Compte8 Rendus Paris, 122, 1896, p. 1451, p. 1513; 124, 1897, p. 334, p. 428. 8»Comptes Rendus Paris, 123, 1896, pp. 374, 405, 683, 737. '"Acta Math., 20, 1896, 143-152. "/bid., 153-7; corrections, 22, 1899, 200. 198 History of the Theory of Numbers. [Chap, vn F. Mertens^- called I'l,. . ., ip the system of indices of n modulo k if n=gi\ . .qj" (mod A:) for the g's of Kronecker.^° Such systems of indices differ from Dirichlet's. C. Moreau^^ set A^ = pV . . . , v = p''~^q^~^ . . . , where p, q,... are distinct primes. Take € = 1 if iV" is not divisible by 4 or if N = 4, but e = 2 if iV is divisible by 4 and A'' > 4. Let \p{N) denote the 1. c. m. of v/e, p — l,q — l,. . . [equivalent to Cauehy's-^ maximum indicator for modulus N]. For A prime to N, A*^^^= 1 (mod N) . If A^ = p'', 2p^ or 4 (so that N has primitive roots), yp{N) =4>{N) [Lucas^^j^ ^j^^^.^ -^ ^ ^^^^^ ^^ values of A^< 1000 and certain higher values for which \p{N) has a given value < 100. A. Cunningham^"* noted that we may often abbre\iate Gauss' method of finding a primitive root of a prime p by testing whether or not the trial root a is a primitive root before computing the residues of all powers of a. The tests are the simple rules to decide whether or not a is a quadratic or cubic residue of p. If a is both a quadratic non-residue and a cubic non- residue of p = 3co+l, and if a^^l for every/ dividing p — 1 except /=p — l, then a is a primitive root. A. Cunningham^^ gave tables showing the residues of the successive powers of 2 when divided by each prime or power of prime < 1000, also companion tables showing the indices x of 2"" whose residues modulo p'' are 1, 2, 3, . . .. The tables are more convenient than Jacobi's Canon-^ (errata given here) for the problem to find the residue of a given number with respect to a given power of a prime, but less convenient for finding all roots of a given order of a given prime. There are given (p. 172) for each power p^< 1000 of a prime p the factors of 0(p^"), the exponent ^ to which 2 belongs modulo p'', and the quotient 0/^. E. Cahen^^ proved that if p is a prime >(32"'^'-l)/2'"+^ and if 5 = 2^+'^p-\-l (7«>0) is a prime, then 3 is a primitive root of q, whereas Tchebychef^^ had the less advantageous condition p>3^^V2'"+^. Other related theorems by Tchebychef are proved. There are companion tables of indices for primes < 200. G. A. Miller^^ appUed the theory of groups to prove the existence of primitive roots of p", to show that the primitive roots of p^ are primitive roots of p", and to determine primitive roots of the prime p. L. Kronecker^^ discussed the existence of primitive roots, defined sys- tems of indices and appHed them to the decomposition of fractions into partial fractions. He developed (pp. 375-388) the theor>^ of exponents to which numbers belong modulo p, a prime, by use of the primitive factor "Sitzungsber. Ak. Wien (Math.), 106, II a, 1897, 259. «Nouv. Ann. Math., (3), 17, 1898, 303. "Math. Quest. Educat. Times, 73, 1900, 45, 47. •'A Binary Canon, showing residues of powers of 2 for divisors under 1000, and indices to residues, London, 1900, 172 pp. Manuscript was described by author. Report British Assoc, i895, 613. Errata, Cunningham.'" "filaments de la th^orie des nombres, 1900, 335-9, 375-390. •'BuU. Amer. Math. Soc, 7, 1901, 350. •'Vorlesungen liber Zahlentheorie, I, 1901, 416-428. Chap. VII] PRIMITIVE RoOTS, EXPONENTS, INDICES. 199 Fd{x) of a;'*— 1 (dividing the last but not x'— 1 iorKd). To every divisor d of p — 1 belong exactly 4>{d) numbers which are the roots of i^d(^)=0 (modp). P. G. Foglini^^ gave an exposition of known results on primitive roots, indices, linear congruences, etc. In applying (p. 322) Poinsot's^ method of finding the primitive roots of a prime p to the case p = 13, it suffices to exclude the residues of the cubes of the numbers which remain after excluding the residues of squares; for, if a; is a residue of a square, (x^)®=l and x^ is the residue of a square. R. W. D. Christie^"" noted that, if 7 is a primitive root of a prime p = 4A; — l,the remaining primitive roots are congruent to p — 7 (n = 1, 2, . . . ) A. Cunningham^°^ noted that 3, 5, 6, 7, 10 and 12 are primitive roots of any prime /^, = 22"+l>5. Also 7^/^^+1 = (mod F^+, >5). E. I. Grigoriev^"^ noted that a primitive root of a prime p can not equal a product of an even number of primitive roots [evident]. G. Wertheim^°^ treated the problem to find the numbers belonging to the exponent equal to the 1. c. m. of m, n, given the numbers belonging to the exponents m and n, and proved the first theorem of Stern. ^^ He dis- cussed (pp. 251-3) the relation between indices to two bases and proved (pp. 258, 402-3) that the sum of the indices of a number for the various primitive roots of w = p" or 2p" equals ^4){m)4> ]0(w) \ ■ If « belongs to the exponent 45 modulo p, the same is true of p — a (p. 266). He gave a table showing the least primitive root of each prime < 6200 and for certain larger primes; also tables of indices for primes < 100. P. Bachmann^°^ gave a generalization (corrected on p. 402) of Stern's^^ first theorem. G. Arnoux^°^ constructed tables of residues of powers and tables of indices for low composite moduli. A. Bindoni^°^ noted that a table showing the exponent to which a belongs modulo p, a prime, can be extended to a table modulo N by means of the following theorems. Let a, 61,..., &„ be relatively prime by twos. A number belonging to the exponent ti modulo bi belongs modulo 6162 ■ ■ -K to the 1. c. m. of ^1, . . . , ^^ as exponent. If ti is the least exponent for which a'''+l=0 (mod bi) and if the ti are all odd, the least t for which a'+l is divisible by 6], ... , 6„ is the 1. c. m. of ^i, . . . , i„. If p is an odd prime not dividing a and if a belongs to the exponent t modulo p, and a' = pg+l, and if p" is the highest power of p dividing q, then a belongs to the exponent lpn~i-u jjiojuio p". Hence if a is a primitive root of p, it is one of p" if s'Memorie Pont. Ac. Nuovi Lincei, 18, 1901, 261-348. ^""Math. Quest. Educat. Times, 1, 1902, 90. "i/6id., pp. 108, 116. "^Kazani Izv. fiz. mat. obsc, BuU. Phys. Math. Soc. Kasan, (2), 12, 1902, No. 1, 7-10. "'Anfangsgrunde der Zahlenlehre, 1902, 236-7, 259-262. ii^Niedere Zahlentheorie, 1, 1902, 333-6. "'Assoc. fran§. av. sc, 32, 1903, II, 65-114. "«I1 Boll, di Matematica Giorn. Sc. Didat., Bologna, 4, 1905, 88-92. 200 History of the Theory of Numbers. [Chap, vii and only if a""^ — 1 is not divisible by p^. If t is even, the least x for which a"+l = (mod p") is l^p""'"". ]\I. Cipolla^°^gave a historical report on congruences (especially binomial), primitive roots, exponents, indices (in Peano's symboUsm). K. P. Nordlund^°^ proved by use of Fermat's theorem that, if rij, . . ., n,. are distinct odd primes, no one dividing a, then A^" = ni"*' . . . n^*"' divides a'-l, where A;=0(iV)/2^-^ R. D. Carmichael^°^ proved that the maximum indicator of any odd number is even; that of a number, whose least prime factor is of the form 4ZH-1, is a multiple of 4; that of p(2p — 1) is a multiple of 4 if p and 2p — 1 are odd primes. A. Cunningham^^° gave a table of the values of v, where {p — l)/v is the exponent to which 2 belongs modulo p"< 10000, the omitted values of p being those for which i' = 1 or 2 and hence are immediately distinguished by the quadratic character of 2 (extension of his Binary Canon^^). A list is given of errata in the table by Reuschle.^^ An announcement is made of the manuscript of tables of the exponents to which 3, 5, 6, 7, 10, 11, 12 belong modulo p"< 10000, and the least positive and negative primitive roots of each prime < 10000 [now in type and extended in manuscript to p"< 22000]. A. Cunningham^ ^^ defined the sub-Haupt-exponent ^i of a base q to modulus m = q°-°y]Q (where 770 is prime to q, and ao^O) to be the exponent to which q belongs modulo r^o- Similarly, let ^2 be the exponent to which q belongs modulo 771, where ^i = 5'''i?i; etc. Then the ^'s are the successive sub-Haupt-exponents, and the train ends with ^,.+1 = 1, corresponding to 77;. = 1 . His table I gives these ^k for bases g = 2, 3, 5 and for various moduli including the primes < 100. Paul Epstein^ ^^ desired a function ^{m), called the Haupt-exponent for modulus m, such that a'''^'"^ = 1 (mod m) for every integer a prime to m and such that this will not hold for an exponent <\p{m). Thus \f/{m) is merely Cauchy's^^ maximum indicator. Although reference is made to Lucas, ^^ who gave the correct value of 4^(ni), Epstein's formula requires modification when m = 4 or 8 since it then gives \p = l, whereas \p = 2. The number x(w, m) of roots of x''= 1 (mod m) is 2dodi . . .d„ if m is divisible by 4 and if H is odd, but is di . . . ci„ in the remaining cases, where, for m = 2'*°pi*i . . .pn'"'*, di is the g. c. d. of jj, and 4>{pi°-^), and do the g. c. d. of fx and 2°""^, when ao>l. The number of integers belonging to the exponent /x = pV-- modulo m is \x{m, p°)-x(m, p°-^)[ \x{m, q^)-x{m, (f-^)\. . .. "^Revue de Math. (Peano), Turin, 8, 1905, 89-117. "8G6teborgs Kungl. Vetenskaps-Handlingar, (4), 7-8, 1905, 12-14. "»Amer. Math. Monthly, 13, 1906, 110. "OQuar. Jour. Math., 37, 1906, 122-145. Manuscript announced in Mess. Math., 33, 1903-4, 145-155 (with list of errata in earUer tables); British Assoc. Report, 1904, 443; I'inter- m^diaire des math., 16, 1909, 240; 17, 1910, 71. CI. Cunningham."^ "iProc. London Math. Soc, 5, 1907, 237-274. "^Archiv Math. Phys., (3), 12, 1907, 134-150. Chap. VII] PRIMITIVE RoOTS, EXPONENTS, INDICES. 201 This formula is simplified in the case tx = \l/{m) and the numbers belonging to this Haupt-exponent are called primitive roots of m. The primitive roots of m divide into families of 0(i/'(m)) each, such that any two of one family are powers of each other modulo m, while no two of different families are powers of each other. Each family is subdivided. In general, not every integer prime to m occurs among the residues modulo m of the powers of the various primitive roots of m. A. Cunningham"^ considered the exponent ^ to which an odd number q belongs modulo 2"*; and gave the values of ^ when m^ 3, and when q = 2^12='= 1 (fi odd), m>3. When g' = 2''=Fl and m>x-\-l, the residue of q^^^^ can usually be expressed in one of the forms 1=f2", 1=f2"=f2^. G. Fontene"^ determined the numbers N which belong to a given exponent p"'~''8 modulo p"", where 5 is a given divisor of p — l, and h^l, without employing a primitive root of p"". li p>2, the conditions are that N shall belong to the exponent 8 modulo p and that the highest power of p dividing N^ — 1 shall he p^, l^h^m. *M. Demeczky"^ discussed primitive roots. E. Landau"^ proved the existence of primitive roots of powers of odd primes, discussed systems of indices for any modulus n, and treated the characters of n. G. A. Miller"^ noted that the determination of primitive roots of g corresponds to the problem of finding operators of highest order in the cyclic group G of order g. By use of the group of isomorphisms of G it is shown that the primitive roots of g which belong to an exponent 2q, where q is an odd prime, are given by —a", when a ranges over those integers between 1 and g/2 which are prime to g. As a corollary, the primitive roots of a prime 2g+l, where q is an odd prime, are — a^, l<a<g+l. A. N. Korkine"^ gave a table showing for each prime p<4000 a primitive root g and certain characters which serve to solve any solvable congruence x^=a (mod p), where g is a prime dividing p — l. Let q" be the highest power of q dividing p — 1. The characters of degree q are the solutions of M« = l, u'' = u, u"' = u',..., (w^"-")' = w(''-2) (mod p) and hence are the residues of the powers of g^p~'^^^^ for k = l,. . ., a. There are noted some errors in the Canon of Jacobi^^ and the table of Burckhardt. Korkine stated that if p is a prime and a belongs to the exponent e = {p — 1)/5, exactly (f){p — l)/<l>{e) of the roots of a;* = a (mod p) are primitive roots of p. K. A. Posse"^ remarked that Korkine constructed his table without knowing of the table by Wertheim,^^and extended Korkine's tables to 10000. "^Messenger of Math., 37, 1907-8, 162-4. "*Nouv. Ann. Math., (4), 8, 1908, 193-216. "^Math 6s Phys. Lapok, Budapest, 17, 1908, 79-86. ii«Handbuch . . .Verteilung der Primzahlen, I, 1909, 391-414, 478-486. ii'Amer. Jour. Math., 31, 1909, 42-4. iisMatem. Shorn. Moskva (Math. Soc. Moscow), 27, 1909, 28-115, 120-137 (in Russian). Cf. D. A. Grave, 29, 1913, 7-11. The table was reprinted by Posse."* iio/bid., 116-120, 175-9, 238-257. Reprinted by Posse.»" 202 History of the Theory of Numbers. [Chap, vii R. D. CarmichaeP^® called a number a primitive X-root modulo n if it belongs to the exponent X(?i), defined in Ch. Ill, Lucas. ""^ The existence of primitive X-roots g is proved. The product of those powers of g which are prhnitive X-roots is = 1 (mod n) if X(n) >2. A method is given to solve X(x) =a, and the solutions tabulated for a ^24. C. Posse^-^ noted that in Wertheim's^'^^ table, the primitive root 14 of 2161 should be replaced by 23, while 10 is not a primitive root of 3851. E. Maillet^^^ described the manuscript table by Chabanel, deposited in the library of the University of Paris, giving the indices for primes under 10000 and data to determine the number having a given index. F. Schuh^^^ showed how to form the congruence for the primitive roots of a prime and gave two further proofs of the existence of primitive roots. He treated binomial congruences, quadratic residues and made applica- tions to periodic fractions to any base. For any modulus n, he found the least m for which x"' = 1 (mod n) holds for every x prime to n, and derived the solutions ?i of 4>{n) =m, i. e., n's having primitive roots. F. Schuh^^^ discussed the solution of a;' = 1 (mod p") with the least com- putation. If X belongs to the exponent q modulo n, the powers of x give a cycle of 0(g) numbers each with the "period" q. The numbers prime to n and having the period q may form several such cycles — more than one if n has no primitive root and q is the maximum period. If n = 2" (a>2), then g = 2* (s^a — 2) and the number of cycles is 1, 3 or 2 according as s = 0, s = 1 or s>l. In the last case, the cj^cles are formed by 2''~^(2fc+l) =f1. When q is even, x is said to be of the first or second kind according as x'''^= — 1 (mod n) or not. If the numbers of a cycle are of the second kind, we get a new cycle of the second kind by changing the signs of the numbers of the first cycle. While for moduli n having primitive roots there exist no numbers of the second kind, when n has no primitive roots and g is a possible even period, there exist at least two cycles of the second kind and of period q. Finally, there is given a table showing the number of cycles of each kind for moduli ^ 150. M. Kraitchik^^^ gave a table showing for each prime p< 10000 a primi- tive root of p and the least solutions of 2""=!, 10"= 1 (mod p). *J. Schumacher^^*^ discussed indices. L. von Schrutka^^^ noted that, if g, r, . . . are the distinct primes dividing p — l, where p is a prime, all non-primitive roots of p satisfy (a;V-l)(xV_i) . . .=0 (mod p). ""Bull. Amer. Math. Soc, 16, 1909-10, 232-7. Also, Theory of Numbers, pp. 71-4. i^iActa Math., 33, 1910, 405-6. i=»L'interm6diaire des math., 17, 1910, 19-20. i23Supplement de Vriend derWiskunde, Culemborg, 22, 1910, 34-114, 166-199, 252-9; 25, 1913, 33-59, 143-159, 228-259. "*Ibid., 23, 1911, 39-70, 130-159, 230-247. '"Sphinx-Oedipe, May, 1911, Num^ro Special, pp. 1-10; errata listed p. 122 by Cunningham and Woodall. Extension to 25000, 1912, 25-9, 39-42, 52-5; errata, 93-4, by Cunningham. "'Blatter Gymnasiaj-Schulwesen, Miinchen, 47, 1911, 217-9, "^Monatshefte Math. Phys., 22, 1911, 177-186. Chap. VII] PRIMITIVE RoOTS, EXPONENTS, INDICES. 203 To this congruence he appHed Hurwitz's^^ method (Ch. VIII) of finding the number of roots and concluded that there are p — l—(f>(p — l) roots. Hence there exist 0(p — 1) primitive roots of p. A. Cunningham and H. J. WoodalP^^ continued to p< 100000 the table of Cunningham"" of the maximum residue indices j^ of 2 modulo p. C. Posse^^^ reproduced Korkine's"^ and his own"^ tables and explained their use in the solution of binomial congruences. C. Krediet^^o treated x*'=l (mod n) of Lucas/^" Ch. Ill, and called x a primitive root if it belongs to the exponent cp. The powers of such a root are placed at equal intervals on a circle for various n's. G. A. Miller^^^ proved by use of group theory that, if m is arbitrary, the sum of those integers < m and prime to m which belong to an exponent divisible by 4 is = (mod m) , and the sum of those belonging to the expo- nent 2 is = — 1 (mod m), and proved the corresponding theorem by Stern^^ for a prime modulus. A. Cunningham^^^ tabulated the number of primes p<10^ for which y belongs to the same exponent modulo p, for y = 2, 3, 5, 6, 7, 10, 11, 12; and the number of primes p in each 10000 to 10^ for which y (2/ = 2 or 10) belongs to the same exponent modulo p. Also, for the same ranges on p and y, the number of primes p for which y''^ 1 (mod p) is solvable, where A; is a given divisor of p — 1 . A. Cunningham^^^ stated that he had finished the manuscript of a table of Haupt-exponents to bases 3, 5, 6, 7, 11, 12 for all prime powers < 15000; also canons giving at sight the residues of z" modulo p'''< 10000 for z = 2, r^l00;2 = 3, 5, 7, 10, 11, r^30. J. Barinaga^^^ considered a number a belonging to the exponent g modulo p, a prime. If a is not divisible by g, the sum of the ath powers of the numbers forming the period of a modulo p is divisible by p. The sum of their products n at a time is congruent to zero modulo p ii n<g, but to =^"1 ii n = g, according as g is even or odd. A. Cunningham^^^ listed errata in his Binary Canon^^ and Jacobi's Canon. ^^ G. A. Miller^^^ employed the group formed by the integers <m and prime to m, combined by multiplication modulo m, to show that, if a number is = ± 1 (mod 2"^), but not modulo 2^+\ where l<7</3, it belongs to the exponent 2^~^ modulo 2^. Also, if p is an odd prime, and A^= 1 (mod p), N belongs to the exponent p^~^ modulo p^ if and only if A^" — 1 is divisible by p^, but not by p^+\ where /3>7^ 1. »8Quar. Jour. Math., 42, 1911, 241-250; 44, 1913, 41-48, 237-242; 45, 1914, 114-125. i^Acta Math., 35, 1912, 193-231, 233-252. ""Wiskundig Tijdskrift, Haarlem, 8, 1912, 177-188; 9, 1912, 14-38; 10, 1913, 40-46, 87-97. (Dutch.) "lAmer. Math. Monthly, 19, 1912, 41-6. "2Proc. London Math. Soc, (2), 13, 1914, 258-272. "'Messenger Math., 45, 1915, 69. Cf. Cunningham."" "<Annaes Sc. Acad. Polyt. do Porto, 10, 1915, 74-6. "^Messenger Math., 46, 1916, 57-9, 67-8. "s/Wd., 101-3. 204 History of the Theory of Numbers. [Chap, vil A. Cunningham^^^ gave five primes p for which there is a maximum number of exponents to which the various numbers belong modulo p. On exponents and indices, see Lebesgue^"'*^ and Bouniakowsky^^^; also Reuschle^^ of Ch. YI, Bouniakowsky"^ of Ch. XIV, and Calvitti^^ of Ch. XX. Binomial Congruences. Bhdscara Achd,rya^*^ (1150 A. D.) found y such that y^ — SO is di\'isible by 7 by solving ?/" = 7c+30. Changing 30 by multiples of 7, we reach a perfect square 16 with the root 4. Hence set 7c+30 = (7n+4)2, c = 7n'+8n-2, y = 7n-\-4. Taking n = 1, we get y = ll. Such a problem is impossible if, after abrading the absolute term (30 above) by the divisor (7 above) and the addition of multiples of the divisor, we do not reach a square. Similarly for the case of a cube, with corresponding conditions for impos- sibihty (§206, p. 265). For y^ = 5e+Q, abrade 6 by the divisor 5 to get the cube 1; adding 43-5, we get 216 = 6^. Hence set y = 5n-\-Q. An anonymous Japanese manuscript^^° of the first part of the eighteenth century gave a solution of x^ — ky = a by trial. The residues Oi, . . ., ak-\ of 1", . . ., (^ — 1)" modulo k are formed; if a^^a, then x = r. It was noted that ak-r = 0'r or k—Qr according as k is even or odd, and that the residue of r" is r times that of r"~^ Matsunaga,^^*^" in the first half of the eighteenth century, solved a}-\-hx= y^ by expressing 6 as a product mn and finding p, q and A so that mp — nq=l, 2pa=A (mod n). Then x={Am — 2a)A/n [and y=a — 7nb]. But if Am= 2a, write A-\-n in place of A and proceed as before. Or write 2a+h in the form bQ+R, whence x=2a+b-{Q+l)R. To solve 69+ llx=y'^, consider the successive squares until we reach 5^=3 (mod 11). Write 2-5+11 in the form 1-11 + 10. Then for a=5, 6= 11, Q= 1, i2= 10, the preceding expression for x becomes 1, whence 5^+11-1 = 6^. Then write 2-6+11 in the form 2-11 + 1. Then 23-(2+l)-l = 20 gives 6"+ 20-11= 16^, and a;= (256-69)/ll= 17. L. Euler^^^ proved that, if n divides p — l, where p is a prime, and if a = c''-\-kp, then (by powering and using Fermat's theorem), a^^~^^^" — l is divisible by p. Conversely, if a'" — 1 is divisible by the prime p = w7i+l, we can find an integer y such that a — ?/" is divisible by p. For, o'"-2/'"'' = (a-2/")Q(^), and the differences of order mn—n of Q(l), Q(2),. . ., Q{mn) are the same >"Math. Quest, and Solutions (Ed. Times), 3, 1917, 61-2; corrections, p. 65. '"Vlja-ganita, §§ 204-5; Algebra, with arith. and mensuration, from the Sanscrit of Brahmegupta and Bhdscara, transl. by H. T. Colebrooke, London, 1817, pp. 263-4. ""Abhand. Geschichte Math. Wiss., 30, 1912, 237. ^'^Ibid., 234-5. i"Novi Comm. Acad. Petrop., 7, 1758-9 (1755), p. 49, eeq., §64, §72, §77; Comm. Arith., 1, 270-1, 273. In Novi Comm., 1, 1747-8, p. 20; Comm. Arith., 1, p. 60, he proved the first statement and stated the converse Chap. VII] BiNOMIAL CONGRUENCES. 205 as those of the term t/'""-" for ?/ = !,..., mn, and hence equal {mn — n)\, so that Q(y) is not divisible by p for some values 1, . . . , mn of y. Euler^^^ recurred to the subject. The main conclusion here and from his former paper is the criterion that, if p = mn-]-l is a prime, x''=a (mod p) has exactly n roots or no root, according as a"^=l (mod p) or not. In particular, there are just m roots of a""^!, and each root a is a residue of an nth power. Euler^^^" stated that, if aq-\-h=p'^, all the values of x making ax-\-b b, square are given by a;= ay'^^2py-\-q. J. L. Lagrange^^^ gave the criterion of Euler, and noted that if p is a prime 4n+3, B'-^'^^^^ — l is divisible by p, so that x=B'''^^ is a root of x^=B (mod p). Given a root ^ of the latter, where now p is any odd prime not dividing B, we can find a root of x^=B (mod p^) by setting x = ^-\-\p, i^-B = poi. Then x^-B = {\^-{-n)p'^ if 2|X+co=MP. The latter can be satisfied by integers X, jjl since 2^ and p are relatively prime. We can pro- ceed similarly and solve x^=B (mod p"). Next, consider ^^=B (mod 2"), for n>2 and B odd (since the case B even reduces to the former). Then ^ = 2z-\-l, ^^ — B = Z-\-\—B, where Z = 4:z{z-\-l) is a multiple of 8. Thus 1—B must be a multiple of 8. Let w>3 and 1-B = 2'^,r>3. If r^n, it suflaces to take 2 = 2""^, where f is arbitrary. If r<n, Z must be divisible by 2'', whence 2 = 2'""^^ or 2*""^^ — 1. Hence w=^{2'-^i:=i=l)-\-p must be divisible by 2"-''. If n-r^r-2, it suffices to take f =^iS divisible by 2""''. The latter is a necessary condition if n-r>r-2. Thus ^ = 2'-^p=F^, w = 2'-\f=t=p). Hence f ±p must be divisible by 2""^'^+^. We have two sub-cases according as the exponent of 2 is ^ or >r — 1; etc. Finally, the solution of x^=B{mod m) reduces to the case of the powers of primes dividing m. For, if / and g are relatively prime and ^^ — Bis divisible by /, and \p^—B by g, then x^—B\b divisible by fg ii x= jif^ ^ = vg^\l/. But the final equality can be satisfied by integers /z, v since / is prime to g. A. M. Legendre^^^ proved that if p is a prime and co is the g. c. d. of n and p — \ = oip', there is no integral root of (1) a:"=j5(modp) unless B^'= 1 (mod p) ; if the last condition is satisfied, there are co roots of (1) and they satisfy (2) x'^^B^ (mod p), where I is the least positive integer for which (3) ln — q{p — \)=o}. For, from (1) and x''-^=l, we get x^''=B\ x^^^-^^^l, and hence (2), by use of (3). Set n = oin'. Then, by (2) and (1), ^n'l^^n^2, 5P''=a;P'"=<rP-l=l (mod p) . "2Novi Comm..Petrop., 8, 1760-1, 74; Opusc. Anal. 1, 1772, 121; Comm. Arith., 1, 274, 487. "^aOpera postuma, I, 1862, 213-4 (about 1771). "^Mem. Acad. R. Sc. Berlin, 23, ann6e 1767, 1769; Oeuvres, 2, 497-504. "^M6m. Ac. R. Sc. Paris, 1785, 468, 476-481. (Cf. Legendre.i^^) 206 History of the Theory of Numbers. [Chap, vii Since In'—qp' = 1, the first gives 5^"'= 1. Hence Conversely, if B^'=l, a^p-i_l=a^p'-_j5p'/ (mod p) has the factor x" — B\ so that (Lagrange^) congruence (2) has co roots. If 4n divides p — 1, the roots of x^"= —1 (mod p) are the odd powers of an integer belonging to the exponent 4n modulo p. Let n divide p — l, and 7n divide {p — l)/n. Let co be the g. c. d. of m , n and set n = cov. Determine positive integers I and q such that lv — qm = l. If 5'"= =•= 1 (mod p), (1) is satisfied by the roots of x'^^B'y (mod p), where y ranges over the roots of ?/"= (=t 1)^ (mod p). For, the last two congruences give x'' = x'"'=B''^y''=B'"^+\=i=iy=B (mod p). Hence by means of the roots of ?/''=±l, we reduce the solution of (1) to binomial congruences of lower degrees. In particular, let n = 2, m = (p — 1)/2, and let 2 be prune to ?«, so that p = 4:a — l,l = a,q = l. Then x^ = B (mod p) requires that 5"* = 1 , so that we have the solutions .t = =*= 5" without trial (Lagrange^^^). Next, if n = 2 and 5^'^+^= —1, the theorem gives x = B'''^^y, where ?/- = — L But we may generalize the last result. Consider x" + c^ = (mod p). Since p must have the form 4a+l, we have p=f^-\-g^. Deter- mine u and z so that c = gu—pz. Then x = fu (mod p). Let a belong to the exponent nw modulo p, where w divides (p — l)/n. Then the roots of B"' = l (mod p) are B = a"*' (m = 1,- • •, w) — 1), and, for a fixed B, the roots of (1) are x = 0'"""+" (m = 0, 1, . . . , n - 1). For, a" belongs to the exponent w, whence B = a'"'. Legendre^^^ gave the same theorems in his text. He added that, know- ing a root 6 of (1), it is easy to find a root of x" = B (mod p"), with the possible exception of the case in which n is divisible by p. Let6"—B = Mp and set a:=^+i4p. Then a;" — 5 is divisible by p^ if M+'nB''-^A=pM', which can be satisfied by integers A, Af' if n is not divisible by p. To solve (1) when p is composite, p = a°6^ . . . , where a, 6, . . . are distinct primes, deter- mine all the roots X of X" = B (mod a°), all the roots ^ of ijl" = B (mod b^), Then if x=\ (mod a°), x=iJi (mod 6^),. . ., x will range over all the roots of (1). Legendre^^^ noted that if p is a prime 8n+5 we can give explicitly the solutions of a:^+a = (mod p) when it is solvable, viz., when a'*"''"^ = 1. For, either «-"+' + 1=0 and x = a"+' is a solution, or a-"+^-l=0 and (9 = a"+^ satisfies d^ — a^O (mod p), so that it remains only to solve x^-\-d' = 0, which was done at the end of his^^* memoir. For p = 8?i+l, let n = a^, where a is a power of 2 and /3 is odd; if 0"= ± 1, x^-\-a = can be solved as in the i"Th(5orie des nombres, 1798, 411-8; ed. 2, 1808, 349-357; ed. 3, 1830, Nos. 339-351; German transl. by Maser, 1893, 2, pp. 15-22. ^"Ibid., 231-8; ed. 2, 1808, pp. 211-219; Maser, I, pp. 246-7. Chap. VII] BiNOMIAL CONGEUENCES. 207 case p = 8n+5; but in general no such direct solution is known, and it is best to represent some multiple of p by the form y'^+az^. If we have found 6 such that ^^+a is divisible by the prime p, not dividing a, we readily solve x^+a = (mod p"). For, from r^-\-as^ is divisible by p^. Now s is not divisible by p. Thus we may take r = sx+p''y, whence x^+a is divisible by p". [Cf. Tchebychef, Theorie der Congruenzen, §30.] The case of any composite modulus N is easily reduced to the preceding (end of Lagrange's^^^ paper). Legendre proved that, if N is odd and prime to a, the number of solutions of a:^+a = (mod N) is 2'"^ where i is the number of distinct prime factors of N; the same is true for modulus 2N. Henceforth let N be odd or the double of an odd number and let d be the g. c. d. of N and a. If d has no square factor, the congruence has 2'"^ roots, where i is the number of distinct odd prime factors of N not dividing a. But if d=o)\(/^, where co has no square factor, the congruence has 2'~V roots where i is the number of distinct odd prime factors of N/d. C. F. Gauss^" treated congruence (1) by the use of indices. However, we can give a direct solution (arts. 66-68) when a root is known to be con- gruent to a power of B. For, by (1) and x = B^, B^B^"". If therefore a relation of the last iy^e is known, a root of (1) is B''. The condition for the relation is l = A:n (mod t), where t is the exponent to which B belongs modulo p. It is shown that t must divide m = (p — l)/n. We may discard from m any factor of n; if the resulting number is m/q, the unique solution k of 1 = 1:11 (mod m/q) is the desired k. [Cf. Poinsot^^^] Gauss (arts. 101-5) gave the usual method of reducing the solution of x^= A (mod m) for any composite modulus to the case of a prime modulus and gave the number of roots modulo p'* in the various possible subcases. His well-known and practical ''method of exclusion" (arts. 319-322) employs successive small powers of primes as moduU. Another method (arts. 327-8) is based on the theory of binary quadratic forms [cf. Smith^^°]. The congruence ax^-\-'bx-\-c=0 (mod m) is reduced (art. 152) to y^=}? — Aac (mod 4am). For each root y, it remains to solve 2ax-{-b=y (mod 4aw). Gauss^^^ showed in a somewhat incomplete posthumous paper that, if t is a prime and f~'^{t — l)=a''¥. . ., where a,h,. . . are distinct primes, the solution of a:"= 1 (mod t") may be made to depend upon the solution of a congruences of degree a, jS congruences of degree h, etc. Use is made of the periods formed of the primitive roots of the congruence, as in Gauss' theory of roots of unity. Legendre^^^ solved x^+a=0 (mod 2'") when a is of the form — 1 =F8a by i"Disquis. Arith., 1801, Arts. 60-65. "sWerke, 2, 1863, 199-211. Maser's German transl. of Gauss' Disq. Arith., etc., 1889, 589-601 (comments, p. 683). "'Theorie des nombres, ed. 2, 1808, pp. 358-60 (Nos. 350-2). Maser, 2, 1893, 25-7. 208 History of the Theory of Numbers. [Chap. VII use of the expansion of (1+2)^'^: M M-3 Vl±8a = l±|2^a-— 7 2V=t— — 2V- . . . =tiV23"a"+ . N = M-3-5 2-4 (2n 2-4-6 ■3) 2-4-6-8. ..2n The coefficient of a" is an integer divisible by 2"^^ Retain only the terms whose coefficients are not divisible by 2'""^ and call their sum 6. Hence every term of 6~-^a is divisible by 2'". Thus the general solution of the proposed congruence is x=2'^~^x'^d. P. S. Laplace^*^" attempted to prove that, if p is a prime and p — l=ae, there exists an integer x<e such that x' — l is not divisible by p. For, if x = e and all earlier values of x make a:* — 1 divisible by p, /=(e^-l)-e^ would be divisible by p. ,{e-lY-l\ + {^^y,{e-2Y-l\-... The sum of the second terms of the binomials is + ... = -(1-1)^ = 0, while the sum of the first terms of the binomials is e ! by the theory of differ- ences, and is not divisible by p since e<p. [But the former equality implies that the last term of / is ( — 1)''(0— 1), whereas the theorem is trivial if x is allowed to take the value 0. Again, nothing in the proof given prevents a from being unity; then the statement that there is a positive integer x<p — l such that x^~^ — 1 is not divisible hyp contradicts Fermat's theorem.] L. Poinsot^^ deduced roots of a;"= 1 (mod p) from roots of unity. M. A. Stern^^ (p. 152) proved that if n is odd and p is a prime, rc"= —1 (mod p) is solvable and the number of roots is the g. c. d. of n and p — l; while, if n is even, it is solvable if and only if the factor 2 occurs in p — 1 to a higher power than in n. G. Libri^^^ gave a long formula, involving sums of trigonometric func- tions, for the number of roots of x^+c=0 (mod p). V. A. Lebesgue^^ applied a theorem on/(a:i, . . ., Xk) = to derive Legen- dre's^^^ condition B^'=l for the existence of roots of (1), and the number of roots. Cf. Lebesgue^^ of Ch. VIII. Erlerus^^ (pp. 9-13) proved that, if pi, . . . , p^ are distinct odd primes, x~=l (mod2''p/'...p/) has 2", 2", 2"+^ or 2"+^ roots according as j/ = 0, 1, 2 or >2. For the last result and the like number of roots of x^=a, see the reports, in Ch. Ill on Fermat's theorem, of the papers by Brennecke^^ and Crelle^* of 1839, Crelle,^^ Poinsot" (erroneous) and Prouhet^^ of 1845, and Schering^''* of 1882. C. F. Arndt^^^ proved that the number of roots of x'= 1 (mod p") for "»Communication to Lacroix, Traitd Calcul Diff. Int., ed. 2, vol. in, 1818, 723. '"Jour, fur Math., 9, 1832, 175-7. See Libri," Ch. VIII. "*Archiv Math. Phys., 2, 1842, 10-14, 21-22. Chap. VII] BiNOMIAL CONGRUENCES. 209 p an odd prime is the g. c. d. of t and (/)(p") ; the same holds for modulus 2p". He found the number of roots of x'^=r (mod m), m arbitrary. By using S</>(0 =5, if i ranges over the divisors of 5, he proved (pp. 25-26) the known result that the number of roots of x"= 1 (mod p) is the g. c. d. 5 of n andp — 1. The product of the roots of the latter is congruent to ( — 1)*"^^; the sum of the roots is divisible by p; the sum of the squares of the roots is divisible bypif 6>2. P. F. Arndt^^^ used indices to find the number of roots of x^ = a. A. L. Crelle^^^gave an exposition of known results on binomial congruences. L. Poinsot^^^ considered the direct solution of x"=A (mod p), where p is a prime and n is a divisor of p — l=nm (to which the contrary case reduces). Let the necessary condition ^""=1 be satisfied. Hence we may replace A by A^+"''^ and obtain the root rc=A^ if l-\-mk = ne is solvable for integers k, e, which is the case if m and n are relatively prime [cf. Gauss^^^]. The fact that we obtain a single root x=A^ is explained by the remark that it is a root common to a:"=A and x"'=l, which have a single common root when n is prime to m. Next, let n and m be not relatively prime. Then there is no root A' if A belongs to the exponent m modulo p. But if A belongs to a smaller exponent m' and if m' is prime to n, there exists as before a root A", where l-\-m'k = ne'. The number of roots of a;"=l (mod N) is found (pp. 87-101). C. F. Arndt^^^ proved that x'=l (mod 2"), n>2, has the single root 1 if t is odd; while for t even the number of roots is double the g. c. d. of i and 2n-2^ The sum of the A:th powers of the roots of x'= 1 (mod p) is divisible by the prime p if A: is not a multiple of t. By means of Newton's identities it is shown that the sum, sum of products by twos, threes, etc., of the roots of x*= 1 (mod p) is divisible by the prime p, while their product is = + 1 or — 1 according as the number of roots is odd or even. If the sum, sum of products by twos, threes, etc., of m integers is divisible by the prime p, while their product is =—( — 1)'", the m integers are the roots of x'"=l (mod p). A. Cauchy^" stated that if I = p\'' . . ., where p, q,... are m distinct primes, and if n is an odd prime, x"= 1 (mod 7) has rf distinct roots, includ- ing primitive roots, i. e., numbers belonging to the exponent n. [But x^= 1 (mod 5) has a single root.] Cauchy^^^ later restricted p, q,. . . to be primes =1 (mod n). Then a:"=l (mod p^) has a primitive root ri, and rc^^l (mod q") has a primitive root 7-2, so that x''^ 1 (mod /) has a primitive root, viz., an integer =ri (mod p*") and =r2 (mod q"), etc.; but no primitive root ii p, q,. . . are not all =1 (mod n). i«3Von den Kubischen Resten, Torgau, 1842, 12 pp. »*Jour. fiir Math., 28, 1844, 111-154. »«Jour. de Math^matiques, (1), 10, 1845, 77-87. ""Archiv Math. Phys., 6, 1845, 380, 396-9. "'Comptes Rendus Paris, 24, 1847, 996; Oeuvres, (1), 10, 299. "sComptes Rendus Paris, 25, 1847, 37; Oeuvres, (1), 10, 331. 210 History of the Theory of Numbers. [Chap, vii Hoen4 Wronski^^^ stated without proof that, if a;'"=a (mod M), a = {-iy+'\hK+{-lY+''rA[M/K,o)Y-^+Mi, x = h + (-lY+'A[M/K, iry-'+Mj, and that M must be a factor of aK"" - \hK-{-iy+^\'". Here the "alephs" A[M/K, o)Y, for r = 0, 1,. . ., are the numerators of the reduced fractions obtained in the development of M/K as a continued fraction. In place of K, Wronski wrote the square of l''^^ = k\. Concerning these formulas, see Hanegraeff,"^ Bukaty,!^^ Dickstein.^^^ Cf. Wronski^^^ of Ch. VIII. E. Desmarest" noted that, if x'^+D=0 (mod p) is solvable, x^+Dy^ = mp can be satisfied by a value of m<3+y>/16 and a value of 2/^3. His proof is not satisfactory. D. A. da Silva''^ (Alasia, p. 31) noted that x^^l (mod m), where m = Pi*'P2^* •• ■ . has the roots 'Zxiqi7n/p{^ where Xi is a root of x^'= 1 (mod Pi"^), Di being the g. c. d. of D and <t){pr), while the g's are integers such that Xqim/p['=\ (mod m). Da Silva^^^" proved that a solvable congruence a:"=r (mod m) can be reduced to the case r prime to m and then to the case m = p'',p a prime > 2. Then, if 5 is the g. c. d. of n and (f){p'')=8di, there is a root if and only if r*'=l (mod p") and hence if and only if r'^=l (mod p"'"^^), where p"' is the g. c. d. of n and p"~"\ while d is the quotient of p — 1 by its g. c. d. with n. H. J. S. Smith^'^" indicated a simplification in Gauss'^" second method of solving x^^A, If r^-\- D= (mod P) is solvable, mP = x^-\-Dy'^ is solvable for some value of ?72< 2V-D/3. Employing all values of m under that limit for which also (i)=S> and finding with Gauss all prime representations of the resulting products by the form x^-\-Dy^, we get ±r=x'/y', x"/y",. . .(mod P), where x', y'; x", y" \. . . denote the sets of solutions of mP = x^-\-Dy^. Eg. Hanegraeff^^^ reduced x"'=r to d"'r=l (mod p) by use of 6x=l. When p/d is developed into a continued fraction, let /x and P^_i be the number of quotients and number of convergents preceding the last. Let v, P^_i be the corresponding numbers for p/O"". Then x^i-iy-'P,_„ r=(-ir^P,_i (mod p). For p a prime, we get all roots by taking 6 = 1,. . . , (p — 1)/2. By starting with d{x — h)^l in place of 6x= 1, we get "'R6forme des Math^matiques, being Vol. i of R6forme du savoir humain, 1847. Wronski's mathematical discoveries have been discussed by S. Dickstein, Bibliotheca Math., (2), 6, 1892, 48-52, 85-90; 7, 1893, 9-14 [on analysis, (2), 8, 1894, 49, 85; (2), 10, 1896, 5]. Bull. Int. Ac. Sc. Cracovie, 1896; Rozprawy, Krakow, 4, 1913, 73, 396. Cf. I'intermd- diaire des math., 22, 1915, 68; 23, 1916, 113, 164-7, 181-3, 199, 231-4; 25, 1918, 55-7. "'<KU. Alasia, Annaes Sc. Acad. Polyt. do Porto, 9, 1914, 65-95. There are many confusing misprints; for example, five at the top of p. 76. ""British Assoc. Report, 1860, 120-, §68; CoU. M. Papers, 1, 148-9. "*Note BUT r^quation de congruence x^=r (mod p), Paris, 1860. Chap. VII] BiNOMIAL CONGRUENCES. 211 x-h={-iy-'P,-u r^{-iy-\dh+irP^.^ (mod p). By taking ^=(1^/')^ and replacing 1 by (-1)^+^ in 0(a;-/i) = l, the last results become the fundamental formula given without proof by Wronski^^^ in his Reforme des Mathematiques. G. L. Dirichlet^^^ discussed the solution of x^^D for any modulus. G. F. Meyer"^ gave an elementary discussion of the solution of x^=b (mod k), for k a prime, power of prime, or any integer. V. A. Lebesgue^^^ employed a prime p, a divisor n of p — l=nn', and a number a belonging to the exponent n' modulo p. Then the roots of a;" = a (mod p) are a"6^, where h is not in the period of a, and 6 is a quadratic non-residue of p if a is a quadratic residue, and 6" is the least power of h congruent to a term of the period of a. If we set 6" = a" (mod p), then must na-\-v^ = l (mod n'). The roots x are primitive roots of p. In the construction of a table of indices, his method is to seek a primitive root giving to ±2 the minimum index (rather than to ±10, used by Jacobi); thus we use the theorem for a= ±2. Lebesgue^'^^ gave reasons why the conditions imposed on h in his pre- ceding paper are necessary. He added that when we have found that x" = a (mod p) leads to a primitive root x = g oi p,\i is easy to solve x'"=r (mod p) when m divides p — 1, by expressing r as a power of g by the equiva- lent of an abridged table of indices. Lebesgue^^*^ noted that the usual method of solution by indices leads to the theorem: If a belongs to the exponent e modulo p, and if n divides p — 1, and we set n = e'm, where e' has only prime factors which divide e, while m is prime to e, then, for every divisor M of m, x'^^a (mod p) has e'(j){M) roots belonging to the exponent M. If a belongs to the exponent e modulo p, there are e0(n) numbers h, not in the period of a, for which 6"= a' (mod p), with n sl minimum. A common divisor of n and i does not divide e. Then the n roots of x"= a (mod p) are a'6", where nt — iu — l = ev, t<e, u<n. This generahzation of his^^'' earlier theorem is used to find the period of a primitive root of p from the period of 2. R. Gorgas"^ stated that, if p is the residue modulo M of the pth term of ](M-l)/2[^. . .,2^ 1^, then p(p-l)=p±m+ikf a, according as ilf = 4m=t:l. Take the lower signs and solve for p ; we get 2p = l±6, 62 = M(4a-l)+4p. Set 4p = Mc+p'. Hence the initial equation x^ = My+p has been replaced by 6^ = M(4a-fc — l)+p' of like form. Let p' be the p'th place from the end. The process may be repeated until we reach an equation P(P — 1) = MA-\-p^—m solvable by inspection. "^Zahlentheorie, 1863, §§32-7; ed. 2, 1871; ed. 3, 1879; ed. 4, 1894. »"Archiv Math. Phys., 43, 1865, 413-36. "^Comptes Rendus Paris, 61, 1865, 1041-4. "'Ihid., 62, 1866, 20-23. "«/6id., 63, 1866, 1100-3. "^Ueber Losung dioph. Gl. 2. Gr., Progr., Magdeburg, 1867. 212 History of the Theory of Numbers. [Chap, vii Ladrasch^^^ obtained known results on x^=a for any modulus. V. Bouniakowsky^"^ gave a method of sohdng q-S'^ ^r (mod P), where P is odd. His first illustration is 3'==±=1 (mod 25). Write the integers ^(25 — 1)/2 in a line. Under the first four wTite in order the integers = (mod 3) ; under the next four write in reverse order those = 1 ; under the last four write in order those = 2. 1* 2* 3* 4* 3 6 9 12 5 6* 7* 8* 10 7 4 1 9* 10 11* 12* 2 5 8 11 Mark with an asterisk 1 in the first line; below it lies 3; mark with an asterisk 3 in the first line; etc. The number 10 of the integers marked with an asterisk is the least solution x of 3""= —1 (mod 25). The sign is determined by the number of integers in the second set marked by an asterisk. The method applies to any P = 6n+1. But for P = 6n+5, we use for the second set of numbers in the second line those =2 (mod 3) in reverse order, and for the third set those =1 in order. If P = 23, we see that each of the 11 numbers in the first line are marked with an asterisk, whence 3^^=-l (mod 23). A like marking occurs for P = 5, 11, 17, 29. For P = 35, 12 numbers are marked, whence 12 is the least x for which 3''=1 (mod 35). Starting with the unmarked number 5, we get the cycle 5, 15, 10, whence 3^= —1 (mod 7); similarly, the cycle 7, 14 gives 3"= —1 (mod 5). For g'-3'^=='=4 (mod 25), we begin with 4 in the second row. Since it hes below 7, we mark 7 with an asterisk in the second row; etc. We use an affix n on the number which is the nth marked by an asterisk. 12 3 4 5 6 7 8 9 10 11 12 3*6 g*3 9*5 12*10 10 7*2^*1 ]^*7 2*^ 5 8*8 11*9 For 5 = 11, we have the entry 8*^ below 11; hence 11-3^=— 4, the sign following from the number of entries ^ 8 in the second set which are marked with an asterisk. Similarly for any 5^ 12, except g = 5, 10. Bukaty^^" discussed the formula of Wronski.^^^ T. N. Thiele^^^ used a mosaic (empty and filled squares on cross-section paper) to test y^=d{T[\od c), where c is an integer or Gauss complex integer a + 5v— 1, employing the graph oi y'^ — cx = d. Dittmar^^^ discussed a;^=r (mod p). Using Cauchy's^* explicit con- gruence for the numbers belonging to a given exponent, he gave the expanded form of the congruence with the roots belonging to the successive exponents 1,. ..,21. "*Von den Kubischen Resten u. Nichtresten, Progr., Dortmund, 1870. i-'Bull. Ac. Sc. St. Pdterebourg, 14, 1870, 356-375. i*'>D6duction et demonstration de trois lois primordiales de la congruence des nombres, Paris, 1873. *""0m Talmonstre," Forhandl. Skandinaviske Naturforskeres, Kjobenhavn, 11, 1873, 192-5. "^Die Theorie der Reste, insbesondere derer vom 3. Grade, nebst einer Tafel der Kubischen Reste aller Primzahlen der Form 6m + 1 zwischen den Grenzen 1 und 100. Progr. Koln Gym., Berlin, 1873. ^li Chap. VII] BiNOMIAL CONGRUENCES. 213 L. Sancery" (pp. 17-23) employed the modulus M = p'' or 2^', where p is an odd prime. Let a belong to the exponent n modulo M. Let A be the g. c. d. of m and 4>{M)/n. Set A=AiA2 where Ai = pi*'p2*' • • • > and p,- is a prime dividing both A and n, and p/< is the power of Pi dividing A. Let b be any divisor of Ag. Then a:"'=a (mod M) has 0(nAi5)/0(n) roots belonging to the exponent nAjS ; the power aAi5 of such a root is congruent to a, where a can be found by means of a linear congruence. Given a number belonging to the exponent nAi5, we can find Ai5 roots of the con- gruence. C. G. Reuschle^^^" tabulated the roots of /=0 (mod p), where p = wX+l and X are primes and / is the maximum irreducible algebraic prime factor of a^ — 1; also the roots of T^Hc^O, r/Hc^^O, TyHc^sO, rf^'n-\-d=Q, for c<13, d= —1 to —26, d=+2 to +21, and for various cubic and quartic congruences. A. Kunerth's method for ^^=c (mod h) will be given in Vol. 2, Ch. XII. E. Lucas^^^^ treated a;^+l=0 (mod p"), where p is a prime >2, for use in the question of the number of satins. Given a^+l=0 (mod p), set {a+i)"' = A+Bi, ^B=l (mod p'"). Then A^ is a root x of the proposed congruence. B. Stankewitsch^^^ proved that if x^^q (mod p) is solvable, p being an odd prime, the positive root <p/2 is =B/A (mod p), where 1-2 i A=Si_,+qSi.3+q%_s+ ...+q^ S^, B = Si+qSi_2+ ■ ■ • +q^ where i = {p — l)/2 and Sk denotes the sum of the products of 1, 2, . . ., i taken k at a time. Let n be a divisor of p — 1. Let F{x) be the g. c. d. modulo p of x" — ! and Il(x''^"- — 1), where a ranges over the distinct prime factors of n. Call f{x) the quotient of x'' — l by i^(a;). Then the roots of f{x) = (mod p) are the primitive roots of x"= 1 (mod p) . [Cf . Cauchy.14] N. V. Bougaief^^^ noted that if p = 8n+5 is a prime and if x^=q (mod p) is solvable, it has the root g(p+3)/8 ^j. (pzl)! g(p+3)/8 according as q2n+i^-^ or -1. If p = 2^Z+l, I odd, and q'=l, it has the root x=q^'+'^/\ [Legendre.^^®] T. Pepin^^^ treated x^= 2 by tables of indices. P. Gazzaniga^^^ gave a generalization of Gauss' lemma (the case n = 8 = 2, 1820 Tafeln Complexer Primzahlen . . . , Berlin, 1875. Errata, Cunningham."* ^^^^ G6om6trie des tigsus, Assoc, fran^., 40, 1911, 83-6; French transl. of his Italian paper in I'Ingegnere Civile, 1880, Turm. is'Moscow Math. Soc, 10, 1882-3, I, 112 (in Russian). ^<^Ibid., p. 103. "»Atti Accad. Pont. Nuovi Lincei, 38, 1884-5, 201. "»Atti Reale Istituto Veneto, (6), 4, 1885-6, 1271-9. 214 History of the Theory of Numbers. [Chap, vii ^ = 0). Separate the residues modulo p of kq, for k = l, 2, . . ., {p — l)/d, into three sets: .': P 5-1 i 0<ri,. . ., r,<-<Si,. . ., s,<-^p<tu. • ., t^<p j and form the differences mi = p — t,. From the set 1,. . ., (p — 1)/5, delete the r, and ?n,; there remain v numbers i\. If ?/, is a root of s,?/,= yi (mod p), then x"=5 (mod p) is solvable if and only if ( — l)"?/!. . .2/„=l (mod p), where 5 is the g. c. d. of n and p — 1 . P. Seelhoff^^^ gave the known cases in which x^=r (mod p) can be solved explicitly [Lagrange, ^^^ Legendre^^^]. In the remaining cases, one uses Gauss' method of exclusion, the process of Desmarest,^^ or, with Seelhoff, use various quadratic residues of p {ibid., p. 306). Here x^=41 (mod 120097) is treated. A. Berger^^^ considered a quadratic congruence reducible to a:^=D (mod 4n), where D=0 or 1 (mod 4). If D is prime to n, the number of roots is ^(D, in) = 2n{l + (f ) } = 2S (f ) f . = 22 (f ) f,, where p ranges over the distinct prime factors of n, while d and di range over the pairs of complementary divisors of n, and f ^ = or 1 according as d has a square factor or not. If g{nm)=g{n)g{m) for all integers n, m, ands'(l) = l, zgy (Z), 4n)^(n) = 22 Q ^(n) -S Q ^(n) -^2 Q ^(n)^ where n ranges over all positive integers. Mean values are found : J,(?)*»«-;b5ti7S.?,©I«-" it=i TT h=i\n/n where A is a fundamental discriminant according to Kronecker, X, Xi are finite for all n's, and p ranges over all primes. G. Wertheim^^^ presented the theory of a;^=a (mod m). R. Marcolongo^®° treated x^-\-P=0 (mod p) in the usual manner when explicit solutions are known. Next, from a particular set of solutions X, y of x^+p'"?/+P = 0, where p is a prime >2, we get the solution =i=x,=x-p'"y[ai.. .o„_i] (mod p"'+') of Xi^-\-p"''^^yi-{-P = 0, where [ai. . .a„_i] is the numerator of next to the last convergent to the continued fraction for p"'/{2x). The method is Serret's, Alg. Sup^r., II. For p = 2 the results obtained are the same as in Dirichlet's Zahlentheorie, §36. i"Zeitschrift Math. Phys., 31, 1886, 378-80. "SQfversigt K. Vetenskaps-Ak. Forhandlingar, Stockholm, 44, 1887, 127-153. Nova Acta regise soc. sc. Upsalensis, (3), 12, 1884. "»Elemente der Zahlentheorie, 1887, 182-3, 207-217. ""Giomale di Mat., 25, 1887, 161-173. Chap. VII] BiNOMIAL CONGRUENCES. 215 F. J. Studnicka^^^ treated at length the solution in integers x, y (y<h) of hx-{-l = y^, discussed by Leibniz in 1716. L. Gegenbauer^^^ gave a new derivation of the equations of Berger^^^ leading to asymptotic expressions for the number of solutions of x^=Z). A. Tonelli^^^ gave a method of solving x^=c (mod p), when p is a prime 4/i+l and some quadratic non-residue g' of p is known. Set p = 2'y-{-l, where y is odd. By Euler's criterion, the power 72^"^ of c and g are con- gruent to +1, —1. Set €0 = or 1, according as the power 72^"^ of c is congruent to +1 or —1. Then For s^3, set ei = or 1 according as the square root of the left member is = -f-lor-l. Then Proceeding similarly, we ultimately get g2eycy=-^l (mod p), e = €o+2€i+ . . . +2'-\_2- Thus a;= ±^'^c^^+^^/2 (mod p). Then Z^=c (mod p^) has the root X=x^'-'c(^^-'^^"'+i^/2 (mod p^). G. B. Mathews'^^ (p. 53) treated the cases in which x^^a (mod p) is solvable by formulas. Cf. Legendre.-^^^ S. Dickstein^^^ noted that H. Wronski^^^ gave the solution rM -l(7r-l) y==hK+{-iy+'+Mi, = /i+(-1)'+'A| ^, ttJ +Mj of 2"— a?/"=0 (mod M) with (iV^)^ in place of K, and gave, as the condition for solvability, a(lV^)2"-l=0(modM). But there may be solutions when the last condition is satisfied by no integer A;. This is due to the fact that the value assigned to y imposes a limitation, which may be avoided by using the same expressions for y, z in a parameter K, subject to the condition aK" — 1 = (mod M). M. F. J. Mann"^'^ proved that, if n=2^XV. . ., where X, m, • • • are dis- tinct odd primes, the number of solutions of x^= 1 (mod n) is GGiGi . . . QiQi . . . , where G= 1 if n or p is odd, otherwise G is the g. c. d. of 2p and 2^~'^, and where Gi, Gi,.., gi, g2,. . are the g. c. d.'s of p with X"~\ m''~\- • •> X — l,jLi— 1,. . ., respectively. A. Tonelli^^^ gave an explicit formula for the roots of x^=c (mod p^), "iCasopis, Prag, 18, 1889, 97; cf. Fortschritte Math., 1889, 30. "^Denkschriften Ak. Wiss. Wien (Math.), 57, 1890, 520. "'Gottingen Nachrichten, 1891, 344-6. I'^BuU. Internat. de I'Acad. Sc. de Cracovie, 1892, 372 (64-65); Berichte Krakauer Ak. Wiss., 26, 1893, 155-9. "^aMath. Quest. Educ. Times, 56, 1892, 24-7. "^AttiR. Accad. Lincei, Rendiconti, (5), 1, 1892, 116-120. 216 History of the Theory of Numbers. [Chap, vii when p is an odd prime, and a quadratic non-residue ^ of p is known. Set p = 2'a+l, where s^ 1 and a is odd. Then y = ap^~^ is odd, and {}>(p^) = 2'y. Tonelli's earlier work for modulus p now holds for modulus p^ and we get x= ± g'^c^'^'^^^^. If s = 1, then e = and the root is that given by Lagrange if X = l. If s = 2, whence p = 4a+l = 8Z+5, the expression for X is given a form free of e = cq: x= ± (c»+3)V^+^^/^ y = ap^-\ A. Tonelli^^^ expressed the root x in a form free of e for every s: 7+1 where the v's are given by the recursion formula v.-H = c''-S?.t^'\ . .t^--,%+k (/i = 2, 3,. . .). Here k is an existing integer such that A;-|-l is a quadratic residue of p, and A: — 1 a non-residue. Thus, if s = 3, 7+1 x=^{c^''+ky {{c-'+ky^C+kl^'c 2 , where we may take A: = —2 if a is not divisible by 3, but A; = —4 if a is divi- sible by 3, while neither a nor 4a +1 are di\'isible by 5. N. Amici^^ proved that a;^*=6 (mod 2"), h odd, k^v — 2, is solvable only when h is of the form 2^'"^^/i+l and then has 2^+^ roots, as shown by use of indices. For (x'")^ = 5, the same condition on b is necessary ; thus it remains to solve x'"=j8 (mod 2') when m is odd. If i3 = 8A;+l or 8A:+3, it has an index to the base 8/z + 3 and we get an unique root. If /3 = 8/j — 3 or 8A: — 1, then x'"= — j3 has a root a by the preceding case, and —a is a root of the proposed congruence. Jos. Mayer^^^ found the number of roots of x^=a (mod p''), for the primes 2, 3, p = 6m='= 1. If fli, Go,. . . are residues of nth powers modulo p, and if g is the g. c. d. of n and p — 1, then 0102- . . = -f-l or —1 (modp), according as p' = (p — l)/g is odd or even. If p' is even, we can pair the numbers belonging to the exponent p' so that the sum of a pair is or p; hence there exists a residue of an nth power = — 1 (mod p) ; but none if p' is odd. K. Zsigmondy^^ obtained by the use of abelian groups known theorems on the number, product and sum of the roots of x*= 1 (mod m). G. Speckmann^^^ considered x^=a (mod p), where p is an odd prime. Set P=(p — 1)/2. When they exist, the roots may be designated P — k, P-\-l-\-k, whose sum is p. The successive differences of P^, (P-|-l)^ (P+2)2,. . . arep, p+2, p+4, . . .. Thesumof 2 = s+l termsof 2,4, 6, . . . is s'^+Ss+2 = z^+z. Adding to the latter the remainder r obtained by di\'iding P^ by p, we must get pn-{-a. Hence in pn-\-a—r we give to n the values i»«Atti R. Accad. Lincei, Rendiconti, (5), 2, 1893, 259-265. "'Ueber nte Potenzreate und binomische Congruenzen dritten Grades, Progr., Freising, 1895. >»^\rcliiv Math. Phys., (2), 14, 1896. 445-8; 15, 1897, 335-6. Chap. VII] BiNOMIAL CONGRUENCES. 217 0, 1,2,... until we reach a number of the form ^-\-z (found by extracting the square root). Then fc = s, so that the roots P-A^, P+l+Zc are found. N. Amici^^^ proved that if neither m nor 6 is divisible by the prime p, and if a is a given root of x'^=h (mod p), and if /3, g are (existing) integers such that i3(/)(p')-p'-i + l=mg, then of^'^"^ is a root of x™=6 (mod y^). Hence we limit attention to the case X = l. Consider henceforth x'^=h (mod p), where p = 2''/i+l is an odd prime, h being odd, and 6 not divisible by p. First, let ¥^8. Then 6''=1 (mod p) is a necessary and sufficient condition for solvabihty and x= ± y^ are roots, where q is such that 2^q — 1 is divisible by /i. If gr is a quadratic non-residue of p, all 2" roots are given by ± 6''gr''', where e = 6i+2e2 4- . . . +2*~^€^,_i, the ei taking the values and 1 independently. Finally, let A:<s. Then two roots ±(3 are determined by the method of TonelU, while all the roots are given by x==^^g'', t = e,+2e2+ . . . +2'-\_„ €^ = or 1. R. Alagna^°° considered a prime p = 4/c+l for which /b is a prime. Since 2 is known to be a primitive root of p, it is easy to write down those powers of 2 which give all the roots of x'^=l (mod p), where d is one of the six divisors 2' or 2'k of p — 1, likewise of x'^^N, since N must be congruent to an even power of 2. For the modulus p^, we may apply the first theorem of Amici or proceed directly. The same questions are treated for a prime 4A;+3 for which 2A; + 1 is a prime. A. Cunningham^"^ treated at length the solution of x^=\ (mod iV'), where iV is a prime, and gave tables showing all incongruent roots when < = 1, 2, N-^ 101, I any admissible divisor of iV — 1 ; also for a few additional f's when N is small. Cunningham^oi" treated a^= 1 (mod q^) and 3.2^= ± 1 (mod p). He^oi*- treated the problem to find 5''=+l or ±a, given a^=\, a''=^h (mod p), where ^ is odd and ^, x, 17 are the least values of their kind; also given a*=l, a'^^^h, a'=^c, to find the least /? and 7 such that h^=c, c^=6 (mod p). W. H. Besant^°^ would solve y^ = ax+h by finding the roots s of s^=6 (mod a). Then y = ar-\-s, x = ar^-\-2rs+{s'^ — h)/a. G. Speckmann^°^ replaced x"=A; (mod p) by the pair of congruences x"~^=r, xr^A; (mod p). In np+/c give to n the values 0, 1, 2, . . . until we find one for which np+k = rx such that, by trial, x"~^=r. The method is, of course, impractical. "'Rendiconti Circolo Mat. di Palermo, 11, 1897, 43-57. ""Rendiconti Circolo Mat. di Palermo, 13, 1899, 99-129. "^Messenger of Math., 29, 1899-1900, 145-179. Errata, Cunmngham226, p. 155. See 13a of Ch. IV. "i^Math. Quest. Educ. Times, 71, 1899, 43-4; 75, 1901, 52-4. 2"&/6td., (2), 1, 1902, 70-2. ^o^Math. Gazette, 1, 1900, 130. '"'Archiv Math. Phys., (2), 17, 1900, 110-2, 120-1. 218 History of the Theory of Numbers. [Chap, vii G. Picou^*^ applied to the case n = 2 Wronski's^^^ formula for the resi- dues of 71 th powers modulo M, M arbitrary. For example, if M = \Qa^\, (h=^Sa)-^ =pa{^h-iy (mod M). [If 8a were replaced by 4a, we would have an identity in h.] P. Bachmann^^ (pp. 344-351) discussed x"'=a (mod p"), p>2, p = 2. G. Arnoux-°^ solved x^^=79 (mod 3-5-7) by getting the residue 2 of 79 modulo 7 and that of 14 modulo 0(7) =6 and solving x'^=2 (mod 7) by use of a table of residues of powers modulo 7. Similarly for moduli 3, 5. Take the product of the roots as usual. M. Cipolla-"'^ generahzed the results of Alagna-*^" to the case of a prime p = 2"'q-\-l, 7n>0, q an odd prime, including unity. For any divisor d of p — 1, the roots of x'^=N (mod p) are expressed as given powers of a primi- tive root a of p. If 2 belongs to the exponent 2''co modulo p, where w is odd, theng'= 1 (mod p) if and only if 2""^ is the highest power of 2 dividing m. Cunningham-"^" found the sum of the roots of (i/"=tl)/(?/±l) = (mod p). M. Cipolla'"^ proved the existence of an integer k such that k~ — q is a quadi'atic non-residue of the prime p not dividing the given integer q. Let Un = h^q\{kWqr-{k-Vqr\, v^=Viik+V¥^r+{k-V¥^r\. By expansion of the binomials it is shown that the roots of x^=q (mod p) are given by =*=W(p_i)/2 and by ±y(p+i)/2. These may be computed by use of Wn=2kWn_i—qWn-2 (mod p) {w = u or v), with the initial values Uq = 1, Wi = p; ^0 = 1? Vi = k. Although u^, y„ are the functions of Lucas, the exposition is here simple and independent of the theory of Lucas (Ch. XVII). M. Cipolla-°^ proved that if 5 is a quadratic residue and k^—q is a quadratic non-residue of an odd prime p, z~^q (mod p^) has the roots ^lVq\{k+Vqr-{k-V~qr\, where r = p^~\p — 1)/2. Other expressions for the roots are ^hq'{(k+V¥^r+{k-V¥^y\, t=ip^-2p^-' + l)/2, s = p^-\p + l)/2. Thus if Zi~=q (mod p), the roots modulo p^ are ^q'zi^^''^ (TonelH^^^). Finally, let n=TLpi^', where the p's are primes >3; take e, = =»=l when Pi=^l (mod 4). There exists a number A of the form k^—q such that ^ML'iatermddiaire dea math., 8, 1901, 162. '^o* Assoc, frang. av. sc, 31, 1902, II, 185-201. »«Periodico di Mat., 18, 1903, 330-5. »'»«Math. Quest. Educ. Times, (2), 4, 1903, 115-6; 5, 1904, 80-1. "'Rendiconto Accad. Sc. Fis. e Mat. Napoli, (3), 9, 1903, 154-163. "8/6id., (3), 10, 1904, 144-150. I Chap. VII] BiNOMIAL CONGRUENCES. 219 (A/pi) = €i,. . ., (A/pJ = e^, where the symbols are Legendre's. Call M the 1. c. m. of pl^~\pi — ei)/2 for {=1,..., v. Then z^=q (mod n) has the root A. Cunningham^°^ indicated how his tables may be used to solve directly x''= —1 (mod p) for n = 2, 3, 4, 6, 12. From p = a^-\-h^, we get the roots re = ± a/6 of a;^=— 1 (modp). Also p = a^+ 6^ = c^+2(i^ gives the roots =i= d{a-\-b) / (ce) and ±c(a±6)/(2de) of x^^—1 (mod p), where e = a or 6. Again, p = A^+SB^ gives the roots iA-B)/{2B), {B+A)/{B-A), and their reciprocals, of x^=l (mod p). M. Cipolla^°^ gave a report (in Peano's symbolism) on binomial con- gruences. M. Cipolla^^° proved that if p is an odd prime not dividing q and if z^=q (mod p) is solvable, the roots are z= ^2{qs,+q\+q\+ . . . +5'^~'%-4+Sp-2) where s,=r+z+...+ m- Then x^=q (mod p'^) has the root z^^ V, e = (p^-2p^-^ + l)/2. For p=l (mod 4), x'^=q (mod p) has the root 4 1 5^S2,_i- 2 q^-%j_s+2 S g^s^^.i (z = ^) • 1=1 y=i i=l \ 1 / M. Cipolla^^^ extended the method of Legendre^^® and proved that x^"'=l-\-TA (mod 2*), for A odd and s^w+2, has a root x = l+2^Aci-22M2^2+- • • + (-l)"'"'2"^A"c„, n=r _^~^ J , where ^_]_ ^(2'"-l)(2-2'"-l) . . .(n^l-2'"-l) are the coefficients in il+zy^'"' = l+c,z-c,z'+c,z'- . . . -(-1)X2"+. . .. 0. Meissner^^^ gave for a prime p = Sn+5 the known root £+3 £-1 ^ = D » oix^=D (modp), D * =i (modp). But if 2)^p-i)/4= _i (mod p), a root is ^](p — 1)/2)!, since the square of the last factor is congruent to ( — l)(p+^)/2 j-^y wrjigon's theorem. Tamarkine and Friedmann^^^ expressed the roots of z^^q (mod p) by a formula, equivalent to Cipolla's,^^° ^osQuadratic Partitions, 1904, Introd., xvi-xvii. Math. Quest. Educ. Times, 6, 1904, 84-5; 7, 1905, 38-9; 8, 1905, 18-9. ""Rendiconto Accad. Sc. Fis. e Mat. Napoli, (3), 11, 1905, 13-19. "i/6id., 304-9. '"Archiv Math. Phys. (3), 9, 1905, 96. 2"Math. Annalen, 62, 1906, 409. 220 History of the Theory of Numbers. [Chap, vii (p-3)/2 z==b2 2 q^''-''-"'s2^+,. m = For, according as 2/^ is or is not =q (mod p), we have y\i-{y^-qy~'\=y or o (mod p). We can express S2m+\ in terms of Bemoullian numbers. A. Cunningham-^^ gave a tentative method of solving x'^=a (mod p). He-^^^" noted that a root Y=2r]^ of Y^=-l leads to the roots of y^=-l (mod p). M. Cipolla^^^ employed an odd prime p and a divisor n of p — l=ni/. If Ti, . . ., r, form a set of residues of p whose nth powers are incongruent, and if ^'=1 (mod p), then x''=q (mod p) has the root k=0 ;=1 Forn = 2, this becomes his^^° earUer formula by taking 1,2,..., (p — 1)/2 as the r's. Next, let p — l=mji, where m and /x are relatively prime and m is a multiple of n. If 7 and 8 belong to the exponents m and /x modulo p, the products 7''5* {r<m/n, s<iJi) may be taken as ri,. . ., r,. According as nk= 1 or not (mod /x), we have y{nk-l)m/n -j^ At= -ufi — „^_i - or Ak=0 (mod p). 7 — i If n is a prime and n" is its highest power dividing p — 1, there exists a number co not an nth power modulo p and we may set m = n'', 7=0)" (mod p). In particular, if n = 2, x^^q has the root _ 1 P+2^-1 2'"-'-l 2 «=o where co is a quadratic non-residue of p. If p=5 (mod 8), we may take CO = 2 and get M. CipoUa^^® considered the congruence, with p an odd prime, x^ =a (mod p"*), r<7n, a necessary condition for which is that h = {a^ — a)/p'"^' be an integer. Determine A by a^ A=h (mod p"*). Then the given congruence has the root axo if Xq is a root of :r'''=l-^p'+' (modp"*). This is proved to have the root '"Math. Quest. Educ. Times, (2), 13, 1908, 19-20. ^^"^Ibid., 10, 1906, 52-3. "»Math. Annalen, 63, 1907, 54-61. >"Atti R. Accad. Lincei, Rendiconti, (5), 16, I, 1907, 603-8. Chap. VII] BiNOMIAL CONGRUENCES. 221 where Ci = l/p*", . . . are given by the expansion ^1—2 ==l—CiZ — C2^ — M. Cipolla^" treated x"=a (mod p"*) where n divides 4>{p^). We may set n = p'v, where v divides p — 1. Determine integers a, j3 such that Then the initial congruence has the root yx^" if y^^=a^ (mod p"*), solved as in his preceding paper, and if Xi is a root of x''=a (mod p"*). The latter has the root ^ A;=0 t = l where t={p — l)/v, pi^rf""' (mod p'"), ri, . . ., r^ being integers prime to p such that their j/th powers are incongruent and form a group modulo p"*. K. A. Posse^^^ gave a simplified exposition of Korkine's^^^ method of solving binomial congruences. Cf. Posse/^^ Schuh.^^^'^ F. Stasi^^^ proved that we obtain all solutions of x^=a^ (mod n), where n is odd and prime to a, by expressing n as a product of two relatively prime factors P and Q in all ways, setting x — a = Pz and finding z from Pz+2a=0 (mod Q). [Instead of his very long proof, it may be shown at once that we may take x — a, x+a divisible by P, Q, respectively.] L. Grosschmid^^° gave for the incongruent roots of x^^r (mod M) an expHcit formula obtained by means of the ideal factors of ilf in a quadratic number-field. L. Grosschmid^^^ treated the roots of quadratic binomial congruences. A. Cunningham^^^ solved x^= —1 (mod p), where p = 616318177 is a prime factor of 2^^ — 1; by using various small moduli, he obtained p = 24561^ + 36161 L. von Schrutka^^^'' used a correspondence between the integers and certain rational numbers to treat quadratic congruences without novelty as to results. The method will be given under the topic Fields in a later volume of this History. Grosschmid^^^ employed the products R and N of all the quadratic residues and non-residues, respectively, ^2n of a prime p = 4n+l. Then R^={-iy+\ iv2=(_i)« (modp). 2"Atti R. Accad. Lincei, Rendiconti, (5), 16, I, 1907, 732-741. "'Charlkov Soobsc. Mat. Obs6 (Report Math.Soc. Charkov), (2), 11, 1910, 249-268 (Russian). "»I1 BoU. Matematica Gior. Sc.-Didat., 9, 1910, 296-300. «20Jour. fur Math., 139, 1911, 101-5. ""Math. 6s Phys. Lapok, Budapest, 20, 1911, 47-72 (Hungarian). 222Math. Questions Educat. Times, (2), 20, 1911, 33-4 (76). 2««Monatshefte Math. Phys., 23, 1912, 92-105. 223Archiv Math. Phys., (3), 21, 1913, 363; 23, 1914-5, 187-8. 222 History of the Theory of Numbers. [Chap, vii Hence ±7? and =»=iV are the roots of x^=— 1 (mod p) according as p = S7n + l or Sm + 5. ,;, U. Concina"^ proved the first result by Legendre.^^ A. Cunningham--^ tabulated the roots of i/*=±2, 2?/"*=±l (mod p), for each prime p< 1000. Cunningham"^ listed the roots of ?/'= ± 1 (mod p"), where l^qp", p being an odd prime ^19, p''<10^, a = l and often also a = 2, q a factor of p— 1. A. Gcrardin and L. Valroff"7 solved 2i/=l (mod p), 1000<p<5300. Cunningham-^^ announced the completion of tables giving all proper roots of ?/'"= 1 (mod p*) for m odd ^15, and of 'y'"= — 1 (mod p*) for m even ^ 14. These tables have since been completed up to p* < 100000 and are now nearly all in type. T. G. Creak^-^ announced the completion of like tables for m = 16 to 50; 52, 54, 56, 63, 64, 72, 75, and 10^<p'^<10^ H. C. Pocklington--^ noted that if p is a prime 8m+5 and a}"'^^=-l, x^=a (mod p) has the roots =»=^(4a)'"'^^ He showed how to use {t-\- u\/DY to solve a;-= —D (mod p=4A-+l), and treated a:^=a. *J. Maximoff^^° treated binomial congruences and primitive roots. *G. Rados-^^ gave a new proof of known criteria for the solvability of x- = D (mod p). He-^^ gave a new exposition of the theory of binomial congruences without using indices. Congruences ^''"^^l (mod p") are treated in Chapter IV. Euler**'' of Ch. XVI solved x-=— 1 (mod p). Lazzarini"^ of Ch. I erred on the number of roots of 2-= —3 (mod n). Many papers in Ch. XX treat x*=a; (mod 10"). The following papers from the first part of Ch. VII treat also binomial congruences: Euler,^ Lagrange,^ Poinsot," Cauchy,^^ Lebesgue," Epstein,i^2 Korkine."^ =«*Periodico di Mat., 28, 1913, 212-6. 22*Messenger Math., 43, 1913-4, 52-3. 2"/Wd., 148-163. Cf. Cunningham .201 227Sphinx-0edipe, 1913, 34; 1914, 18-37, 73. 228Messenger Math., 45, 1915-6, 69. 2^Proc. Cambridge Phil. Soc, 19, 1917, 57-9. 2^Bull. Soc. Phys.-Math. Kasan, (2), XXI. 23iMath. 4s Term^s Ertesito, 33, 1915, 758-62. ^Ibid., 34, 1916, 641-55. CHAPTER VIII. HIGHER CONGRUENCES. A Congruence of Degree n has at most n Roots if the Modulus p is a Prime. J. L. Lagrange^ proved that, if a is not divisible by the prime p, ax'^+fex""^ + . . . is divisible by p for at most n integers x between p/2 and — p/2. For, let a, jS, . . . , p, (7 be n+1 such distinct integers. Then the quotient of a(a:'^-a")+6(a:'^-i-a"-^)+ . . . by a: — a is a polynomial aa:"~^ + . . . which is divisible by p when x=^,. . .,a. Proceeding as before, we finally have a{p—<7) divisible by p, which is impossible. L. Euler^ noted that a:" — 1 is divisible by a prime p for not more than n integers x, 0<x<p. For, if x = a, is such an integer, then x — a divides x^ — l—mp, where m is a suitable integer; the quotient / is of degree n — 1. If a: = 6 is a second such integer, x — h divides/— m'p. Proceeding as in alge- bra, we obtain the theorem stated. [The argument is applicable to any polynomial of degree n in x.] A. M. Legendre^ noted that P^(x — a)Q-\-pA has only one more root than Q. C. F. Gauss'* proved the theorem by assuming that there is a congruence ox"4- . . . = (mod p) with more than n roots a, . . ., and that every con- gruence of degree I, Kn, has at most I roots. Substituting y+a for x, we obtain a congruence a?/"-f- ... =0 with more than n roots, one of which is zero. Removing the factor y, we obtain a?/"~^+. . . = with more than w — 1 roots, contrary to hypothesis. Gauss^ noted that if a is a root of ^=0 (mod p), then ^ is divisible by x — a modulo p. li a, b,. . . are incongruent roots, ^ is divisible modulo p by the product (x — a){x — b).... Hence the number of roots does not exceed the degree of ^. A. Cauchy^ made the proof by use of X=(x — a)Xi (mod p), identically in x, where the degree of Xi is one less than the degree of X. A. L. Crelle^ and S. Earnshaw^ gave Lagrange's proof. Crelle^ proved that if ei, . . ., e„ are n distinct roots, ' ax^^-i- . . . = a(x — ei) . . .{x — e^+pN. iMem. Ac. BerUn, 24, ann6e 1768 (1770), p. 192; Oeuvres, 2, 1868, 667-9. ='Novi Comm. Ac. Petrop., 18, 1773, p. 93; Comm. Arith., 1, 519-20. »M^m. Ac. Roy. Sc, Paris, 1785, 466; TWorie des nombres, 1798, 184. ♦Disq. Arith., 1801, Art. 43. ^Posthumous paper, Werke, 2, p. 217, Art. 338 (p. 214, Art. 333). Maser's German translation of Gauss' Disq. Arith., etc., 1889, p. 607 (p. 604). "Exercices de Math., 4, 1829, 219; Oeuvres, (2), 9, 261; Comptes Rendus Paris, 12, 1841, 831-2; Exercices d'Analyse et de Phys. Math., 2, 1841, 1-40, Oeuvres, (2), 12. 'BerUn Abhand., Math., 1832, p. 34. Cambridge Math. Jour., 2, 1841, 79. •BerUn Abhand., Math., 1843, 50-54. 223 224 History of the Theory of Numbers. [Chap.viii 'J L. Poinsot^" gave the proof due to Crelle.' ■ J. A. Gninert^^ proceeded by induction from n — 1 to n, making use of the first part of Lagrange's proof. D. A. da Silva^' gave a proof. Number of Roots of Higher Congruences. G. Libri^" found that /(a:, ?/, . . .) = (mod m) has I 1 b d -22 Wl x=o v=t ;--! 2A-7rf . . 2Uj\ .ALi COS \-x sm \ Lfc=o Tnn m \ sets of solutions such that a^x^6, c^?/^d, .... The total number of sets of solutions is 1 ^ ^ r, , 27r/, 47r/ , , Am-X)'KJ\ — 2 2 . . . O +COS — ^+cos — ^+ . . . +COS 2^^ '-^ V 7^1=0 v=o y m m m \ V. A. Lebesgue^^ proved that if p is a prime we obtain as follows the residue modulo p of the number S>k of sets of solutions of F{xx, . . ., x„) = (mod p), in which each x, is chosen from 0, 1,. . ., p — 1, and F is a poly- nomial with integral coefficients. Let 2A be the sum of the coefficients of the terms Ax^ ■ ■ x/ of the expansion of F^~'^ in wliich each of the exponents a, . . . , ^ is a multiple > of p - 1 . Then Sk= ( - 1) *"^^ 2 A (mod p) . Henceforth, let p = hm+l. First, let F = x"'—a. In F^~^ the coefficient of a;""'?-!-"^ is (p-^)(-a)"=a" (mod p). The exponent of x will be a multiple >0 of p — 1 only when n = k(p — l)/d, for A: = 0, 1,. . ., d — 1, where c? is the g. c. d. of m and p-L Thus 51=20*^^"^^'''^ (mod p), while evidently Si<p. According as a'-^~^^^'^=l or not, we get Si=d or 0. Next, let F = x"'-ay"'-h. Set c = ay"'-\-h. In (x"'-c)p-^ we omit the terms in which the exponent of x is not a multiple >0 of p — 1 and also the ^rn(p-i) jjq|. containing y. Since the arithmetical coefficient is =1 as in the first case, we get In this, we replace c'''' by those terms of (ay"* +6)'''' in which the exponents are multiples >0 of p — 1, viz., SGB^"^") kh-lhUh Set 2/ = 1, and sum for A* = 1, . . . , w — 1 ; we get —S2 (mod p). It is shown otherwise that *S2 is a multiple <mp of m. To these two cases is reduced the solution of (1) ^ = 01X1"*+. . .-\-a^k"=a (mod p = hm-\-l). "Jour, de Math^matiques, 10, 1845, 12-15. "Klugel'8 Math. Worterbuch, 5, 1831, 1069-71. "Proprietades . . .Congruencias binomias, Lisbon, 1854. Cf. C. Alasia, Rivista di fisica, mat. e sc. nat., 4, 1903, p. 9. "Mdm. divers Savants Ac. Sc. de I'Institut de France (Math.), 5, 1838, 32 (read 1825). Jour. fur Math., 9, 1832, 54. To be considered in vol. n. "Jour, de Math., 2, 1837, 253-292. Cf. vol. 3, 113; vol. 4, 366. Chap. VIII] NuMBER OF RoOTS OF CONGRUENCES. 225 Denote by P the sum of the first / terms of F and by Q the sum of the last k—f terms. Let gr be a primitive root of p. Let P° be the number of sets of solutions of P=0 (mod p); P^'^ the number for P^g^ (mod p); Q^ and Q^'^ the corresponding numbers for Q=0, Q=g'. Then the number of sets of solution of P=Q (mod p) is P°Q°+/iS:=rP^*^Q^*\ Hence we may deduce the number of sets of solutions of P=0 from the numbers for P = Aand Q= -A. For P= a, we employ P = P, Q = /x'"and get P° = P° -\-{'p — l)P^^\ which determines the desired P''^\ The theory is applied in detail to (1) for m = 2, k arbitrary, and for w = 3, 4, k = 2. Finally, the method of Libri^^ is amplified. Th. Schonemann^^ noted that, if Sk is the sum of the A;th powers of the roots of an equation x"+ . . . =0 with integral coefficients, that of x"" being unity, and if >S(p_i)«=n (mod p) for <= 1, 2, . . ., w, where p is a prime >n, the corresponding congruence a;"+ . . . = (mod p) has n real roots. A. L. Cauchy^^ considered F{x) = Q (mod M), with M=AB. . ., where A, B,. . . are powers of distinct primes. If F{x) = (mod A) has a roots, F(x) = (mod B) has /S roots, etc., the proposed congruence has a/3. . . roots in all. For, if a, 6, . . . are roots for the moduli A, B,. . . and X=a (mod A), X=b (mod P), . . . , then X is a root for modulus M. P. L. Tchebychef^° proved that, if p is a prime, a congruence /(x)=0 (mod p) of degree m<p has m roots if and only if the coefficients of the remainder obtained by dividing x^—x hj f{x) are all divisible by p. Ch. Hermite^^ proved the theorem: If fx and jjl' are the numbers of sets of solutions of 4>{x, y)=0 for the respective moduli M and M', which are relatively prime, the number of sets of solutions modulo MM' is /jl/j,'. If 0=0 is solvable for a prime modulus p, it will be solvable modulo p" if have no common sets of solutions. In this case, the number of sets of solutions modulo p" is p"~V if tt is the number for modulus p. Similar results are said to hold for any number k of unknowns. If ikf is a product of powers of the distinct primes pi, . . ., p„, and if tt, is the number of sets of solutions of the congruence modulo Pi, then the number of sets for modulus M is M' k~l TTi-.-TTn For a:^+A|/^=A (mod M), we have Xj = p, — (— A/p»), where (a/p) is =«= 1 according as a is a quadratic residue or non-residue of p. JuUus Konig gave a theorem in a seminar at the Technische Hochschule in Budapest during the winter, 1881-2, which was published in the following paper and that by Rados.^^ "Jour, fiir Math., 19, 1839, 293. "Comptes Rendus Paris, 25, 1847, 36; Oeuvres, (1), 10, 324. "Theorie der Congruenzen, in Russian, 1849; in German, 1889, §21. "Jour, fiir Math., 47, 1854, 351-7; Oeuvres, 1, 243-250. 226 History of the Theory of Numbers. [Chap, viii G. Raussnitz*^ proved the theorem, due to Konig: Let (2) f{x) =aoX^-2+OiX''-H . . . +ap_2, where the a's are integers and ap_2 is not divisible by the prime p. Then f{x)=0 (mod p) has real roots if and only if the cyclic determinant (3) D = Oo Oi 02 ... ap_3 ap_2 «! a2 03 ... ap_2 Oo Op_20o Oi ... Op_4 Op.s is divisible by p. In order that it have at least k distinct real roots it is necessary that all p — k rowed minors of D be divisible by p. If also not all p — k — 1 rowed minors are divisible by p, the congruence has exactly k distinct real roots. The theorem is applicable to any congruence not ha\'ing the root zero, since we may then reduce the degree to p — 2 by Fermat's theorem. Gustav Rados-'* proved Konig's theorem, using the fact that a system of p — l linear homogeneous congruences modulo p in p — 1 unknowns has at least k sets of solutions linearly independent modulo p if and only if the p — k rowed minors are divisible by p. L. Kronecker^^ noted that, if p is a prime, the condition for the existence of exactly p—m — 1 roots of (2), distinct from one another and from zero, is that the rank of the system (3') (a,+,) (i,A: = 0, l,...,p-2) modulo p is exactly m, where Os+p_i = a^. The same is the condition for the existence of a {p—m — l)-fold manifold of sets of solutions of the system of linear congruences 2'a,+,0,= (mod p) ( A = 0, 1 , . . . , p - 2) . fc=0 L. Kronecker^^ gave a detailed proof of his preceding results, noted that the rank is ?« if not all principal t?? -rowed minors are divisible by p while all VI -{-1 rowed minors are, and added that Co+Ci.t+ . . . +Cp_2a:^~^=0 (mod p) has exactly s roots 7^0 if one and the same linear homogeneous congruence holds between every set of p—s (but not fewer) successive terms of the periodic series Cq, Ci, . . . , Cp_2, Cq, Ci, . . . . L. Gegenbauer^^ proved Kronecker's version of Konig's theorem. Gegenbauer^^ noted that Kronecker's theorems imply the corollary: "Math, und Naturw. Berichte aus Ungam, 1, 1882-3, 266-75. "Jour, fiir Math., 99, 1886, 258-60; Math. Termea Ertesito, Magj'ar Tudon Ak., Budapest, 1, 1883, 296; 3, 1885, 178. »Jour. fur Math., 99, 1886, 363, 366. **Vorlesungen liber Zahlentheorie, 1, 1901, 389-415, including several additions by Hensel (pp. 393, 399, 402-3). "Sitzungsber. Ak. Wiss. Wien (Math.), 95, II, 1887, 165-9, 610-2. *^Ibid., 98, Ila, 1889, p. 32, foot-note. Cf. Gegenbauer.« Chap. VIII] NuMBER OF RoOTS OF CONGRUENCES. 227 There exist exactly p—m—2 roots of (2), distinct from one another and from zero, if and only if there exist exactly p — m — 2 distinct linear homo- geneous functions p-2 Xak,hah {k = l,. . .,p-m-2) h=0 which remain divisible by p after applying all cyclic permutations of the ah, so that sV»a,„^0 (mod p) filo/iV.'.^.T™-!)- A simple proof of this corollary is given. L. Gegenbauer^^ noted that the niunber of roots of /(a^)=0 (mod k) is since D{k) = 1 or according as /(a:) is divisible by k or not. Let ki,...,ka be a series of increasing positive integers and g{x) any function. In the first equation take k = ki, multiply by g{ki) and sum iorl = l,. . .,8. Revers- ing the order of the summation indices I, x in the new right-hand member, we get i \f(x), h\g{k) = S G, G=XD{ix)g{f,), 1=1 1=0 where in G the summation index n takes those of the values ki,...,ki which exceed x. Thus G represents the sum G{f{x)] ki,..., k^; x) of the values oi fifx) when ij, ranges over those of the numbers ki,. . ., ks which exceed x and are divisors oif{x). In particular, if ^(x) = 1, G becomes the number x}/ of the k's which exceed x and divide /(a:). Let f{x)=vi^nx. Then f{x) = (mod k) has {k, n) roots or no root according as m is or is not divisible by the g. c. d. {k, n) of k and n; let {k, 71] m) denote {k, n) or in the respective cases. Then S {ki, n; m) g{ki) = s G(m^nx; ki,. . ., k^; x). '-1 x=0 Let G{a, h) denote the sum of the values of g(ii) when n ranges over all the divisors >6 of a; xl/{a, h) the number of divisors >b of a. Taking ki = l ioT 1=1,. . ., d, we deduce S 5-1 S (Z, n; m)^(0= S {G{m^nx, x)-G{m^nx, b)\. 1=1 z=0 For gil) = 1, this reduces to Lerch's^°° relation (16) in Ch. X. Again, a b 2 {G{m+nx, x — 1) —Gim+nx,h+x)\ = S {G{m—nfx,tx) — Giin—nfx,ti-\-a)\, X-l u=0 "SitzuDgsberichte Ak. Wiss. Wien (Math.), 98, Ila, 1889, 28-36. 228 History of the Theory of Numbers. [Chap, viii which for g{x) = l yields the first formula of Lerch. Next, if the A;'s are primes and g is a prime distinct from them, 2 Gix'^-q] k,,.. ., k,; x)= 2 (fc,-l, n; q)g{ki). x=0 /=1 Finally, he treated f{x) of degree d = A-j — 2, whose constant term is prime to each A-, and coefficient of x''~' is divisible by the prime k^ if i<ks — k^. Gegenbauer^" noted that, if p — l—n is the rank of the system (3) modulo p, the congruence, satisfied by the distinct roots 5^0 of (2) and by these only, is given symboUcally by (-X--V \dai daj fli+fc I =0 (mod p) {%, k = 0,. . ., p-2). He obtained easily Kronecker's"^ form of the last congruence. He gave necessary and sufficient conditions, expressed in terms of a comphcated determinant and its /z — l successive derivatives with respect to Op_2, in order that (2) and a second congruence of degree p — 2 shall have jx common roots ?^0, and found the congruence satisfied by these ji common roots. He deduced determinantal expressions for the sum o-^ of the rth powers of the roots of (2), and for the coefficients in terms of the cr's. Michael Demeczky^^ would employ Euclid's process to find the g. c. d. G{x) modulo p of (2) and x'^—x. If G{x) = (mod p) is of degree v it has V real roots and these give all the real roots of (2) . Multiple roots are then treated. The case of any composite modulus is known to reduce to the case of p', p a prime. If (2) has X distinct real roots, not multiple roots, we can derive X real roots of /(a;) = (mod p'). If pi, . . . , p„ are distinct primes and if /(x) = (mod p,) has X, real roots, then/(x) = (mod pi. . .p„) has Xi. . .X„ real roots, and is satisfied by every integer x if the former are. Various sets of necessary and sufficient conditions are found that f{x) = (mod m=np'<) shall have m distinct real roots; one set is that/(x)=0 (mod p'<) identically for each i. L. Gegenbauer^^ proved that a congruence modulo p, a prime, of degree p — 2 in each of n variables has a set of solutions each ^0 if and only if p divides the determinant of a cycUc matrix A« A' .. A'- -1 ' A"-- -2 A^ A' A^ .. where A" is itself a cyclic matrix in B^,. . ., B'^~^; etc., until we reach matrices in the coefficients of the congruence. An upper limit is found for '"Sitzungsber. Ak. Wiss. Wien (Math.), 98, Ila, 1889, 652-72. "Math. u. Naturw. Berichte aus Ungarn, 8, 1889-90, 50-59. Math. 68 Term^s Ertesito, 7, 1889, 131-8. »=Sitzungsber. Ak. Wiss. Wien (Math.), 99, Ila, 1890, 799-813. Chap. VIII] NUMBER OF RoOTS OF CONGRUENCES. 229 the number of sets of solutions each not divisible by p. He proved that s EzlL n S ttjXj ^ + S a,+jX,+j-{-h=0 (mod p) y=i i=i has p""^*"^ sets of solutions. Of these, have each x^^O, where r is the number of the 2' integers 6 =1=01 ±02='= ...=*= a, which are divisible by p. The number of sets of solutions of s Pui n l^QjXj 2 +i:a,+jXs+j+b=0 (modp) i=i 3=1 is expressed in terms of the functions used for quadratic congruences. *E. Snopek^^ gave a generalization of Konig's criterion for the solva- bility of a congruence modulo p. L. Gegenbauer^^ proved that if the p congruences S Zk^x^-^-'=0 (mod p) (X = 0, 1,. . ., p-1) A: = have in common at least p—p distinct roots not divisible by p then all p-rowed determinants in the matrix (^^^x) are divisible by p. The converse is proved when a certain condition holds. By specialization, Konig's theorem is obtained. Gegenbauer^^ proved that, if r is less than the prime p and ii Zq,. . ., 2r_i are incongruent and not divisible by p, the system of linear congruences (4) s' h+,y,^0 (mod p) (p = 0, 1,. . ., p-2) has all its sets of solutions of the form (5) 2/*^'sa,2,* (A: = 0, l,...,p-2) x=o or not, according as the matrix (bk+p), k = r, r+1, . . ., p — 2; p = 0, . . ., p — 2, has a p— r — 1 rowed determinant prime to p or not. Next, if (6) S 6;fcX^=0 (mod p) A:=0 has exactly r distinct roots Zq, . . ., z^-i each not divisible by p, every sys- tem of solutions of (4) is given by (5), and conversely. By combining this theorem of Kronecker's with the former, we obtain Kronecker's form of Konig's theorem. "Prace Mat. Fiz., Warsaw, 4, 1893, 63-70 (in PoUsh). '^Sitzungsber. Ak. Wiss. Wien (Math.), 102, Ila, 1893, 549-64. "Monatshefte Math. Phys., 5, 1894, 230-2. Cf. Gegenbauer." 230 History of the Theory of Numbers. [Chap, viii K. Zsigmondy^^ proved that, if p is a prime, there are exactly ,K«, k) = p- - Q p-' + g) p'-' -...+(- 1)» g) congruences x"+ . . . = (mod p) not having as roots k given distinct num- bers. Also, ^(n, k)=p4^(n-l, k) + {-ir(^^, rPin, k + l)=^|^{n, A:)-^(n-l, k). If n'^k, \J/{n, ^)=p"~''(p — 1)^'. For n = k, \p{n, k) is the number \J/{n) of congruences of degree n with no root. The number with exactly i roots is {'])\l/{n — i). There are {^V)\l/ii—r) distinct matrices (3) of rank i such that Qr-i is the first one of Qq, ai,. . . not divisible by p. K. Zsigmondy^^ considered a function $(/) of a polynomial f{x) such that $ is unaltered when the coefficients of f{x) are increased by integral multiples of the prime p. Let/t^'^(x), i = l,. . ., p'', denote the polynomials of degree k which are distinct modulo p and have unity as the coefficient of x''. It is stated that p" p- r,n-2 i, i' J = 1 where a takes those values 1, 2, . . ., p" for which /°^(x)=0 (mod p) does not have as a root one of the given incongruent numbers ai, . . ., a/, while, in the outer sums on the right, i, i',. . . range over the combinations of 1, . . ., s without repetitions. Zsigmondy^^ had earlier given the preceding formula for the case in which tti,. . ., a, denote 0, 1,. . ., p — \. Then taking <I>(/) = 1, we get the number of congruences of degree n with no root (Zsigmondy^^) . Taking $(/) =/, we see that the sum of the congruences of degree n with no root is = (mod p), aside from specified exceptions. Taking $(/)=co-^, where co is a pth root of unity, and n^p, we see that the system /j;^(x) takes each of the values 1, . . . , p — 1 (mod p) equally often. Zsigmondy^^ proved his^^'^^ earher formulas, obtained for an integral value of X the number of complete sets of residues modulo p into which fall the values of the fH (^) not having prescribed roots, and investigated the system 5„ of the least positive residues modulo p of the left members of all congruences of degree n having no root. In particular, he found how often the system B^ contains each residue, or non-residue, of a gth power. He investigated (pp. 19-36) the number of polynomials in x which take k prescribed residues modulo p for k given values of x. 3«Sitzung8ber. Ak. Wiss. Wien (Math.), 103, Ila, 1894, 135-144. »'Monatshefte Math. Phys., 7, 1896, 192-3. "Jahresbericht d. Deutschen Math. Verein., 4, 1894-5, 109-111. "Monatshefte Math. Phys., 8, 1897, 1-42. Chap. VIIIJ NuMBER OF RoOTS OF CONGRUENCES. 231 L. Gegenbauer'*" proved that (2) has as a root a quadratic residue or non-residue of the prime p if and only if the respective determinant P = \ a^+i+o^+i+x \, N = \ a^+i-a^+i+, \ (i, /i = 0, . . ., tt-I) be divisible by p, where 7r= (p — 1)/2. From this it is proved that (2) has exactly ir—r distinct quadratic residues (or non-residues) of p as roots if and only if P (or N) and its tt — 1— r successive derivatives with respect to a,_i+ap_2 have the factor p, while the derivative of order tt— r is prime to p. These residues satisfy the congruence where K = P or N, while the j^th power of the sign of differentiation repre- sents the ^'th derivative. A second set of conditions is obtained. Con- gruence (2) has exactly tt — I—k distinct quadratic residues as roots if and only if the determinants of type P with now i = 0, . . ., k, k+1 and fi = 0,. . . , K, T, are divisible by p for r = /c+l, . . ., tt — 1; while p is not a factor of the determinant of type P with now i, fx = 0,. . ., k. These residues are the roots of «■ S I a^+i+a^+i+^ I x'~^~^=0 (mod p), T=(C where ^ = 0, . . ., k, and /x = 0, . . ., k — 1, t in the determinants. For non- residues We have only to use the differences of a's in place of sums. S. O. Satunovskij^^ noted that, for a prime modulus p, a congruence of degree n (n<p) has n distinct roots if and only if its discriminant is not divisible by p and Sp+q=S g+i (mod p) ior q = l,. . ., n — 1, where Sk is the sum of the kth powers of the n roots. A. Hurwitz^^ gave an expression for the number A^ of real roots of f{x)=aQ-\-aiX-\- . . . -\-arX''=0 (mod p), where p is a prime. By Fermat's theorem, -1 N=X \l-f{xy-'\ (modp). a;=l Letf{xy-'^ = Co-\-CiX+ .... Then N is determined by Ar+l=Co+Cp_i+C2(,-i)+ . . . (mod p). Letf(xi, X2) be the homogeneous form of f(x). Let A be the number of sets of solutions of f{xi, a:2) = (mod p), regarding {xi, X2) and (x/, X2) as the same solution if Xi=pxi, X2=pX2 (mod p) for an integer p. Then A-l= -0^-1-0^71+2^^:1^1 ao'^o. . Mr^r (mod p), tto ! . . . a^ ! "Sitzungsber. Ak. Wiss. Wien (Math.), 110, Ila, 1901, 140-7. "Kazani Izv. fiz. mat. Obsc. (Math. Soc. Kasan), (2), 12, 1902, No. 3, 33-49. Zap. mat. otd. Obsc, 20, 1902, I-II. "Archiv Math. Phys., (3), 5, 1903, 17-27. 232 History of the Theory of Numbers. [Chap, viii where the summation extends over the sets of solutions ^ of ao+ai+- • .+a, = p — 1, ai + 2a2+. . .+ra^=0 (mod p-1). The right member is an invariant modulo p oi f{xi, x^ with respect to all linear homogeneous transformations on Xi, Xo with integral coefficients whose determinant is not divisible by p. The final sum in the expression for .4 — 1 is congruent to A^+1. If r = 2, p>2, the invariant is congruent to the power (p — 1)/2 of the discriminant a^—^aQa^ of/. *E. Stephan^^ investigated the number of roots of linear congruences and systems of congruences. H. Kuhne^ considered J{x)=x"'-\- . . .-\-a„ with no multiple irreducible factor and with a^ not a multiple of the prime p. For n<m, let ^ = x'*+ . . . + 6„ have arbitrary coefficients. The resultant R{f, g) is zero modulo p if and only if / and g have a common factor modulo p. Thus the number of all ^'s of degree n which have no common factor with / modulo p is p„, where P,.^^\R{J, gYr (mod p':), co = p''-np-l), the summation extending over the p" possible ^'s. He expressed p„ as a sum of binomial coefficients. For any two binary forms 4>, \p of degrees w, n, it is shown that is invariant modulo p" under linear transformations with integral coeffi- cients of determinant prime to p ; Ji is Hurwitz's^" invariant. M. Cipolla^^ used the method of Hurwitz^^ to find the sum of the kth powers of the roots of a congruence, and extended the method to show that the number of common roots of /(x) = 0, g{x) = (mod p), of degrees r, s, is congruent to —XCjKi, where i, j take the values for which 0<i^s{p-l), 0<;^r(p-l), i+j=0 (mod p), the Cs being as with Hurwitz, and similarly g{xy-' = Ko+K^x+.... The number of roots common to n congruences is given by a sum. L. E. Dickson^^ gave a two-fold generalization of Hurwitz's^^ formula for the number of integral roots of f{x) = (mod p) . The first generalization is to the residue modulo p of the number of roots which are rational in a root of an irreducible congruence of a given degree. A further generaliza- tion is obtained by taking the coefficients Oi of f{x) to be elements in the Galois field of order p'* (cf. Galois^^ etc.). Then let N be the number of roots of f{x) = which belong to the Galois field of order P = p"'". Then "Jahresber. Staatsoberrealsch. Steyer, 34, 1903-4, 3-40. "Archiv Math. Phys., (3), 6, 1904, 174-6. «Periodico di Mat., 22, 1907, 36-41. «BuU. Amer. Math. Soc, 14, 1907-8, 313. Chap. VIII] NuMBER OF RoOTS OF CONGRUENCES. 233 N=N* (mod p), where A''*+l is derived from either of Hurwitz's two sums for iV+1 by replacing p by P. The same replacement in Hurwitz's expres- sion for A — 1 leads to the invariant A* — l, where A* is congruent modulo p to the number of distinct sets of solutions in the Galois field of order p""* of the equation /(xi, 0:2) =0. G. Rados^'^ considered the sets of solutions of fix, y) = i\a\i^^ x^'-^+ai'' x''-^+ . . . +05,^^2)2/^"'"'= (mod p) for a prime p. Let Ak denote the matrix of D, in (3), with a^ replaced by ap\ Let C denote the determinant of order (p — iy obtained from D by replacing ak by matrix A^. Then/=0 has a solution other than x^y=0 if and only if C is divisible by p ; it has exactly r sets of solutions other than x=y=0 if and only if C is of rank (p — 1)^— r. To obtain theorems including the possible solution x=i/=0, use cf>{x, y) = S W^ x^-i+af V-2+ . . . +af.,)y''-^-'^Q (mod p), a = k=0 CL2 03 \ap_i+aoai Op-3 Op-2 o,p-\ \ ap_2 Op.i+floO ap_i+ao Oi ap_3 ap_2 / and tt/fc derived from a by replacing a^ by al^\ Let y be the determinant derived from la| by replacing a^ by matrix a^; and by a matrix whose p"^ elements are zeros. Then ^=0 has a set of real solutions if and only if 7=0 (mod p) ; it has r sets of solutions if and only if y is of rank p^—r. *P. B. Schwacha^^ discussed the number of roots of congruences. *G. Rados^^ treated higher congruences. Theory of Higher Congruences, Galois Imaginaries. C. F. Gauss/'' in a posthumous paper, remarked that "the solution of congruences is only a part of a much higher investigation, viz., that of the factorization of functions modulo p. Even when ^(x) = has no real root, ^ may be a product of factors of degrees ^2, each of which could be said to have imaginary roots. If use had been made of a similar freedom which younger mathematicians have permitted themselves, and such imaginary roots had been introduced, the following investigation could be greatly condensed." As the later work of Serret^^ shows, such imaginaries can be *'Arm. Sc. ficole Normale Sup., (3), 27, 1910, 217-231. Math. 6s Term^s firtesito (Report of Hungarian Ac), Budapest, 27, 1909, 255-272. "Ueber die Existenz und Anzahl der Wurzeln der Kongruenz Sc<x* = (mod w), Progr. Wilher- ing, 1911, 30 pp. «Math. 6s. Term6s Ertesito, Budapest, 29, 1911, 810-826. "Werke, 2, 1863, 212-240. Maser's German translation of Gauss' Disq. Arith., etc., 1889, 604-629. 234 History of the Theory of Numbers. [Chap, viii introduced in a way free from any logical objections. Avoiding their use, Gauss began his investigation by showing that two polynomials in xwith integral coefficients have a greatest common divisor modulo p, which can be found by Euclid's process. It is understood throughout that p is a prime (cf. Maser, p. 627). Hence if A and B are relatively prime poly- nomials modulo p, there exist two polynomials P and Q such that PA+QB^l {mod p). " Thus if A has no factor in common with B or C modulo p, we find by mul- tiplying the preceding congruence by C that A has no factor in common with the product BC modulo p. If a polynomial is divisible by A, B, C,. . ., no two of which have a common factor modulo p, it is divisible by their product. A polynomial is called prime modulo p if it has no factor of lower degree modulo p. Any polynomial is either prime or is expressible in a single way as a product of prime polynomials modulo p. The number of distinct polynomials x''+aa:"~^+ . . . modulo p is evidently p". Let (n) of these be prime functions. Then p'^-l^d{d), where d ranges over all the divisors of n (only a fragment of the proof is preserved). It is said to follow easily from this relation that, if n is a product of powers of the distinct primes a, 6, ... , then n(n)=p"-2p"/''+2:p"/''^- .... The rth powers of the roots of an equation P = with integral coefficients are the roots of an equation Pr = of the same degree with integral coeffi- cients. If r is a prime, P^=P (mod r). A prime function P of degree m, other than x itself, divides x' — l for some value of vKp"". If v is the least such integer, j^ is a divisor of p*" — 1. Hence P divides (1) x^"-^-l. The latter is congruent modulo p to the product of the prime functions, other than x, whose degrees are the various divisors of m. If P = x"'—Ax"'~^-{'Bx"'~^— ... is a prime function modulo p, the re- mainders by dividing the sum, the sum of the products by twos, etc., of ^ ^p ~p' ^p"*~^ by P are congruent to A, B, etc., respectively. If V is not divisible by p and if m is the least positive integer for which ^"•=1 (mod v), each prime function dividing x" — ! modulo p is a divisor of (1) and its degree is therefore a divisor of m. Let 6 be a divisor of m, and 5', 8", ... the divisors <d oi 8; let ai be the g. c. d. of v and p^-1, fx' the g. c. d. of V and p*' — 1, . . . and set X' =iilii.' , \" =m/m", • • • • Then the num- ber of prime divisors modulo p of degree 6 of a:" — 1 is iV/5, if A^ is the num- ber of integers <y. which are divisible by no one of X', X", .... A method of finding all prime functions dividing j"— 1 is based on periods of powers of X with exponents < v and prime to v (pp. 620-2). Chap. VIII] HiGHER CONGRUENCES, GaLOIS ImAGINARIES. 235 If X has been expressed as a product of relatively prime factors modulo p, we can express X as a product of a like number of factors mod p" con- gruent to the former factors modulo p. There is a fragment on the case of multiple factors. C. G. J. Jacobi" noted that, if g is a prime 6n — 1, x*+^=l (mod q) has q — 1 imaginary roots a + 6V— 3, where a +36^=1 (modg), besides the roots ±1. E. Galois®^ employed imaginary roots of any irreducible congruence F(a;) = (mod p), where p is a prime. Let i be one imaginary root of this congruence of degree v. Let a be one of the p" — 1 expressions a-\-aii+a2i^-\- . . . +a^-ii''~^ in which the a's are integers <p, not all zero. Since each power of a can be expressed as such a polynomial, we have a" = 1 for some positive integer n. Let n be a minimum. Then 1, a, . . . , a"~^ are distinct. Multiply them by a new polynomial (3 ini; we get n products distinct from each other and from the preceding powers of a. If 2n<p'' — 1, we use a new multiplier, etc. Hence n divides p" — 1, and (2) 0^"-^ = !. [This is known as Galois's generalization of Fermat's theorem.] It follows that there exist primitive roots a such that a^p^l if e<p'' — L Any primi- tive root satisfies a congruence of degree v irreducible modulo p. Every irreducible function F{x) of degree v divides x^''—x modulo p. Since jF(x)[^"=F(xO modulo p, the roots of F{x)=0 are All the roots of x^' = x are polynomials in a certain root ^, which satisfies an irreducible congruence of degree v. To find all irreducible congruences of degree v modulo p, delete from x^'' — x all factors which it has in common with x^^—x, iJL<v. The resulting congruence is the product of the desired ones; the factors may be obtained by the method of Gauss, since each of their roots is expressible in terms of a single root. In practice, we find by trial one irreducible congruence of degree v, and then a primitive root of (2); this is done for p = 7, v = 3. Any congruence of degree n has n real or imaginary roots. To find them, we may assume that there is no multiple root. The integral roots are found from the g. c. d. of F{x) and x^~^ — l. The imaginary roots of the second degree are found from the g. c. d. oi F{x) and x^'~^ — l; etc. V. A. Lebesgue^^ noted that, if p is a prime, the roots of all quadratic "Jour, fiir Math., 2, 1827, 67; Werke, 6, 235. "Sur la tMorie des nombres, Bulletin des Sciences Mathlmatiques de M. Ferussac, 13, 1830, 428. Reprinted in Jour, de Math^matiques, 11, 1846, 381; Oeuvres Math. d'Evariste Galois, Paris, 1897, 15-23; Abhand. Alg. Gleich. Abel u. Galois, Maser, 1889, 100. "Jour, de Math^matiques, 4, 1839, 9-12. 236 History of the Theory of Numbers. [Chap, viii congruences modulo p are of the form a-}-h\/n, where n is a fixed quadratic non-residue of p, while a, b are integers. But the cube root of a non-cubic residue is not reducible to this form a-\-hy/n. The p+1 sets of integral solutions of y^ — nz'^=a (mod p) yield the p-|-l real or imaginary roots x^y-[-zy/n of x^^=a (mod p). The latter congruence has primitive roots if = 1. Th. Schonemann^ built a theory of congruences without the use of Euclid's g. c. d. process. He began wath a proof by induction that if a function is irreducible modulo p and divides a product AB modulo p, it divides A or B. ■Much use is made of the concept norm NJ^ of f{x) with respect to <i>{x), i. e., the product /(j8i) . . ./(i3^), where i3i,. . ., jS^ are the roots of 4>{x)=0; the norm is thus essentially the resultant of / and 0. The norm of an irreducible function with respect to a function of lower degree is shown by induction to be not divisible by p. Hence if / is irre- ducible and Nf^=0 (mod p), then/ is a divisor of modulo p. A long dis- cussion shows that if ai, . . . , a„ are the roots of an algebraic equation /(x)=a:"-f . . . =0 and if /(a:) is irreducible modulo p, then niii]z— 0(a,)[ is a power of an irreducible function modulo p. If a is a root of /(x) and f{x) is irreducible modulo p, and if 4>{a) =^(a)-f-pi?(a), we write (p^^/ (mod p, a); then (f>{x)—\p{x) is divisible by J{x) modulo p. If the product of two functions of a is =0 (mod p, a), one of the functions is =0. If /(x) =x'*-f ... is irreducible modulo p and if /(a) =0, then /(a;) = (x-a)(x-a''). . .(x-a''""'), aP"-^=l (mod p, a), n-l P"~' x^ — 1 = n )x— 0,(a)|- (mod p, a), where 4>i is a polynomial of degree n — 1 in a with coefficients chosen from 0, 1,. . ., p — 1, such that not all are zero. There exist (^(p" — 1) primitive roots moduhs p, a, i. e., functions of a belonging to the exponent p" — 1. Let F{x) be irreducible modulis p, a, i. e., have no divisor of degree ^ 1 modulis p, a. Let F{^) = 0, algebraically. Two functions of ^ with coeffi- cients involving a are called congruent modulis p, a, j3 if their difference is the product of p by a polynomial in a, /3. It is proved that F{x)^ix-^){x-^n . ..{x-^^'"'-'"'), /3^'""^1 (mod p, a, ^). If v<n, n being the degree of f(x), and if the function whose roots are the (p"— l)th powers of the roots of /(x) is ^0 (mod p) for x = l, then /(a;) is irreducible modulo p. Hence if ??? is a divisor of p — 1 and if g^ is a primitive root of p, and if k is prime to m, then x"* — ^* is irreducible modulo p. If p<m, m being the degree of F{x), and if the function whose roots are the (p*^— l)th powers of the roots of F{x) is ^0 (mod p, a) for x = l, then "Grundziige einer allgemeinen Theorie der hohem Congruenzen, deren Modul eine reelle Primzahl ist, Progr., Brandenburg, 1844. Same in Jour, fiir Math., 31, 1846, 269-325. Chap. VIII] HiGHER CONGRUENCES, GaLOIS ImAGINARIES. 237 F{x) is irreducible modulis p, a. Hence if m is a divisor of p"— 1, and if ^(a) is a primitive root of a;^"-^=l (mod p, a), and if k is prime to m, then x^ — g^ is irreducible modulis p, a. If F(x, a) is irreducible modulis p, a, and if at least one coefficient satisfies ^p'-i^j (mod p, a) if and only if j' is a multiple of n, then 1^(0^)= n F(a:, a^) (mod p, a) y=o has integral coefficients and is irreducible modulo p. If G{x) is of degree mn and is irreducible modulo p, and G(a) = 0, alge- braically, and if ^(a) is a primitive root of a;^'"''=l (mod p, a), then X(a:)^n (x-F), < = r^ 6 = ^-^, y=o p — 1 has integral coefficients and is irreducible modulo p. The last two theorems enable us to prove the existence of irreducible congruences modulo p of any degree. First, (x'"-"-'-l)/{x'"'"'-'-l) is the product of the irreducible functions of degree p" modulo p. To prove the existence of an irreducible function of degree Zp", where I is any integer prime to p, assume that there exists an irreducible function of each degree <Zp", and hence for the degree a = ylp", where A=(f){l)<l. Let a be a root of the latter, and r a primitive root of x^~^= 1 (mod p, a), where P = p'^. Since I divides P — 1 by Euler's generalization of Fermat's theorem, x^ — r is irreducible modulis p, a. Hence by the theorem preceding the last, JI]Zq{x^ —r'^) is irreducible modulo p. Since its degree is Ip^'A, the last theorem gives an irreducible congruence of degree Zp". Every irreducible factor modulo p of x^"~^ — 1 is of degree a divisor of n. Conversely, every irreducible function of degree a divisor of n is a factor of that binomial. If n is a prime, the number of irreducible functions modulo p of degree n" is (p"'— p""^" )/n\ If n is a product of powers of distinct primes A, B,. . ., say four, the number of irreducible congruences of degree n modulo p is _ipABCD_pABC_ _pBCD\pABi A.pCD_pA_ _pD] Tv where p = p"/(^sc'Z)) Replacing p by p"*, we get the number of irreducible congruences of degree n modulis p, a, where a is a root of an irreducible congruence of degree m. If n is a prime and p belongs to the exponent e modulo n,/= {x'' — \)/{x — \) is congruent modulo p to the product of (n — l)/e irreducible functions of 238 History of the Theory of Numbers. [Chap, viii degree e modulo p. Hence if p is a primitive root of n, / is irreducible modulo p, and therefore with respect to each of the infinitude of primes p+7«. Thus/ is algebraically irreducible. Schonemann^^ considered congruences modulo p"*. If g{x) is not divis- ible by p, and/=x''+ ... is irreducible modulo p"* and \i A{x) is not divisible by/ modulo p, then/g'=A5 (mod p'") implies that B{x) is divisible by/ modulo p"*. If /=/i, ^=6^1 (mod p) and the leading coefficients of the four functions are unity, while / and g have no common factor modulo p, then /f/^/i^iCinod p'") impHes /=/i, g=g\ (mod p"*). He proved the final theorem of Gauss. ^° Next, {x—aY-\-'pF{x) is irreducible modulo p^ if and only if F{a)^0 (mod p) ; an example is ^ = {x-ir-'+pF{x), F(l) = l. Henceforth, let/(a;) be irreducible modulo p and of degree n. If f{xY-\-'pF{x) is reducible modulo p", then (p. 101) /(x) is a factor of F(a;) modulo p. If /(a) = and g(a) ^0 (mod p, a), then g'^ 1 (mod p'", a), where e = p'"~Hp''- 1). If the roots of G{z) are the (p'"~^)th powers of the roots of /(x), then G{z)^{z-^){z-n...{z-^'''-') (mod p-, a). If M is any integer and if F{x) has the leading coefficient unity, we can find z and w such that {x^—iy is divisible by F(x) modulo M. A. Cauchy^® noted the uniqueness of the factorization of a function f{x) with integral coefficients into irreducible factors modulo p, a prime. An irre- ducible function divides a product only when it divides one factor modulo p. A common divisor of two functions divides their g. c. d. modulo p. Cauchy^^ employed an indeterminate quantity or symbol i and defined f{i) to be not the value of the polynomial f{x) for x = i, but to be a-\-hi if a-\-hx is the remainder obtained by dividing /(a:) by x^+1. In particular, if /(x) is x^+1 itself, we have i^+1 =0. Similarly, if w(x) = is an irreducible congruence modulo p, a prime, let i denote a sjmiboHc root. Then 0(i);/'(t) = O implies either </)(i) = or yp{i) = (mod p). At most n integral functions of i satisfy /(x, i) = (mod p), if the degree of / in a: is nKp. If our co(x) divides x" — 1, but not x"* — 1, m<n, modulo p, where n is not a divisor of p — 1, call i a symbolic primi- tive root of x''=l (mod p). Then rc"-l=(.T-l)(x-i) . . . (x-i"-^) If s is a primitive root of n and if n — l=gh, and p''= 1 (mod n), equals a function of x with integral coefficients, while every factor of x" — 1 modulo p with integral coefficients equals such a product. «Jour. fur Math., 32, 1846, 93-105. "Comptes Rendus Paris, 24, 1847, 1117; Oeuvres, (1). 10, 308-12. "Comptes Rendus Paris, 24, 1847, 1120; Oeuvres, (1), 10, 312-23. Chap. VIII] HiGHER CONGRUENCES, GaLOIS ImAGINARIES. 239 G. Eisenstein^^ stated that if /(a;) = is irreducible modulo p, and a is a root of the equation /(x) = of degree n, and if ao, ai,. . . are any integers, K = ao-\-aia-\-. . . . +a„_ia"~-^ is congruent modulis p, a to one and but one expression 5 = 60/3+61^^+62/3^'+ . . . +6„_i/3^"-\ where the 6's are integers and /5 is a suitably chosen function of a. Hence the p" numbers B form a complete set of residues modulis p, a. If co is a primitive nth root of unity, and if (/)(X) =a+ajV+co2V'+ . . . +cu^"-i^V""\ the product 0(X)0(X') ... is independent of a if X+X'+ ... is divisible by n. Th. Schonemann^^ proved the last statement in case n is not divisible by p. To make K = B, raise it to the powers p, p^,. . ., p"~^ and reduce by j8^"=/3 (mod p, a). This system of n congruences determines /3 uniquely if the cyclic determinant of order n with the elements hi is not divisible by p; in the contrary case there may not exist a (3. The statement that the expressions B form p" distinct residues is false if jS is a root of a congruence of degree <n irreducible modulo p; it is true if /3 is a root of such a con- gruence of degree n and if i8+/3^+ . . . +/3^""'^0 (mod p, a). J. A. Serref^" made use of the g. c. d. process to prove that if an irre- ducible function F{x) divides a product modulo p, a prime, it divides one factor modulo p. Then, following Galois, he introduced an imaginary quantity i verifying the congruence F{i) = (mod p) of degree v>l, but gave no formal justification of their use, such as he gave in his later writings. However, he recognized the interpretation that may be given to results obtained from their use. For example, after proving that any polynomial a{i) with integral coefficients is a root of a^''=a (mod p), he noted that this result, for the case a = i, may be translated into the following theorem, free from the consideration of imaginaries: If F{x) is of degree v, has integral coefficients, and is irreducible modulo p, there exist polynomials f{x) and x(x) with integral coefficients such that x''''-x=f{x)F(x) +px{x). The existence of an irreducible congruence of any given degree and any prime modulus is called the chief theorem of the subject. After remarking that Galois had given no satisfactory proof, Serret gave a simple and ingeni- ous argument; but as he made use of imaginary roots of congruences without giving an adequate basis to their theory, the proof is not conclusive. 'sjour. fur Math., 39, 1850, 182. «'Jour. fiir Math., 40, 1850, 185-7. "Cours d'algdbre sup6rieure, ed. 2, Paris, 1854, 343-370. 240 History of the Theory of Numbers. [Chap, viii R. Dedekind"^ developed the subject of higher congruences by the methods of elementary number theory without the use of algebraic prin- ciples. As by Gauss^° he developed the theory' of the g. c. d. of functions modulo p, a prime, and their unique factorization into prime (or irreducible) functions, apart from integral factors. Two functions A and B are called congruent modulis p, M, \i A—B is divisible by the function AI modulo p. We may add or multiply such congruences. If the g. c. d. of A and B is of degree d, Aij=B (mod p, M) has p'^ incongruent roots y{x) modulis p, M. Let <t){M) denote the number of functions which are prime to M modulo p and are incongruent modulis p, M. Let ii be the degree of M. A pri- mary function of degree a is one in which the coefficient of a:" is = 1 (mod p). If D ranges over the incongruent primary divisors of M, then 20(Z))=p''. If M and A^ are relatively prime modulo p, then 4){MN) =4){M)<t){N). If A is a prime function of degree a, 0(A'') =p*'(l -~ Vp")- If Af is a product of powers of incongruent primary prime functions a,. . ., p, *W=p-(i-l)...(i-l). If F is prime to M modulo p, F'^^-^^^= 1 (mod p, M), which is the generaliza- tion of Fermat's theorem. Hence if A is prime to M, the above Unear con- gruence has the solution y=BA'^~^. If P is a prime function of degree tt, a congruence of degree n modulis p, P has at most n incongruent roots. Also (3) y''-'-l^Il{y-F)^(mod p, P), identically in y, where F ranges over a complete set of functions incongruent moduhs p, P and not divisible by P. In particular, l+nF=0 (mod p, P), the generalization of Wilson's theorem. There are (/)(p'— 1) primitive roots modulis p, P. Hence we may em- ploy indices in the usual manner, and obtain the condition for solutions of ?/"=A (mod p, P), where A is not divisible by P. In particular, A is a quadratic residue or non-residue of P according as ^(p'-i)/2^_^^ or -1 (mod p, P). His extension of the quadratic reciprocity law will be cited under that topic. A function A belongs to the exponent p with respect to the prime func- tion P of degree tt if p is the least positive integer for which A^''=A (mod p, P). Evidently p is a divisor of tt. Let N{p) be the number of incon- gruent functions which belong to an exponent p which divides w. Then p^='ZN'{d), where d ranges over the divisors of p. By the principle of inversion (Ch. XIX), isr(p) =p''-2p''/''+2:p''/''*-2p''/'''^+ . . ., where a, 6, . . . are the distinct primes di\'iding p. Since the quotient of this sum by its last term is not divisible by p, we have A^(p)>0. "Jour, fiir Math., 54, 1857, 1-26. ■^ Chap. VIII] HiGHER CONGRUENCES, GaLOIS ImAGINARIES. 241 The product of the incongruent primary prime functions modulo p whose degree divides tt is congruent modulo p to Then, if xpip) is the number of primary prime functions of any degree p, l!id\l/{d)=p', where the summation extends over all divisors d of tt. A com- parison of this with XN{d)=p'' above shows that N{p)=p\l/{p). Another proof is based on the fact that (y-A){y-A')...{y-A''-') is congruent modulis p, P to a polynomial in y with integral coefficients which is a prime function. Moreover, if in (3) we associate the linear factors in which the F's belong to the same exponent, we obtain a factor of the left member which is irreducible modulo p. The product of the incongruent primary prime functions of degree m (m being divisible by no primes other than a, 6, . . . ) is congruent modulo p to \m\-'n.\m/ab\ . . . Il\m/a\'Il\m/abc\ H. J. S. Smith^^ gave an exposition of the theory. E. Mathieu,^^ in his famous paper on multiply transitive groups, gave without proof the factorization (p. 301; for m = l, p. 275) h{z^"'''-z)=u\(hzy"'^''-''+ihzy'"^''-''+ . ..+(hzy"'+hz+a}, a where a ranges over the roots of a^'^^a, while /i^"*"=/i; and (p. 302; for m = l,p. 280) /^(gp'"" - z) =n(;i^ V" -hz-^), where jS ranges over the roots of If 12 is a root of a congruence of degree n whose coefficients are roots of z'^"'=z and whose first member is prime to z^'^ — z, then (p. 303) all the roots of z^""'=z are given by ^o+^i^+- • .+^„-il2''~\ where the A's satisfy Z^ =Z. J. A. Serret,'^^ in contrast to his^° earher exposition, here avoided at the outset the use of Galois imaginaries. An irreducible function of degree V modulo p divides x^—x modulo p if and only if v divides ^t. A simple "British Assoc. Reports, 1860, 120, §§69-71; Coll. M. Papers, 1, 149-155. "Jour, de Math6matiques, (2), 6, 1861, 241-323. "M6m. Ac. Sc. de I'Institut de France, 35, 1866, 617-688. Same in Cours d'algfebre sup6- rieure, ed. 4, vol. 2, 1879, 122-189; ed. 5, 1885. 242 History of the Theory of Numbers. [Chap, viii proof is given for Dedekind's^^ final theorem on the product of all irreducible functions of degree m modulo p. A function F{x) of degree v, irreducible modulo p, is said to belong to the exponent n if n is the least positive integer such that x" — 1 is divisible by F{x) modulo p. Then n is a divisor of p" — 1, and a proper divisor of it, since it does not divide p" — 1 for }x<v. Let n be a product of powers of the distinct primes a, b,. . .. Then the product of all functions of degree v, irreducible modulo p, which belong to an exponent n which is a proper divisor of p" — 1, is congruent modulo p to n(a:"/"-l)-n(x"/"'"^-l)... and their number is therefore (f>{n)/v. By a skillful analysis, Serret obtained theorems of practical importance for the determination of irreducible congruences of given degrees. If we know the N irreducible functions of degree /jl modulo p, which belong to the exponent 1= (p" — l)/<i, then if we replace x by x^, where X is prime to d and has no prime factor different from those which divide p" — 1, we obtain the N irreducible functions of degree Xfi which belong to the exponent \l, exception being made of the case when p is of the form 4ih — I, fi is odd, and X is divisible by 4. In this exceptional case, we may set p = 2H — l, i'^2, t odd; X = 2^s, j^2, s odd. Let k be the least of i, j. Then if we know the A^/2^~^ irreducible functions of odd degree ju modulo p which belong to the exponent I and if we replace x by x^, where X is of the form indicated, is prime to d and contains only primes dividing p" — 1, we obtain N/2^~^ functions of degree Xju each decomposable into 2^~'^ irreducible factors, thus giving A'' irreducible functions of degree \fx/2''~^ which belong to the exponent XL Apply these theorems to x — g% which belongs to the exponent (p — l)/d if ^ is a primitive root of p and if d is the g. c. d. of e and p — 1 ; we see that x^—g^ is irreducible unless the exceptional case arises, and is then a product of 2^"~^ irreducible functions. In that case, irreducible trinomials of degree X are found by decomposing x" —g% where i' = 2'~^X. If a is not divisible by p, a:'' — x — a is irreducible modulo p. There is a development of Dedekind's theory of functions modulis p and F(x), where F{x) is irreducible modulo p. Finally, that theory is considered from the point of view of Galois. Just as in the theory of congruences of integers modulo p we treat all multiples of p as if they were zero, so in congruences in the unknown X, (?(X, a:) = (mod p, F(a:)), we operate as if all multiples of F{x) vanish. There is here an indeter- minate X which we can make use of to cause the multiples of F{x) to vanish if we agree that this indeterminate x is an imaginary root i of the irreducible congruence F(a:) = (mod p). From the theorems of the theory of func- tions modulis p, F{x), we may read off briefer theorems involving i (cf. Galois^2)^ Chap. VIII] HiGHER CONGRUENCES, GaLOIS ImAGINARIES. 243 Harald Schiitz^^ considered a congruence Z'^+aiX"-i+ . . . +a„=0 (mod Mix)) in which the a's and the coefficients of M are any complex integers (cf. Cauchy,®^ for real coefficients). Let ai,..., a„ be the roots of the corresponding algebraic equation. Let M = have the distinct roots III,. . ., jirn- Then the congruence has n"* distinct roots. For, let X — a^ =fi{x) have the factor x—jii, for i = 1, . . ., m. Taking i>l, we have fi{^)=fl{^)+0.p-(lpi' Set X = 111. Then the right member must vanish. Using these and /i (/^i) = , we have m independent linear relations for the coefficients of /i(x). C. Jordan^® followed Galois in employing from the outset a symbol for an imaginary root of an irreducible congruence, proved the theorems of Galois, and that, if j, ji, . . . are roots of irreducible congruences of degrees p", q^,. . . where p, q, . . . are distinct primes, their product jji ... is a root of an irreducible congruence of degree p^q^ .... A. E. Pellet^^ stated that, if t is a root of an irreducible congruence of degree v modulo p, a prime, the number of irreducible congruences of degree Vi whose coefficients are polynomials in i is — jp""! — Sp'"'i/5i+Sp'"'i/9i«2— ...+( — l)"'p''V9i- ■ -Sm } if qi,..., qm are the distinct primes dividing vi. Of these congruences, 4){n)/vi belong to the exponent n if n is a proper divisor of (p")"' — L Any irreducible function of degree ix modulo p with integral coefficients is a product of 5 irreducible factors of degree ix/b with coefficients rational in i, where b is the g. c. d. of fx, v. In an irreducible function of degree vi and belonging to the exponent n and having as coefficients rational functions of i, replace x by x^, where X contains only prime factors dividing n; the resulting function is a product of 2^~^D/n irreducible functions of degree \nvi/{2^~^D) belonging to the exponent \n, where D is the g. c. d. of \n and p""* — 1, and 2^~^ is the highest power of 2 dividing the numerators of each of the fractions (p'"''+l)/2 and Xn/(2Z)) when reduced to their lowest terms. Let gf be a rational function of i, and m the number of distinct values among g, g^, g^ ,. . .. If neither g-{-g^-{- . . . +9'^"*" nor v/m is divisible by p, then x^ — x — g is irreducible; in the contrary case it is a product of linear functions. Hence if we replace a; by x^ — x in an irreducible function of degree /x having as coefficients rational functions of i, we get a new irreducible function provided the coefficient of x''~^ in the given function is not zero. ^^Untersuchungen liber Functionale Congruenzen, Diss. Gottingen, Frankfurt, 1867. ^«Trait^ des substitutions, 1870, 14-18. "Comptes Rendus Paris, 70, 1870, 328-330. 244 History of the Theory of Numbers. [Chap, viii [Proof in Pellet.®^] In particular, if p is a primitive root of a prime n, we have the irreducible function, modulo p, (xP-a;)"-l x^'-x-l C. Jordan^^ listed irreducible functions [errata, Dickson,^"'^ p. 44]. J. A. Serret'^^ determined the product F„ of all functions of degree p" irreducible modulo p, a prime. In the expansion of (^ — 1)" replace each power ^^" by x" ; denote the resulting polynomial in x by X^. Then X.,rn^\{^-irl=ie"'-^y, X.m^X^^'^-X (mod p). Hence Vn = Xpr^/Xpn-i. Moreover, X,+i = (^-l)''+^ = $a-ir-(^-l)''^Z/-X, (modp). Multiply this by the relations obtained by replacing /ibyju+1,- • .,iJi+v — l. Thus X,+.^Z,(X/-i-l)(X,^;J -1) . ..{X:+l,-l) (mod p). Take /i = p"~\ /x+i' = p". Hence F„^' "ff" A (mod p), A = Xjnii+x-i-l. X=l Each /x decomposes into p — 1 factors X—g where ^ = 1,..., p — 1. The irreducible functions of degree p" whose product is A are said to belong to the Xth class. When x is replaced by x^—x, X^ is replaced by X^+i since ^' is replaced by ^'(^ — 1) and hence (^ — 1)" by (^ — l)""*"^; thus A is replaced by A+i) while the last factor in F„=nA is replaced by Xpn —1, which is the first factor in Vn+i- Hence if F{x) is of degree p" and is irreducible modulo p and belongs to the Xth class, F{x^—x) is irreducible or the product of p irreducible functions of degree p" according as X= or <p'*— p"""\ For n = l, the irreducible functions of the Xth class have as roots poly- nomials of degree X in a root of i^ — i=l, which is irreducible modulo p. Hence if we eliminate i between the latter and f{i) = x, where f{i) is the general polynomial of degree X in i, we obtain the general irreducible function of degree p of the Xth class. For any n, the determination of the irreducible functions of degree p" of the first class is made to depend upon a problem of elimination (Algebre, p. 205) and the relation to these of the functions of the Xth class, X>1, is investigated. G. Bellavitis'^" tabulated the indices of Galois imaginaries of order 2 for each prime modulus p = 4n+3^63. Th. Pepin^" proved that x^ — ny^=l (mod p) has p + 1 sets of solutions 'Kllomptes Rendus Paris, 72, 1871, 283-290. "Jour, de Mathdmatiques, (2), 18, 1873, 301^, 437-451. Same as in Cours d'alg^bre sup^rieure, ed. 4, vol. 2, 1879, 190-211. "f-Atti Accad. Lincei, Mem. Sc. Fis. Mat., (3), 1, 1876-7, 778-800. ««Atti Accad. Pont. Nuovi Lincei, 31, 1877-8, 43-52. Chap. VIII] HiGHER CONGRUENCES, GaLOIS ImAGINARIES. 245 X, y selected from 0, 1, . . . , p — 1, provided n is a quadratic non-residue of the prime p. Then x-\-y\/n is a root of p+^=l (mod p), which therefore has p + 1 complex roots, all a power of one root. There is a table of indices for these roots when p = 29 and p = 41. [Lebesgue.^^] A. E. Pellet^^ considered the product A of the squares of the differences of the roots of a congruence /(x) = (mod p) having no equal roots. Then A is a quadratic non-residue of p if f{x) has an odd number of irreducible factors of even degree, a quadratic residue if f{x) has no irreducible factor of even degree or has an even number of them. For, if 5i, . . ., 5^ are the values of A for the various irreducible factors of f{x), then A=a^di. . .5j (mod p), where a is an integer. Hence it suffices to consider an irreducible congruence /(a:) = (mod p). Let v be its degree and i a root. In v-i i-i !/=n n {x'' -x^) 1=1 A; = replace x by the v roots; we get two distinct values if v is even, one if ;^ is odd. In the respective cases, i/^=A (mod p) is irreducible or reducible. R. Dedekind^^ noted that, if P(x) is a prime function of degree / modulo p, a prime, a congruence F{x) = (mod p, P) is equivalent to the congruence F(a) = (mod tt), where tt is a prime ideal factor of p of norm p^, and a is a root of P(a) = (mod tt). A. E. Pellet^^ denoted by/(a;)=0 the equation of degree ^(A;) having as its roots the primitive A;th roots of unity, and by /i(i/) =0 the equation derived by setting y = x-\-\/x. If p is a prime not dividing k, f{x) is con- gruent modulo p to a product of <f>{k)/v irreducible factors whose degree v is the least integer for which p" — ! is divisible by k. li fi{y) = (mod p) has an integral root a,f(x) is divisible modulo p by x^ — 2ax-\-l. Either the latter has two real roots and f{x) and fi{y) have all their roots real and p — 1 is divisible by k, or it is irreducible and f(x) is a product of quadratic factors modulo p and the roots oifi{y) are all real and p+1 is divisible by k. If k divides neither p+l nor p — l,fi{y) is a product of factors of equal degree modulo p. [Cf. Sylvester,^^ etc., Ch. XVI.] Let A; be a divisor f^ 2 of p + L Let X be an odd number divisible by no prime not a factor of k, and relatively prime to{p+l)/k. Then x^^ — 2ax^ + 1 is irreducible modulo p [Serret,^^ No. 355]. Also, if h is not divisible by p F={x-\-by''-2aix^-b^)^+ix-by^ is irreducible modulo p; replacing x^ by y, we obtain a function of degree X irreducible modulo p. If /c is a divisor ?^2 of p — 1 and if X is odd, prime to {p — l)/k and divisible by no prime not a factor of k, F decomposes modulo p into two irreducible functions of degree X. The function /(x^) is either irreducible or the product of two irreducible factors of degree v. In the respective cases, the product A of the squares of "Comptes Rendus Paris, 86, 1878, 1071-2. "Abhand. K. Gesell. Wiss. Gottingen, 23, 1878, p. 25. Dirichlet-Dedekind, Zahlentheorie, ed. 4, 1894, 571-2. s^Comptes Rendus Paris, 90, 1880, 1339-41. 246 History of the Theory of Numbers. [Chap, viii the differences of the roots of /(x") = is a quadratic non-residue or residue of p [Pellet^^j. Let Ai be the like product for J{x). Then A = (-l)'2-' /(0)Ai". Hence /(ax- +6) is irreducible if ( — l)7(6)/a'' is a quadratic non- residue and then /(ax '+6) is irreducible modulo p for every i and even v. 0. H. Mitchell^ gave analogues of Fermat's and Wilson's theorems moduhs p (a prime) and a function of x. A. E. Pellet^ considered the exponent n to which belongs the product P of the roots of a congruence F(x) = of degree v irreducible modulo p. If g is a prime factor of n, F(x') is irreducible or the product of q irreducible factors of degree v modulo p according as q is not or is a divisor of {p — \)/n. In particular, F{x^) is irreducible modulo p if, for v even, X contains only- prime factors of n not dividing {p — \)/n; for v odd, we can use the factor 2 in X only once if p = 4mH-l. Let i be a root of F(x) = 0, I'l a root of an irreducible congruence Fi(x) = (mod p) of degree v^ prime to v. Then ill is a root of an irreducible congruence G(x) = (mod p) of degree vvx. F{x) belongs to the exponent Nn modulo p, where n is prime to (p'— 1) -^\{p — \)N\. Let qi be a prime factor of .V not dividing p — 1. Then G(x'') is irreducible or decomposes into qi irreducible factors of degree vvi according as qi is not or is a divisor of (p'' — l)/N. Thus G(x^) is irreducible if X contains onlj' prime factors of N dividing neither p — 1 nor {p'' — l)/N. 0. H. MitchelP^ defined the prime totient of /(x) to mean the number of polynomials in x, incongruent modulo p, of degree less than the degree of (x) and having no factor in conmion with / modulo p. Those which contain S, but no prime factor of / not contained in S, are called >S-totitives of/. C. Dina^^ proved known results on congruences moduhs p and F{x). A. E. Pellet^^ proved that, if ju distinct values are obtained from a rational function of x with integral coefficients by replacing x successively by the 77i roots of an irreducible congruence modulo p, then ^i is a divisor of m and these /jl values are the roots of an irreducible congruence. Thus if A is a rational function of any number of roots of congruences irreducible modulo p, and p is the number of distinct values among A, A^, A^\. . ., these values satisfy an irreducible congruence modulo p. If A belongs to the exponent n modulo p, then v is the least positive integer for which p''= 1 (mod n). He proved a result of Serret's^^ stated in the following form: If, in an irreducible function F{x) modulo p of degree v and exponent n, x is replaced by x^, where X contains only primes dividing n, then F(x^) is a product of irreducible factors of degree vq and exponent n\, where q is the least integer for which ^"^=1 (mod n\). He proved the first theorem of Pellet^ and the last one of Pellet." "Johns Hopkins University Circulars, 1, 1880-1, 132. "Comptps Rendus Paris, 93, 1881, 1065-6. Cf. Pellet." "Amer. Jour. Math., 4, 1881, 25-38. •^Giomale di Mat., 21, 1883, 234-263. For comments on 263-9, see the chapter on quadratic reciprocity law. ««Bull. Soc. Math. France, 17, 1888-9, 156-167. Chap. VIII] HiGHER CONGRUENCES, GaLOIS ImAGINARIES. 247 E. H. Moore^^ stated that every finite field (Korper) is, apart from nota- tions, a Galois field composed of the p" polynomials in a root of an irreducible congruence of degree n modulo p, a prime. E. H. Moore^° proved the last theorem and others on finite fields. K. Zsigmondy^^ noted that the number of congruences of degree n modulo p, having no irreducible factor of degree i, is P"-({)p"-^+(2)p"-'' where / is the number of functions of degree i irreducible modulo p. G. Cordone^^ noted that if a function is prime to each of its derivatives with respect to each prime modulus Pi,..., Pn and is irreducible modulo M = piK . .pn", it is irreducible with respect to at least one of Pi,..., Pn- If F(x) is not identically =0 modulo pi, nor modulo p2, etc., and if it divides a product modulo M and is prime to one factor according to each modulus Pi,. . ., Pn, then F(x) divides the other factor modulo M. Let F{x) be a function of degree r irreducible with respect to each prime Pi, . . ., Pn, while /(x) is not divisible by F{x) with respect to any one of the p's, then (pp. 281-8) l/(x)[-^^^^^l (mod M, F(x)), 0,(M)=M'-(l-^,y . .(l-^), <}>r{M) being the number of functions Cix''~^+ . . . -\-Cr, in which the c's take such values 0, 1,. . ., M — 1 whose g. c. d. is prime to M. Let A be the product of these reduced functions modulis M, F{x). Then (pp. 316-8), A= — 1 (mod M, F) if M = p^, 2p^ or 4, where p is an odd prime, while A= + l in all other cases. Borel and Drach^^ gave an exposition of the theory of Galois imaginaries from the standpoint of Galois himself. H. Weber^^ considered the finite field (Congruenz Korper) formed of the p" classes of residues modulo p of the polynomials, with integral coefficients, in a root of an irreducible equation of degree n. He proved the generaliza- tion of Fermat's theorem, the existence of primitive roots, and the fact that every element is a square or a sum of the squares of two elements. Ivar Damm^* gave known facts about the roots of congruences modulis p, /(x), where /(x) is irreducible modulo p, without exhibiting the second modulus and without making it clear that it is not a question of ordinary congruences modulo p. Let e be a fixed primitive root of the prime p. Then the roots of every irreducible quadratic congruence are of the form a± hoi, where co^ = e. Let k^^^ = e, ki = k^. 89Bull. New York Math. Soc, 3, 1893-4, 73-8. •"Math. Papers Chicago Congress of 1893, 1896, 208-226; University of Chicago Decennial Publications, (1), 9, 1904, 7-19. "El Progreso Matemdtico, 4, 1894, 265-9. "Introd. th^orie des nombres, 1895, 42-50, 58-62, 343-350. •'Lehrbuch der Algebra, II, 1896, 242-50, 259-261; ed. 2, 1899, 302-10, 320-2. "Bidrag till Laran om Kongruenser med Primtalsmodyl, Diss., Upsala, 1896, 86 pp. 248 History of the Theory of Numbers. [Chap, viii Analogous to the definition of trigonometric functions in terms of expo- nentials, he defined quasi cosines and sines by and Tqx as their quotient. Their relations are discussed. He defined pseudo cosines and sines by Cpx = Cq[{p - l)x] = e-'Cq2x, Spx = -e-'Sq2x. For each prime p<100, he gave (pp. 65-86) the (integral) values of e^, ind x, Cqx, Sqx, Tqx, Cpx, Spx for x=l, 2,. . ., p + 1. L. E. Dickson^^ extended the results of Serret^* to the more general case in which the coefficients of the functions are poljTiomials in a given Galois imaginary (i. e., are in a Galois field of order p"). For the corresponding generaUzation of the results of Serret^^ on irreducible congruences modulo p of degree a power of p, additional developments were necessary. To obtain the irreducible functions of degree p in the GFlp'^'] which are of the first class, we need the complete factorization, in the field, hiz^'-z-v) =U{h''z''-hz-^) where hv is an integer and /S ranges over the roots of all of whose roots are in the field. For the case v = this factorization is due to ]Mathieu." Thus K^z^—hz—^ is irreducible in the field if and only if B^O. In particular, if /3 is an integer not divisible by p, z^ — z—^ is irreducible in the GF[p"] if and only if n is not divisible by p. R. Le Vavasseur^^ employed Galois imaginaries to express in brief no- tation the groups of isomorphisms of certain tj-pes of groups, for example, that of the abelian group G generated by n independent operators ai, . . ., a„, each of period a prime p. If i is a root of an irreducible congruence of degree n modulo p, and if j = ai-\-ia2+ . . .+i''~^a„, he defined a^ to be Oi"' . . . an"". Then the operators of G are represented by the real and imaginary powers of a. A. Guldberg^^ considered linear differential forms A d^y , , dy , ^y=^>:d^.-^----^^^di+^oy, ^4th integral coefficients. The product of two such forms is defined by Boole's sjTnbolic method to be d'' 6} d Ay'By={au-^-\- • • • +«o)(^/^+ • • • +^^+^o)2/. "BuU. Amer. Math. Soc.,3, 1896-7, 381-9. "M^m. Ac. Sc. Toulouse, (9), 9, 1897, 247-256. "^Comptes Rendus Paris, 125, 1897, 489. Chap. VIII] HiGHER CONGRUENCES, GaLOIS ImAGINARIES. 249 If the product is =Cy (mod p), Ay and By are called divisors modulo p of Cy. Let A?/ be of order n and irreducible modulo p. Then Ay is con- gruent modulis p, A?/ to one and but one of the p" forms (4) s'c,^ (c, = 0, l,...,p-l). If It is any one of these forms (4) and if e = 'p^ — \, Guldberg stated the analogue of Fermat's theorem dfu ^='M (mod p, ^y), but incorrectly gave the right member to be unity [cf. Epsteen/''^, Dickson^"^]. L. Stickelberger^^ considered F{x) =x^+aix''~^-\- . . . with integral coeffi- cients, such that the product D of the squares of the differences of the roots is not zero. Let p be any prime not dividing D. Let v be the number of factors of F{x) which are irreducible modulo p. He proved by the use of prime ideals that (f)=(-i)»-. where the symbol in the left member is that of Legendre [see quadratic residues]. L. E. Dickson^^ proved the existence of the Galois field GF[p''] of order p" by induction from r = n to r = qn, by showing that (a:^"'-a:)/(x^"-x) is a product of factors of degree q belonging to and irreducible in the (?F[p"]. Any such factor defines the GF[p'"']. L. Kronecker^°° treated congruences modulis p, P{x) from the stand- point of modular systems. F. S. Carey ^°^ gave for each prime p< 100 a table of the residues of the first p + 1 powers of a primitive root a-\-hj of z^~^=l (mod p) where /=»' (mod p), V being an integral quadratic non-residue of p. The higher powers are readily derived. While only the single modulus p is exhibited, it is really a question of a double modulus p and x^—v. Methods of "solving" 2?"-!=! are discussed. In particular, for n = 3, there is given a primitive root for each prime p< 100. L. E. Dickson^"^ gave a systematic introductory exposition of the theory, with generalizations and extensions. M. Bauer^"^ proved that, if /(a;) =0 is an irreducible equation with inte- gral coefficients and leading coefficient unity, w a root, D its discriminant, d = D/k^ that of the domain defined by w, p sl prime not dividing k, x>l, «8Verhand. I. Internat. Math. Kongress, 1897, 186. "^BuU. Amer. Math. Soc, 6, 1900, 203-4. looVorlesungen iiber Zahlentheorie, I, 1901, 212-225 (expanded by Hensel, p. 506). i"Proc. London Math. Soc, 33, 1900-1, 294-310. lo'Linear groups with an exposition of the Galois field theory, Leipzig, 1901, pp. 1-71. lo^Math. Naturw. Berichte aus Ungarn, 20, 1902, 39-42; Math. 6s Phyp. Lapok, 10, 1902, 28-33. 250 History of the Theory of Numbers. [Chap, viii then f{x) is congruent modulo p" to a product of Fi(j),. . ., F,(a:), each irreducible modulo p°, such that F,(x)=/,(a:)'« (mod p), where /(x)= 11/. (x)*' (mod p), and /.(x) is irreducible modulo p. There is an example of an irreducible cyclotomic function reducible with respect to every prime power modulus. P. Bachmann^^ gave an exposition of the general theory. G. Arnoux^"^ exhibited in the form of tables the work of finding a primi- tive root of the GF\1^] and of the GF[5^], and tabulated the reducible and irreducible congruences of degrees 1, 2, 3, modulo 5. S. Epsteen^°^ proved the result of Guldberg,^" and developed the theory of residues of hnear differential forms parallel to the theory of finite fields, as presented by Dickson. ^°- L. E. Dickson ^°^ noted that the last mentioned subjects are identical abstractly. Let the irreducible form A?/ be d'^y dy To the element (4) we make correspond the element 2c,z* of the Galois field of order p", where z is a root of the irreducible congruence 5„2"+...+5i2+5o=0 (mod p). Since product relations are preserved by this correspondence, the p" resi- dues (4) define a field abstractly identical with our Galois field. Dickson^"^'' proved that x'"=.t (mod vi = p") has p and only p roots if p is a prime and hence does not define the Galois field of order m as occasionally stated. A. Guldberg^"^'' employed the notation of finite differences and wrote n m n m Fy, = 2 afi%, Gy, = 2 hj9%, Fy^.Gy, = 2 afi\ 2 bfi%, t=0 »=0 t=0 »=0 where 6y^ = yjc+i, ^'?/x = 2/1+2. • • •> symbolically. To these linear forms with integral coefficients taken modulo p, a prime, we may apply EucUd's g. c. d. process and prove that factorization is unique. Next, let 6„, be not di\'isible by p, so that Gy^ is of order m. With respect to the two moduU p, Gy^, a complete set of p"* residues of hnear forms is a„,_i?/:r+m-i+ ■ • • +ao2/x (^1 = 0, 1,..., p-1). Amongst these occur <}>{Gy^) =p"'{l- 1 /p'"') .. .{1-1 /p"''^) forms Fy^. prime to Gyj^ if Wi, . . . , niq are the orders of the irreducible factors of Gyx modulo p, and FyJ'^^'^^-^^y, (mod p, Gy,) In particular, if Gy^ is irreducible and of order m, Fyr-'^yAmodp,Gy,). i i«Niedere Zahlentheorie, 1, 1902, 363-399. »»Assoc. frang. av. sc, 31, 1902, II, 202-227. i«BuU. Amer. Math. Soc, 10, 1903-4, 23-30. '"/bid., pp. 30-1. "'"Amer. Math. Monthly, 11, 1904, 39-40. ^ >»"f'.\iinaU di Mat., (3), 10, 1904, 201-9. ■ Chap. VIII] HIGHER CONGRUENCES, GaLOIS ImAGINARIES. 251 W. H. Bussey^°^ gave for each Galois field of order < 1000 companion tables showing the residues of the successive powers of a primitive root, and the powers corresponding to the residues arranged in a natural order. These tables serve the same purposes in computations with Galois fields that tables of indices serve in computations with integers modulo p", where p is a prime. G. Voronoi^°^ proved the theorem of Stickelberger.^^ Thus, for n = 3, (D/p) = — 1 only when v = 2. Hence a cubic congruence has a single root if (D/p) = —1, and three real roots or none if {D/p) = +1. P. Bachmann^^° developed the general theory from the standpoint of Kronecker's modular systems and considered its relation to ideals (p. 241). M. Bauer^^^ employed a polynomial /(z) of degree n irreducible modulo p, and another one M{z) of degree less than that of f{z) and not divisible by/(2) modulo p. Then if {t, a) = l, the equation /(2)+p«M(2)=0 is irreducible. The case a = 1 is due to Schonemann^^ (p. 101). G. Arnoux,^^^ starting with any prime m and integer n, introduced a symbol i such that i^^^ 1 (mod m) and such that i, i^, . . ., i^ are distinct, where s = m" — 1, without attempting a logical foundation. If /(x) is irre- ducible modulo m and of degree n, there is only a finite number of distinct residues of powers of x modulis/(x), m] let x^ andx'^'^^have the same residue. Thus x^ — 1 is divisible by/(x) modulo m. It is stated (p. 95) without proof that p divides s. "Call a a root of /(a:) = 0. To make a coincide with the primitive root i of a;^= 1, we must take p = s, whence every such primitive root is a root of an irreducible congruence of degree n modulo m." Follow- ing this inadequate basis is an exposition (pp. 117-136) of known properties of Galois imaginaries. L. I. Neikirk^^^ represented geometrically the elements of the Galois field of order p" defined by an irreducible congruence /(x) =rc"+aia:"-^H- . . . -fa„=0 (mod p). Let j be a root of the equation f{x) = and represent Ci/~^+ . . . +c„_ii-Fc„ (c's integers) by a point in the complex plane. The p"" points for which the c's are chosen from 0, 1, . . ., p — 1 represent the elements of the Galois field. G. A. Miller^^^ listed all possible modular systems p, 4>{x), where p is a prime and the coefficient of the highest power of x is unity, in regard to which a complete set of prime residues forms a group of order ^12. If 4>{x) is the product of k distinct irreducible functions 4>i, . . . , (f)^ modulo p, "SBuU. Amer. Math. Soc, 12, 1905, 21-38; 16, 1909-10, 188-206. lO'Verhand. III. Internat. Math. Kongress, 1905, 186-9. ""AUgemeine Arith. d. Zahlenkorper, 1905, 81-111. "iJour. fur Math., 128, 1905, 87-9. "'Arithm^tique Graphique, Fonctiona Arith., 1906, 91-5. "'BuU. Amer. Math. Soc, 14, 1907-8, 323-5. "«Archiv Math. Phys., (3), 15, 1909-10, 115-121. 252 History of the Theory of Numbers. [Chap, viii the residues prime to p, (t>{x) constitute the direct product of the groups with respect to the various p, 0.(a:). Not every abelian group can be repre- sented as a congruence group composed of a complete set of prime residues with respect to Fj, . . ., Fx, where the F's are functions of a single variable. Mildred Sanderson^ ^^ employed two moduU m and P{y), the first being any integer and the second any polynomial in y with integral coefficients. Such a polynomial /(?/) is said to have an inverse /i (y) if //i= 1 (mod m, P). If P{y) is of degree r and is irreducible with respect to each prime factor of m, a function f{y), whose degree is <r, has an inverse moduhs m, P{y), if and only if the g. c. d. of the coefficients of /(?/) is prune to m. For such an/, /" = 1 (mod vi, P), where n is Jordan's function Jr{fn) [Jordan, ^"^ Ch. V]. In case m is a prime, this result becomes Galois'^^ generalization of Fermat's theorem. The product of the n distinct residues having inverses moduHs m, P{y), is congruent to —1 when m is a power of an odd prime or the double of such a power or when r= 1, w = 4; but congruent to + 1 in all other cases — a two-fold generahzation of Wilson's theorem. There exists a polynomial P{y) of degree r which is irreducible with respect to each prime factor of m. Then if A{y), B{y) are of degrees <r and their coefficients are not all divisible by a factor of m, there exist polynomials a(i/), ^{y), such that aA+/3B=l (mod m, P). Several writers^^^ discussed the irreducible quadratic factors modulo p of {x'^—\)/{x^ — l), where A' = 1 or 2, p is a prime, a a divisor of p-fl. G. Tarry^^^ noted that, if f=q (mod m), where 5 is a quadratic non- residue of the prune m, the Galois imaginary a+hj is a primitive root if its norm {a-\-'bj){a — hj) is a primitive root of m and if the ratio a:h and the analogous ratios of the coordinates of the first m powers of a-\-hj are incon- gruent. L. E. Dickson^ ^^ proved that two polynomials in two variables with integral coefficients have a unique g. c. d. modulo p, a prime. Thus the unique factorization theorem holds. G. Tarr>'^^^ stated that Ap is a primitive root of the GF[p^] if the norm of A = a-\-'bj is a primitive root of p and if the imaginar>^ p belongs to the exponent p+1. The 0(p-f 1) numbers p are found by the usual process to obtain the primitive roots of a prime. U. Scarpis^-° proved that an equation of degree v irreducible in the Galois field of order p" has in the field of order p"*" either v roots or no root according as v is or is not a di\dsor of m [Dickson^°^, p. 19, lines 7-9]. Cubic Congruences. A. Cauchy^^° solved y'^-\-By-\-C={) (mod p) when it has three distinct '"Annals of Math., (2), 13, 1911, 36-9. "•L'interm^diaire dea math., 18, 1911, 195, 246; 19, 1912, 61-69, 95-6; 21, 1914, 158-161; 22, 1915, 77-8. Sphinx-Oedipe, 7. 1912, 2-3. "'Assoc, fran^. av. sc, 40, 1911, 12-24. "sBuU. Amer. Math. Soc, (2), 17, 1911, 293-4. "'Sphinx-Oedipe, 7, 1912, 43^, 49-50. ""AnnaU di Mat., (3), 23, 1914, 45. ""Exercices de Math., 4, 1829, 279-292; Oeuvres, (2), 9, 326-333. Chap. VIII] CuBIC CONGRUENCES. 253 integral roots y^, 2/2, Vs, and p is a prime = 1 (mod 3), and B^O (mod p) . Set ^Vi = yi+ry2-\-r%, ^V2 = yi+r^y2+ry3, r^+r+ 1=0 (mod p). The roots of u^-\-Cu—B^/27= (mod p) are Ui = ^i^, U2 = ^2^. After finding Vi from Vi^=Ui (mod p), we get V2= —B/{3vi), and determine the y's from '2yi=0 and the expressions for 3^1, Sv2. Thus 2/i=yi+z;2, y2=r\+rv2, ys^rvi+r^ (mod p). Since by hypothesis the cubic congruence has three distinct integral roots, the quadratic has two distinct integral roots, whence p-l P-l 7^2 ^3 "^"^7 ' +(~^+^7 ' ^2' ^ ' ^^ (modp). Conversely, if the last two conditions are satisfied, the cubic congruence has three distinct real roots provided p=l (mod 3), B^O (mod p). G. Oltramare^^^ found the conditions that one or all of the roots of x^+Spx+2q=0 (mod fx) given by Cardan's formula become integral modulo fx, a prime. Set D = q^-\-p\ a=-q+VD, T=-q-VD, First, let /x be a prime 6n — 1 . If D is a quadratic residue of /x, there is a single rational root — 2g/(p+(r^"+T^"). If Z) is a quadratic non-residue of fx, there are three rational roots or no root according as the rational part M of the development of o-^"~^ by the binomial theorem satisfies or does not satisfy ilfp^+g^O (mod //) ; if also /x = 18m+ll and there are three rational roots, they are 2M^, -^(M±iW-3i)), if (72"*+i = ikf 4-iVVD; with a like result when m = 18m+5. Next, let /x = 6n+l. If Z) is a quadratic non-residue of ^x, there is one rational root or none according as the rational part M of the development of o-^" is or is not such that (2M-l)2(ikf+l)=-2gVp3 (modM), and if a rational root exists it is 2q/ \ p {2M — 1 ) } . If Z) is a quadratic residue of IX, there are three rational roots or none according as cr^''= 1 (mod /x) or not. When there are three, they are given explicitly if ^t=18m-|-7 or 18m + 13, while if // = 18m + l there are sub-cases treated only partially. G. T. Woronoj^^^ (or Voronoi) employed Galois imaginaries a -{-hi, where i^—N=0 (mod p) is irreducible, p being an odd prime, to treat the solution of x^—rx — s=0 (mod p). I'lJour. fiir Math., 45, 1853, 314-339. '"Integral algebraic numbers depending on a root of a cubic equation (in Russian), St. Peters- burg, 1894, Ch. I. Cf. Fortschritte Math., 25, 1893-4, 302-3. Cf. Voronoi."' 254 History of the Theory of Numbers. [Chap, viii If 4r^ — 27s^ is a quadratic non-residue of p, the congruence has one and only one root; but if it is a residue, there are three roots or no root. G. Cordone^^^ gave simpler proofs of Oltramare's^^^ theorem II on the case fjL = Qn — I, gave theorems to replace VII and VIII, and proved that the condition in IX is sufficient as well as necessary. Ivar Damm^^ found when Cardan's formula gives three real roots, one or no real root of a cubic congruence, and expressed the roots by use of his quasi sine and cosine functions. For the prime modulus p = 3nH-l, f=x^-\-ax-\-b is irreducible if c = ^isrea.,(-|+cf.*l. If p = 3n — 1, it is irreducible if c and ( — 6/2+0)" are both imaginary. There are given (p. 52) explicit expressions for h such that / is irreducible. J. Iwanow^^ gave another proof of the theorem of Woronoj.^^^ Woronoj ^^^ gave another proof of the same theorem and stated that the congruence has the same number of roots for all primes representable by a binary quadratic form whose determinant equals — 4r^+27s^. G. Arnoux^^*^ gave double-entry tables of the roots of the congruences x''+6x*+a=0 (mod m), and solved numerical cubic congruences by in- terpreting Cardan's formulas. G. Arnoux^^^ treated x^+6x+a=0 (mod m) by use of Cardan's formula. For m = 1 1 , he gave a table of the real roots for a ^ 10, 6^ 10, and the residues of ^-4+27 When R is a quadratic residue, the cube roots of — a/2=t y/R are found by use of a table for the Galois field of order 11^ defined by r=2 (mod 11), and the cubic is seen to have a real and two imaginary roots involving i. If jR is a quadratic non-residue, there are three real roots or none. Like results are said to hold when m — 1 is not divisible by 3. If w= 1 (mod 3), there is a single real root if 7? is a quadratic non-residue ; three real or three imaginary roots of the third order if ^ is a residue. L. E. Dickson'^^ proved that, if p is a prime >3, x^-\-^x-\-b=0 (mod p) has no integral root if and only if —4/3^ — 276^ is a quadratic residue of p^ say = 81)u^ , and if §( — 6+/xV — 3) is not congruent to the cube of any y+zy/ — 3, where y and z are integers. The reducible and irreducible cubic congruences are given explicitly. Necessary and sufficient conditions for the irreducibility of a quartic congruence are proved. '"Rendiconti Circolo Mat. di Palermo, 9, 1895, 221-36. "^BuU. Ac. Sc. St. Petersburg, 5, 1896, 137-142 (in Russian). '^Natural Sc. (Russian), 10, 1898, 329; of. Fortschritte Math., 29, 1898, 156. '3« Assoc, franc;, av. sc, 30, 1901, II, 31-50, 51-73; corrections, 31, 1902, II, 202. '"Assoc, frang. av. sc, 33, 1904, 199-230 [182-199], and Amoux'", 166-202. '"BuU. Amer. Math. Soc, 13, 1906, 1-8. Chap. VIII] CUBIC CONGRUENCES. 255 D. Mirimanoff^^^ noted that the results by Arnoux"^'^" may be com- bined by use of the discriminant D= —4b^ — 27a^= —3-Q^R in place of R, since — 3 is a quadratic residue of a prime p = Sk-{-l, non-residue ofp = Sk — l, and we obtain the result as stated by Voronoi.-^^^ To find which of the values 1 or 3 is taken by v when D is a quadratic residue, apply the theorem that if /(x) = (mod p) is an irreducible con- gruence of degree n and if Xq is one of its imaginary roots (say one of the roots of the equation f{x) = 0) , the roots are p pn—l Hence a function unaltered by the cycHc substitution (xqXi. . .Xn-i) has an integral value modulo p. Take w = 3, D=d^, a a root 5^1 of 2^=1 (mod p), and let M = (xo+aXi+a^X2)^. If p=l (mod 3), a is an integer, and M is an integer if ^^ = 1, while M is the cube of an integer if v = S. Thus we have Arnoux's criterion :^^^ v = S if ilf or-| ( — 9a+V — 3d) is a cubic residue modulo p. If p= —1 (mod 3) J/ = 3 if and only if ilf^^l (mod p), where k = {p'^ -l)/3. For quartic congruences, we can use (a^o — a:i-|-a:2 — a^s)^. R. D. von Sterneck^"*" noted that if p is a prime >3 not dividing A, and if k = SAC — B^^O (mod p), then the number of incongruent values taken by Ax^+Bx^+Cx-\-D is i{2p+(-3/p)j ; but, if k=0, the number is p if p = Sn — l, (p+2)/3 if p = 3n+l. Generalization by Kantor.^^^ C. Cailler^^^ treated x^+px-\-q=0 (mod I), where I is a prime >3. By the algebraic method leading to Cardan's formula, we write the congruence in the form (1) x^-Sabx+abia+b)^0 (mod /), where a, h are the roots of z^-{-Sqz/p—p/S=0 (mod T), whence z={xQ-\-aXi+ a^X2) V (9p) , a^ + a + 1 = (mod I) . Let A = 4p^+27g^. If 3A is a quadratic residue of i, a and b are distinct and real. If 3A is a non-residue, a and b are Galois imaginaries r±s\/iV^ where N is any non-residue. For a root a: of (1), Use is made of a recurring series S with the scale of relation [a +6, —ab] to get 2/0, Vi,.. .. Write Q = (3A/Z). If ; = 3m-l, Q = l, then If l = 3m-\-l, Q = l, the congruence is possible only when the real number a/b is a cubic residue, i. e., if 2/^=0 in S; let a/b belong to the exponent 3m=f1 modulo I, whence '"L'enseignement math., 9, 1907, 381-4. "oSitzungsber. Ak. Wiss. Wien (Math.), 116, 1907, Ila, 895-904. "'L'enseignement math., 10, 1908, 474r-487. 256 History of the Theory of Numbers. [Chap, viii w= I 7 I J x= or > \o/ 2/2^-1 2/m according as the upper or lower sign holds. If l==Sm-\-l, Q= —1, then 2/3m+2=0, (rl =-, realx=-; . \0/ y2m+l li l = Sm — l, Q= —1, there are three real roots if and only if a/b is a cubic residue of I, viz., 2/^=0; when real, the roots may be found as in the second case. Cailler^^^ noted that a cubic equation X = has its roots expressible rationally in one root and VA, where A is the discriminant (Serret's Algebre, ed. 5, vol. 2, 466-8). Hence, if p is a prime, X=0 (mod p) has three real roots if one, when and only when A is a quadratic residue of p. If p = 9m^l, his^^^ test shows that x^ — 3xH-l = (mod p) has three real roots, but no real root for other prime moduli 5^3. The function F{x) =x^-{-x- — 2x — l for the three periods of the seventh roots of unity is divisible by the primes 7m=*= 1 (then 3 real roots, Gauss®", p. 624) and 7, but by no other primes. E. B. Escott"^ noted that the equation F(x) =0 last mentioned has the roots a, ^ = a^—2, 7=/3^— 2, so that F{x) = (mod p) has three real roots if one real root. To find the most general irreducible cubic equation with roots a, (3, y such that ^=/(a), y=m, a=f{y), we may assume that/(x) is of degree 2. For /(a) = a^—n, we get (2) x^-\-ax'^-{a'-2a-\-3)x-ia^-2a^-\-da-l)=0, with ^ = a^ — c, y=j3^—c, a=y^—c, c = o^ — a+2. The corresponding con- gruence has three real roots if one. To treat/(a)=a^+^a+Z, add k/2 to each root. For the new roots, jS' = a'^ — n, as in the former case. To treat /(a) = ta^-\-ga-\-h, the products of the roots by t satisfy the preceding relation. L. E. Dickson^^ determined the values of a for which the congruence corresponding to (2) has three integral roots. Replace x by z—a; we get z^-2az^+{2a-S)z-\-l = (mod p). If one root is z, the others are 1 — 1/z and 1/(1—2). Evidently a is rational in z. If —3 is a quadratic non-residue of p, there are exactly (p — 2)/S values of a for which the congruence has three distinct integral roots. If — 3 is a residue, the number is (pH-2)/3. A second method, yielding an explicit congruence for these values of a, is a direct application of his^^® general criteria for the nature of the roots of a cubic congruence. T. Hayashi^^^ treated cyclotomic cubic equations with three real roots by use of Escott's^'*^ results. "«L'interm^diaire des math., 16, 1909, 185-7. '"/bid., (2), 12, 1910-11, 149-152. '"Annals of Math., (2), 11, 1909-10, 86-92. >«/6id., 189-192. Chap. VIII] MISCELLANEOUS RESULTS ON CONGRUENCES. 257 Miscellaneous Results on Congruences. Linear congruences will be treated in Vol. 2 under linear diophantine equations, quadratic congruences in two or more variables, under sums of four squares; ax''+hy''-}-cz''=0, under Fermat's last theorem. Fermat^^^ stated that not every prime p divides one of the numbers a+1, a^+l,a^+l,. . .. For, if /c is the least value for which a^' — 1 is divis- ible by p and if k is odd, no term a^-{-l is divisible by p. But if k is even, ^fc/2_|_2 jg divisible by p. Fermat^^^ stated that no prime 12n±l divides S'^+l, every prime 12n=t5 divides certain S'^+l, no prime 10n±l divides 5""+!, every prime lOn^S divides certain 5"^+!, and intimated that he possessed a rule relating to all primes. See Lipschitz.^^*^ A. M. Legendre^^° obtained from a given congruence x"=ax'*~^+- ■ • (mod p), p SiTi odd prime, one having the same roots, but with no double roots. Express x^^"^'^^ in terms of the powers of a; with exponents <n, and equate the result to +1 and to —1 in turn. The g. c. d. of each and the given congruence is the required congruence. An exception arises if the proposed congruence is satisfied by 0, 1, . . ., p — 1. Hoen4 de Wronski^^^ developed {ni-\- . . .-\-nJ"', replaced each multi- nomial coefficient by unity, and denoted the result by A[ni+ . . .+nj\"*. Thus A[ni+n2f = ni^+nin2+n2^. SetiV„ = ni+ . . . +n„. Then (pp.65-9), (1) A[N^-nX-A[N^-nX={n,-n,)A[NT~'=0 (mod n,-n,). Let (ni. . .nS)m be the sum of the products of ni,. . ., n„ taken m at a time. Then (p. 143), if A[Nj' = l, (2) A[NJ={n,.. .nJ,A[N^Y-'-in,.. .nJ^AWJ-' + in,.. .n^)sA[NJ-'- . . .+{-iy+\n,. . .nJ),A[Nj. He discussed (pp. 146-151) in an obscure manner the solution of Xi=X2 (mod X), where the X's are polynomials in ^ of degree v. Set N^ = ni-\- . . . H-n„_2+np+ng. Let the negatives of Ui,..., n„_2, Up be the roots of P = Po-\-PiX+ . . .+P^^2^"~^-\-x"~^ = 0; the negatives of ni,..., n^-2, "riq the roots of Q = Qo+ . . .+x"~^ = 0. We may add fiX and ^2^ to the members of our congruence. It is stated that the new first member may be taken to be A[A^„— nj"", whence by (2) X,-\-^,X = P^_2A[N^-nr-'-P.-3A[N^-nX-^+ . . ., and the A's may be expressed in terms of the P's by (2). Similarly, ^2+^2^ niay be expressed in terms of the Q's. By (1), X = nq—np = Q^_2 — P„_2. Since P = 0, Q = have co — 2 roots in common, we have further conditions on the coefficients Pi, Qi. It is argued that w — 3 of the latter "^Oeuvres, 2, 209, letter to Frenicle, Oct. 18, 1640, i^Oeuvres, 2, 220, letter to Mersenne, June 15, 1641. i"M6m._Ac. Sc. Paris, 1785, 483. "^Introduction a la Philosophic des Math^matiques et Technie de I'Argorithmie, Paris, 1811. He used the Hebrew aleph for the A of this report. Cf. Wronski^^' of Ch. VII. 258 History of the Theory of Numbers. [Chap, viil remain arbitrary, and that ^ is a function of them and one of the n's, which has an arbitrary rational value. A. Cauchy^" noted that if / and F are polynomials in x, Lagrange's interpolation formula leads to polynomials u and v such that uJ-\-vF = R, where i? is a constant [provided / and F have no common factor]. If the coefficients are all integers, R is an integer. Hence R is the greatest of the integers di\-iding both / and F. For /= x^—x, we may express i2 as a prod- uct of trigonometric functions. If also F{x)= (x"+l)/(a:+l), where n and p are primes, R=0 or ±2 according as p is or is not of the form nx-\-\. Hence the latter primes are the only ones dividing x^+l, but not x-\-\. Cauchy^^^ proved that a congruence /(x) = (mod p) of degree m<p is equivalent to (x— r)'</)(x) = 0, where 4> is of degree m—i, if and only if /(r)^0, /'(r) = 0,. . ., r-'\r)^Q (mod p), where p is a prime. The theorem fails if m'^p. He gave the method of Libri (M^moires, I) for solving the problem: Given"/(a:) = (mod p) of degree m^p and with exactly m roots, and/i(x) of degree l^m, to find a polynomial </)(a;), also with integral coeflficients, whose roots are the roots common to/ and /i. He gave the usual theorem on the number of roots of a binomial congruence and noted conditions that a quartic congruence have four roots. Cauchy^^ stated that if 7 is an arbitrary modulus and if ri, . . ., r„, are roots of /(x)=0 (mod 7) such that each difference Vi—Tj is prime to 7, then f{x)={x-ri) . . .(x-rJQ(x) (mod 7). If in addition, m exceeds the degree of /(x), then/(x) = (mod 7) for every x. A congruence of degree n modulo p^, where p is a prime, has at most n roots unless every integer is a root. If /(r) = (mod 7) and if in the irre- ducible fraction equal to _ /(r) the denominator is prime to 7, then r— r7 is a root of /(a:)=0 (mod P). V. A. Lebesgue^^^ wrote a/b=c (mod p) if h is prime to p and a=bc (mod p), and a/b=c/d (mod p) \i h, d are prime to p and ad=bc (mod p). J. A. Serret^^^ stated and A. Genocchi proved that, if p is a prime, the sum of the mth. powers of the p" polynomials in x, of degree n — 1 and with integral coefficients <p, is a multiple of p if m<p'' — 1, but not if m = p'' — 1. J. A. Serret^^^ noted that all the real roots of a congruence f{x) = (mod p), where p is a prime, satisfy \j/{x)^0, where \f/ is the g. c. d. of f{x) andxP~^-l. '"Exercices de Math., 1, 1826, 160-6; Bull. Soc. Philomatique; Oeuvres, (2), 6, 202-8. ^"Exercices de Math., 4, 1829, 253-279; Oeuvres, (2), 9, 298-326. j i"Compte8 Rendus Paris, 25, 1847, 37; Oeuvres, (1), 10, 324-30. ~\ '"Nouv. Ann. Math., 9, 1850, 436. i^Nouv. Ann. Math., 13. 1854, 314; 14, 1855, 241-5 "'Cours d'alg^bre sup6rieure, ed. 2, 1854, 321-3. Chap. VIII] MISCELLANEOUS RESULTS ON CONGRUENCES. 259 N. H. AbeP^^ proved that we can solve by radicals any abelian equation, i. e., one whose roots are r, 0(r), 0^(r) = <t)[<i){r)], . . ., where </> is a rational function. H. J. S. Smith^^^ concluded that when the roots of a congru- ence can be similarly expressed modulo p, its solution can evidently be reduced to the solution of binomial congruences, and the expressions for the roots of the corresponding equation may be interpreted as the roots of the congruence. For the special case a:"=l, this was done by Poinsot in 1813-20 in papers discussed in the chapter on primitive roots. M. Jenkins^^^" noted that all solutions of a^=l(mod x) are x= Un=UiU2 . . .Un, where Ui is any divisor of any power of a — 1; u^ any divisor prime to a — 1, of any power of a'" — 1;. . .; u,, any divisor, prime to a^"-2 — 1, of any power of a^^-'^ — l. For a*+l = (mod x), modify the preceding by taking odd factors of a+1 instead of factors of a — 1. J. J. Sylvester^®° proved that if p is a prime and the congruence /(a:) = (mod p) of degree n has n real roots and if the resultant of f{x) and g{x) is divisible by p, then g{x)^0 has at least one root in common with /(a;) = 0. There are exactly p — 1 real roots of x^~^=l (mod p^). A. S. Hathaway^^^ noted the known similarity between equations and congruences for a prime modulus. He^^^ made abstruse remarks on higher congruences. G. Frattini^^^ proved that x^ — Dy'^=\ and x'^ — Dy^=\ are each solvable when the modulus is a prime p>5 and Dp^O. If d = B^—AC^O, then Ax'^-i-2Bx^y+Cy"=\ (mod p) is solvable since dx'^+XC can be made con- gruent to a square and hence to {Cy-{-Bx^y. Likewise for ax'^-\-2bx-\-c=y'^. A. Hurwitz^^^ discussed the congruence of fractions and the theory of the congruence of infinite series. If (/)(x) =ro-\-riX-{- . . . +r„a:V^-+ • • • and if yp{x) is a similar series with the coefficients s^, then and \l/ are called congruent modulo m if and only if Vn^s^ (mod m) for n = 1, 2, . . . . G. Cordone^^^ treated the general quartic congruence for a prime modulus ji by means of a cubic resolvent. The method is similar to Euler's solution of a quartic equation as presented by Giudice in Peano's Rivista di Matematica, vol. 2. For the special case x'^-\-%Hx^-\-K=0 (mod /x), set t = {ix — l)/2, r^ = 9H^ — K; then if K is a quadratic residue of /jl, there are four rational roots or none according as ( — 3/f+r)'=+l or not; but if K is a non-residue, there are two rational roots or none according as one of the congruences (-3H+r)'=4-l, i-ZH-ry^-l is satisfied or not. i^sjour. fur Math., 4, 1829, 131; Oeuvres, 1, 114. "sReport British Assoc. 1860, 120 seq., §66: Coll. M. Papers, 1, 141-5. "9aMath. Quest. Educ. Times, 6, 1866, 91-3. ""Amer. Jour. Math., 2, 1879, 360-1; Johns Hopkins University Circulars, 1, 1881, 131. Coll. Papers, 3, 320-1. i"Johns Hopkins Univ. Circulars, 1, 1881, 97. "^Amer. Jour. Math., 6, 1884, 316-330. "^Rendiconti Reale Accad. Lincei, Rome, (4), 1, 1885, 140-2. i"Acta Mathematica, 19, 1895, 356. «6Rendiconti Circolo Mat. di Palermo 9, 1895, 209-243. 260 History of the Theory of Numbers. [Chap, viii R. Lipschitz^^^ examined Fermat's^^^ statement and proved that the primes p for which a' +1 = (mod p) is impossible are those and only those for which a solution u of w" =a (mod p) is a quadratic non-residue of p and for which X^A.', where 2^ is the highest power of 2 dividing p — 1. Cases when a''+l = is impossible and not embraced by Fermat's rule are a = 2, p = 89, 337; a = S, p = 13; a=-2, p = 281; etc. L. Kronecker^^^ called /(.r) an invariant of the congruence k=k' (mod m), if the latter congruence implies the equality /(/v) =/(//). If also, conversely, the equality implies the congruence, f{x) is called a proper (or characteristic) invariant, an example being the least positive residue of an integer modulo m. It is shown that every invariant of k=k' (mod m) can be represented as a symmetric function of all the integers congruent to k modulo m. G. Wertheim^^^ proved that a^+l = (mod p) is impossible if a belongs to an odd exponent modulo p [Fermat^'^^]. E. L. Bunitzky^*^^ (Bunickij) noted that, for any integer M, the con- gruences f(a+kh)=rk (mod M) (A: = 0, 1,. . ., n) hold if and only if the coefficients Ak of /(x) satisfy the conditions k\h''Ak=A% (mod M) {k = 1, . . . , n). If k is the least value of x for which xlh"" is divisible by M, and if the g. c. d. of M and h is k<m, where m is a divisor of M, then if /(a;)=0 (mod M) has the roots a, a-\-h,..., a-\-{k — l)h, it has also the roots a-\-jh {j = k,k-\-l,...,m-l).^ G. Biase^^'' called a similar to h in the ratio m:n modulo k if the remainders on dividing a and h by k are in the ratio m:n. Two numbers similar to a third in two given ratios modulo k are similar to each other modulo k in a ratio equal to the quotient of the given ratios. The problem^ '^^ to find n numbers whose n^ — n differences are incon- gruent modulo n^ — n+1 is possible for n = 6, but not for n = 7. R. D. von Sterneck^^° proved that, if A is not divisible by the odd prime p, Ax'^+Bx^+C takes \p{2AB, p) incongruent values (when x ranges over the set 0, 1, . . ., p — 1) if 5 is not divisible by p, while if B is divisible by p, it takes (p+3)/4 or (p-fl)/2 values according as p = 4n-l-l or p = 471 — 1. In terms of Legendre's symbol, >««Bull. des Sc. Math., (2), 22, I, 1898, 123-8. Extract in Oeuvres de Fermat, 4, 196-7. >"Vorlesungen iiber Zahlcntheorie, I, 1901, 131-142. "^AnfangsRTunde der Zahlenlehre, 1902, 265. ''"Zap. mat. otd. Obsc. (Soc. of natur.), Odessa, 20, 1902, III- VIII (in Russian); cf. Fortschr. Math., 33, 1902, p. 205. •^"Il Boll. Matematica Gior. Sc. Didat., Bologna, 4, 1905, 96. i"L'interm6diaire des math., 1906, 141; 1908, 64; 19, 1912, 130-1. Amcr. Math. Monthly, 13, 1906, 215; 14, 1907, 107-8. Chap. VIII] MISCELLANEOUS RESULTS ON CONGEUENCES. 261 E. Landau^'^ proved that, if /(x) =0 is an equation with integral coeffi- cients and at least one root of odd multiplicity, there exist an infinitude of primes p = 4:n — l such that /(a;) = (mod p) has a root. R. D. von Sterneck^^^ found the number of combinations of the ith. class (with or without repetition) of the numbers prime to p of a complete set of residues modulo p^ whose sum is congruent to a given integer modulo p^, p being a prime. E. Piccioli^^'* gave known theorems on adding and multiplying con- gruences. C. Jordan^^^ found the number of sets of integers aik for which the determinant |aa-| of order n is congruent to a given integer modulo M. C. Krediet^^^ gave theorems on congruences of degree n for a prime modulus analogous to those for an algebraic equation of degree n, including the question of multiple roots. The determination of roots is often sim- plified by seeking first the roots which are quadratic residues and then those which are non-residues. The exposition is not clear or simple. G. Rados"^ proved that, if p is a prime, fix) = ao^^-^+ • • ■ +«p-2= 0, g{x) = hx^'-^-i- . . . +bp_2= (mod p) have a common root if and only if each Ri=0 (mod p), where *(w) =Rou''-^+Riu''-^-\- . . . +Rp-i aou+bo aiU+bi ... ap_2W+6p_2 aiu+bi a2U-\-b2 . . . aoU-{-bo ap_2U+bp-2 aou+bo ... ttp-s^+^p-s For ^=/', let ^(u) become DoU^~^-\- . . . +i)p_2; thus/(a:)^0 (mod p) has a multiple root if and only if each A=0 (mod p). Each of these theorems is extended to three congruences. Finally, if f(x) and f'(x) are relatively prime algebraically, there is only a finite number of primes p for which the number of roots of /= (mod p'') exceeds the degree of /. G. Frattini^^^ proved that if p and q are primes, q a divisor of p — 1, every homogeneous symmetric congruence in q variables is solvable modulo p by values of the variables distinct from each other and from zero except when the degree of the congruence is divisible by q. C. Grotzsch^'^ noted that if a is a root of a^'^^a (mod p), where a is prime to p, then x=a (mod p^ — p) is a root, and proved that if d is the g. c. d. of ind a and p — 1 and if ind a>0, it has exactly ^''^Handbuch . . .Verteilung der Primzahlen, 1, 1909, 440. ^"Sitzungsber. Ak. Wiss. Wien (Math.), 118, 1909, Ila, 119-132. '^"11 Pitagora, Palermo, 16, 1909-10, 125-7. "5Jour. de Math., (6), 7, 1911, 409-416. '^^Wiskundig Tijdschrift, Haarlem, 7, 1911, 193-202 (Dutch). i"Ami. sc. ecole norm, sup., (3), 30, 1913, 395-412. "speriodico di Mat., 29, 1913, 49-53. "'Archiv Math. Phys., (3), 22, 1914, 49-53. 262 History of the Theory of Numbers. [Chap, viii iV = 0(p-l)+2^^) roots incongruent modulo p(p — 1), where 5 ranges over all divisors > 1 of ^. If ind a = 0, the number of such roots is p — l-\-N, where now 5 ranges over the di\asors >1 of p — 1. A. Chdtelet^^° noted that divergences between congruences and equa- tions are removed by not Umiting attention to the given congruence fix) = of degree n, but considering simultaneously all the polynomials g{x) derived from f{x) by a Tschirnhausen transformation ky — (i>{x), where k is an integer and </> has integral coefficients and is of degree n — 1. *M. Tihanyi^^"" proved a simple congruence. R. Kantor^^^ discussed the number of incongruent values modulo m taken by a polynomial in n variables, and especially for ax^-\- . . .+d modulo p', generalizing von Sterneck.^"*" The solvabiUty of x^+9a;+6=0 and x^-\-y{y-\-l) = (mod p) has been treated.^«2 A. Cunningham^^ announced the completion, in conjunction with Woodall and Creak, of tables of least solutions {x, a) of the congruences T^=^y% rV==*=l (mod p*< 10000), r = 2, 10; ?/ = 3, 5, 7, 11. T. A. Pierce^^^ gave two proofs that /(a;) = (mod p) has a real root if and only if the odd prime p divides 11(1— a^^"^), where a^ ranges over the roots of the equation f{x) = 0. Christie^^ stated that P(F+1) = 1 (mod p) if t= 2 sin 18° and p is any odd prime. Cunningham gave a proof and a generalization. *G. Rados^^^ found the congruence of degree r having as its roots the r distinct roots ?^0 of a given congruence of degree p — 2 modulo p, a prime. i8'€omptes Rendus Paris, 158, 1914, 250-3. """Math, es Phys. Lapok, Budapest, 23, 1914, 57-60. i8iMonatshefte Math. Phys., 26, 1915, 24-39. J '»2Wiskundige Opgaven, 12, 1915, 211-2, 215-7. '"Messenger Math., 45, 1915-6, 69. '"Annals of Math., (2), 18, 1916, 53-64. i«*Math. Quest. Educ. Times, 71, 1899, 82-3. '»Math. is Term6s firtesito, 33, 1915, 702-10. CHAPTER IX. DIVISIBILITY OF FACTORIALS AND MULTINOMIAL COEFFICIENTS. Highest Power of a Prime Dividing ml Genty^ noted that the highest power of 2 dividing (2'*) ! is 2^""^ and the quotient is 3"-'(5-7)"-2(9-lM3-15)"-^(17. . .31)""^ . .(2"-l). In general if P = 2"'+2"'+. . .+2% where the n's decrease, the highest power of 2 dividing P! is 2^-^ A. M. Legendre^ proved that if p" is the highest power of the prime p which divides m !, and if [x] denotes the greatest integer ^ x, where s = ao+ • • • +«« is the sum of the digits of m to the base p: Th. Bertram^ stated Legendre's result in an equivalent form. H. Anton* proved that, U n = vp+a, a<p, v<p, and p is a prime, = (p — l)'a!y! (mod p), n! P while, if v = vp-\-a, a'<p, v'<p, ■^,= {p-iy+^'a\a\v\v\ (modp). D. Andr^^ stated that the highest power p" of the prime p dividing n! is given expHcitly by }i=^lZi[n/p^] and claimed that merely the method of finding ii had been given earHer. He appHed this result to prove that the product of n consecutive integers is divisible by n!. J. Neuberg*' determined the least integer m such that m\ is divisible by a given power of a prime, but overlooked exceptional cases. L. Stickelberger^ and K. HenseP gave the formula [cf. Anton*]. (2) ^^(-irao!ai!...aj(modp). F. de Brun^ wrote g[u] for the exponent of the highest power of the prime p dividing u. He gave expressions for ■^ rP{n;k)=Uf\ g[rP(n;k)] 3 = 1 in terms of the functions h{a; k) = l*+2*-f . . . +a^. A special case gives (1). ^Hist. et M6m. Ac. R. Sc. Inscript. et Belles Lettres de Toulouse, 3, 1788, 97-101 (read Dec. 4, 1783). ''Th^orie des nombres, ed. 2, 1808, p. 8; ed. 3, 1830, I, p. 10. 'Einige Satze aus der Zahlenlehre, Progr. Coin, Berlin, 1849, 18 pp. *Archiv Math. Phys., 49, 1869, 298-9. "Nouv. Ann. Math., (2), 13, 1874, 185. ^Mathesis, 7, 1887, 68-69. Cf. A. J. Kempner, Amer. Math, Monthly, 25, 1918, 204-10. 'Math. Annalen, 37, 1890, 321. sArchiv Math. Phys., (3), 2, 1902, 294. »Arkiv for Matematik, Astr., Fysik, 5, 1904, No. 25 (French). 263 264 History of the Theory of Numbers. [Chap, ix R. D. CarmichaeP" treated the problem to find m, given the prime p and s = ^ai, in Legendre's formula; a given solution m-, leads to an infinitude of solutions m-zv'', k arbitrary. Again, to find 771 such that p"*~' is the highest power of p>2 dividing m\, we have m — t={m — s)/{p — l), and see that m has a hmited number of values; there is always at least one solution m. Carmichael" used the notation H\y\ for the index of the highest power of the prime p dividing y, and evaluated ;i=i/{n(xa+c)|, where a, c are relatively prime positive integers. Set Co = c and let % be the least integer such that iVa+Cr-i is divisible by p, the quotient being Cr. Let ei = [^4^']' ^'=[^]' ^>i- t-i Then /i= 2(6^+1), where t is the least subscript for which r=l Ct{a-\-Ct){2a-^Ct) . . . (cta+Ct) is not divisible by p. It follows that where R is the index of the highest power of p not exceeding 7i — 1 . If n is a power of p, /i = (n — l)/(p — 1). But if n = 8kp''-\- . . .-\-dip-\-8o, 8k9^0, and at least one further 8 is not zero, ^SftSfc+^. <r = 6.+ ...+6o. P — 1 p — 1 In case the first x for which xa-\-c is divisible by p gives c as the quotient, all the Cr are equal and hence all the v; then , _ rn — 1—i+pl . rn — l—i — ip-\-p^l , rn — l—i — ip — ip'^-i-p^l . L p J"^L p' J"^L w J"^'" The case a = c = l yields Legendre's^ result. The case a = 2, c = 1, gives Hll.3.5. . .(2„-l)( = [?^^] + [?^^^>. . .. E. Stridsberg^^ wrote H^ for (1) and considered Trt = a{a-\-m) . . .{a-{-'mt), where a is any integer not divisible by the positive integer m. Let p be a prime not dividing m. Write a^ for the residue of aj modulo m. He noted that, if pj=l (mod m), "BuU. Amer. Math. Soc, 14, 1907-8, 74-77; Amer. Math. Monthly, 15, 1908, 15-17. ''Ibid., 15, 1908-9, 217. "Arkiv for Matematik, Astr., Fysik, 6, 1911, No. 34. 1 Chap. IX] DIVISIBILITY OF FACTORIALS, MULTINOMIAL COEFFICIENTS. 265 is an integer, and wrote L^ for its residue modulo p^+\ Set :.= P He proved that tt^ is divisible by p*, where s = Hi+Sj;io2^. If t„ is the first one of the numbers tq, ti, . . . which is <p — 1, tt^ is divisible by p", k A. Cunningham^^ proved that if / is the highest power of the prime z dividing p, the number of times p is a factor of p"! is the least of the numbers ^„ 2^«-^+i-l for the various primes z dividing p. W. Janichen^^ stated and G. Szego proved that i:tx{n/d)v{d)=ci>{n)/{v-l), summed for the divisors d of n, where v{d) is the exponent of the highest power of p (a prime factor of n) which divides d\, for /x as in Ch. XIX. Integral Quotients Involving Factorials. Th. Schonemann^^ proved, by use of symmetric functions of pth roots of unity, that if b is the g. c. d. of fjL,v,'..., 8-{m-l)l .ii/l = integer, {m—fx+v-\- . . .). fJLlVl He gave (p. 289) an arithmetical proof by showing that the fractions obtained by replacing 8 by fi, v, .. . are integers. A. Cauchy^^ proved the last theorem and that ^ -^ ^ = integer, {m = a-\- . . .+k). a\. . .k\ D. Andre^*^ noted that, except when n = l, a = 4, n(n + l). . .(na — 1) is not or is divisible by a" according as a is a prime or not. E. Catalan^^ found by use of elliptic functions that {m-\-n-\)\ (2m) ! (2n) ! m!n! m\n\{m-\-n)\ are integers, provided m, n are relatively prime in the first fraction. ^^L'intermediaire des math., 19, 1912, 283-5. Text modified at suggestion of E. Maillet. "Archiv Math. Phys., (3), 13, 1908, 361; 24, 1916, 86-7. »8Jour. fur Math., 19, 1839, 231-243. "Comptes Rendus Paris, 12, 1841, 705-7; Oeuvres, (1), 6, 109. 20N0UV. Ann. Math., (2), 11, 1872, 314. "/fttd., (2), 13, 1874, 207, 253. Arith. proofs, Amer. Math. Monthly, 18, 1911, 41-3. 266 History of the Theory of Numbers. [Chap, ix P. Bachmann^- gave arithmetical proofs of Catalan's results. D. Andr^-^ proved that, if aj, . . . , a„ have the sum N and if k of the a's are not di\isible by the integer >1 which divides the greatest number of the a's, then (iV— A;)! is di\'isible by ai!. . .a„!. J. Bourguet^^ proved that, if k^2, {kmi)l ikmo)\...{knh)\ — , . ^ , ; r: = mteger. Wi!. . .nikl (wi+. . .+mk)l M. WeilP^ proved that the multinomial coefficient (tq) ! -r- {q\y is divisible by tl WeilP^ stated that the following expression is an integer : (a+i8+ ■ • ■ +pg+Pigi+ ■ ■ • +rst) ! am. . .{piyq\{p,\y^q,\. . .{r\ns\yt\' WeilP^ stated the special case that {a-\-^-\-pq-\-rs)\ is divisible by a\^\{q\rp\{s\yrl D. Andr^-^ proved that (tq) ! -r- (g!)' is divisible by (<!)* if for every prime p the sum of the digits of q to base p is ^k. Ch. Hermite'^^ proved that n! divides m{7n+k){m+2k) . . . lm+{n-l)k]k''-\ C. de PoUgnac^" gave a simple proof of the theorem by WeilP^ and expressed the generalization by Andr^^^ in another and more general form. E. Catalan^^ noted that, if s is the number of powers of 2 having the sum ^+^' (2a)! (26)! a!6!(a+6)! is an even integer and the product of 2' by an odd number. E. Catalan^^ noted that, if n = a+6+ . . . -{-t, n\{n+t) a\h\...tl is divisible by a+t, h+t,. . ., a-\-h-\-t,. . ., a-\-b+c+t,. . .. E. Ces^ro^^ stated and Neuberg proved that (p) is divisible by n(n — 1) if p is prime to n(n — 1), and p — l prime to n — 1; and divisible by (p + 1) X(p+2) if p-\-l is prime to n+1, and p+2 is prime to (n + l)(n+2). "Zeitschrift Math. Phys., 20, 1875, 161-3. Die Elemente der Zahlentheorie, 1892, 37-39. «Bull. Soc. Math. France, 1, 1875, 84. "Nouv. Ann. Math., (2), 14, 1875, 89; he wrote r(n) incorrectly for n!; see p. 179. 'MDomptes Rendus Paris, 93, 1881, 1066; Mathesis, 2, 1882, 48; 4, 1884, 20; Lucas, Th^orie des nombres, 1891, 365, ex. 3. Proof by induction, Amer. M. Monthly, 17, 1910, 147. »«Bull. Soc. Math. France, 9, 1880-1, 172. Special case, Amer. M. Monthly, 23, 1916, 352-3. "Mathesis, 2, 1882, 48; proof by Li6nard, 4, 1884, 20-23. "Comptes Rendus Paris, 94, 1882, 426. "Faculty des Sc. de Paris, Cours de Hermite, 1882, 138; ed. 3, 1887, 175; ed. 4, 1891, 196. Cf. Catalan, M6m. Soc. Sc. de Li6ge, (2), 13, 1886, 262-^ ( = Melanges Math.); Heine."" »"Comptes Rendus Paris, 96, 1883, 485-7. Cf . Bachmann, Niedere Zahlentheorie, 1, 1902. 59-62. «Atti Accad. Pont. Nouvi Lincei, 37, 1883-4, 110-3. "Mathesis, 3, 1883, 48; proof by Cesiro, p. 118. "Ibid., 5, 1885, 84. Chap. IX] DIVISIBILITY OF FACTORIALS, MULTINOMIAL COEFFICIENTS. 267 E. Catalan^^ noted that e::r)'(?)"*e)-«" F. Gomes Teixeira^^ discussed the result due to Weill. ^® De Presle^® proved that {k-\-l)(k+2)...{k+hl) . ^ nw ^ ^^ ^^^'' being the product of an evident integer by {hl)\/{U{h\y}. , E. Catalan^^ noted that, if n is prime to 6, (2n-4)! nl{n-2)\ H. W. Lloyd Tanner^^ proved that = integer. = integer. {\,\...\,\ng\y L. Gegenbauer stated and J. A. Gmeiner^^ proved arithmetically that, if n=Sjrja_,iaj2. • Oys, the product m{m+k)(m-h2k) . . . {m+{n-l)k}k''-' is divisible by where m, k, n, an,- ■ ■, o,rs are positive integers. This gives Hermite's'^' result by taking r = s = l. The case m = A; = l,s = 2, is included in the result by Weill.26 Heine^^" and A. Thue^° proved that a fraction, whose denominator is k\ and whose numerator is a product of k consecutive terms of an arithmetical progression, can always be reduced until the new denominator contains only such primes as divide the difference of the progression [a part of Her- mite's^^ result]. F. Rogel'*^ noted that, if P be the product of the primes between (p — 1)/2 and p + 1, while n is any integer not divisible by the prime p, (n-l)(n-2). ..{n-p-^l)P/p=0 (mod P). S. Pincherle^^ noted that, if n is a prime, P={x+l){x+2) . ..(x+n-l) is divisible by n and, if x is not divisible by n, by n !. If n = Up", P is divisible ="Nouv. Ann. Math., (3), 4, 1885, 487. Proof by Landau, (4), 1, 1901, 282. 35Archiv Math. Phys., (2), 2 1885, 265-8. ^eBuU. Soc. Math. France, 16, 1887-8, 159. "M6m. Soc. Roy. Sc. Li^ge, (2), 15, 1888, 111 (Melanges Math. III). Mathesis, 9, 1889, 170. "Proc. London Math. Soc, 20, 1888-9, 287. ^QMonatshefte Math. Phys., 1, 1890, 159-162. "a Jour, fur Math., 45, 1853, 287-8. Cf. Math. Quest. Educ. Times, 56, 1892, 62-63. "Archiv for Math, og Natur., Kristiania, 14, 1890, 247-250. "Archiv Math. Phys., (2), 10, 1891, 93. "Rendiconto Sess. Accad. Sc. Istituto di Bologna, 1892-3, 17. 268 History of the Theory of Numbers. [Chap, ix by n ! if and only if divisible by IIp"'''^, where /3 is the exponent of the power of p dividing (n — 1)!. G. Bauer^^ proved that the multinomial coefficient (n+ni+n2+. . .)' -7- {7i!7?i! . . . } is an integer, and is even if two or more n's are equal. E. Landau^^ generalized most of the preceding results. For integers Qij, bij, each ^ 0, and positive integers Xj, set Then / is an integer if and only if m n t=i t=i for all real values of the Xj for which O^Xj^l. A new example is (4m)!(4n)! a m!n!(2m+n)!(m+2n)!~^^^^^^' | P. A. MacMahon^^ treated the problem to find all a's for which is an integer for all values of n; in particular, to find those "ground forms" from which all the forms may be generated by multiplication. For m = 2, the ground forms have (ai, a2) = (1, 0) or (1, 1). For m = 3, the additional ground forms are (1, 1, 1), (1, 2, 1), (1, 3, 1). For ?7i = 4, there are 3 new ground forms; for m = 5, 13 new. J. W. L. Glaisher^® noted that, if Bp{x) is Bernoulli's function, i. e., the polynomial expression in x for F~^+2^"^+ . . . + (x — 1)^"^ [Bernoulli^^"'' of Ch. V], x{x-\-l) . . .{x-\-p — l)/p=Bp{x)—x (mod p). He gave (ibid., 33, 1901, 29) related congruences involving the left member and Bp_i{x). Glaisher^^ noted that, if r is not divisible by the odd prime p, and l = kp+t, 0^t<p, l{r+l){2r+l) . . . {(p-l)r+i)/p^-|[^]^+A:} (mod p), where [t/p]r denotes the least positive root of px=t (mod r). The residues mod p^ of the same product l{r-\-l) . . . are found to be complicated. E. Maillet*^ gave a group of order t\{q\y contained in the sjrmmetric group on tq letters, whence follows Weill's^^ result. «SitzunKsber. Ak. Wiss. Miinchen (Math.), 24, 1894, 34&-8. "Nouv. Ann. Math., (3), 19, 1900, 344-362, 576; (4), 1, 1901, 282; Archiv Math. Phys., (3), 1, 1901, 138. Correction, Landau." «Trans Cambr. Phil. Soc, 18, 1900, 12-34. "Proc. London Math. Soc, 32, 1900, 172. ^'Messenger Math., 30, 1900-1, 71-92. *»Mem. Pr6s. Ac. Sc. Paris, (2), 32, 1902, No. 8, p. 19. Chap. IX] DIVISIBILITY OF FACTORIALS, MULTINOMIAL COEFFICIENTS. 269 M. Jenkins^^" counted in two ways the arrangements of n = 4>f-\-'yg-\- . . . elements in 4> cycles of / letters each, 7 cycles of g letters, . . . , where/, g, ... are distinct integers > 1, and obtained the result fct>\g'y\... \2\ Sr4\ ' ' ' ^^ ^ n\)' C. de Polignac*^ investigated at length the highest power of n! dividing (nx) \/{x\y. Let rip be the sum of the digits of n to base p. Then {x-\-n)p = Xp+np-k{p -1) , {xn)p = Xp-np-k'{p-l), where k is the number of units "carried" in making the addition x-\-n, and k' the corresponding number for the multiplication x-n. E. Sch6nbaum^° gave a simplified exposition of Landau's first paper.^^ S. K. Maitra^i proved that (n - 1) (2n - 1) . . . { (n - 2)n - 1 } is divisible by (n — 1) ! if and only if n is a prime. E. Stridsberg^^ gave a very elementary proof of Hermite's^^ result. E. Landau^^ corrected an error in his^'* proof of the result in No. Ill of his paper, no use of which had been made elsewhere. Birkeland^^ of Ch. XI noted that a product of 2^k consecutive odd in- tegers is^l (mod 2^). Among the proofs that binomial coefficients are integers may be cited those by: G. W. Leibniz, Math. Schriften, pub. by C. I. Gerhardt, 7, 1863, 102. B. Pascal, Oeuvres, 3, 1908, 278-282. Gioachino Pessuti, Memorie di Mat. Soc. Italiana, 11, 1804, 446. W. H. Miller, Jour, fiir Math., 13, 1835, 257. S. S. Greatheed, Cambr. Math. Jour., 1, 1839, 102, 112. Proofs that multinomial coefficients are integers were given by: C. F. Gauss, Disq. Arith., 1801, art. 41. Lionnet, Complement des elements d'arith., Paris, 1857, 52. V. A. Lebesgue, Nouv. Ann. Math., (2), 1, 1862, 219, 254. Factorials Dividing the Product of Differences of r Integers. H. W. Segar^" noted that the product of the differences of any r distinct integers is divisible by (r — l)!(r — 2)!. . .2!. For the special case of the integers 1, 2, . . ., n, r+1, the theorem shows that the product of any n consecutive integers is divisible by n!. A. Cayley®^ used Segar's theorem to prove that m{m — n) . . .{m — r — ln)-rf is divisible by r! if m, n are relatively prime [a part of Hermite's-^ result]. Segar®^ gave another proof of his theorem. Applying it to the set ^8aQuar. Jour. Math., 33, 1902, 174-9. "Bull. Soc. Math. France, 32, 1904, 5-43. "Casopis, Pras, 34, 1905, 265-300 (Bohemian). "Math. Quest. Educat. Times, (2), 12, 1907, 84-5. ^^Acta Math., 33, 1910, 243. "Nouv. Ann. Math., (4), 13, 1913, 353-5. soMessenger Math., 22, 1892-3, 59. "Messenger Math. 22, 1892-3, p. 186. Cf. Hermite." ^Hbid., 23, 1893-4, 31. Results cited in I'interm^diaire des math., 2, 1895, 132-3, 200; 5, 1898, 197; 8, 1901, 145. 270 History of the Theory of Numbers. [Chap, ix a, a-\-N,. . ., a+A^", we conclude that the product of their differences is divisible by n!(n — 1)!. . .21 = p. But the product equals p=iN-ir-' (ir--ir-\ . .{N^-'^-iyiN"-'-!), multiplied by a power of A^. Hence, if N is prime to n!, P is divisible by v; in any case a least number X is found such that N^P is divisible by ;'. It is shown that the product of the differences of mi,. . ., m^ is divisible by k\{k — l)\. . .2! if there be any integer p such that Wi+p, . . ., nik+p are relatively prime to each of 1, 2, . . . , A;. It is proved that the product of any n distinct integers multiplied by the product of all their differences is a multiple of n!(n-l)!. . .2!. E. de Jonquieres^^ and F. J. Studnicka^ proved the last theorem. E, B. Elliott^^ proved Segar's theorem in the form: The product of the differences of n distinct numbers is di\'isible by the product of the differences of 0, 1,..., n — 1. He added the new theorems: The product of the differences of n distinct squares is divisible by the product of the differences of 0", 1",..., (n — 1)"; that for the squares of n distinct odd numbers, multiplied by the product of the n numbers, is divisible by the product of the differences of the squares of the first n odd numbers, multiplied by their product. Residues of Multinomial Coefficients. Leibniz^' '^ of Ch. Ill noted that the coefficients in (ZaY—Za^ are di\'isible by p. Ch. Babbage^^ proved that, if n is a prime, (^n-/) — 1 is divisible by n^, while ("p") — 1 is divisible by p if and only if p is a prime. G. Libri^'' noted that, if m = 6p-hl is a prime, 2^p-^-^ep-l-(^^P~^y+(^P~^y'- ..=0 (mod m). E. Kummer^^ determined the highest power p^ of a prime p dividing ^; ^1 > A = ao+aip-{-. . .-{-aip\ B=ho+hip-{- . . .+bip\ where the a, and 6, belong to the set 0, 1,. . ., p — 1. We may determine Ci in this set and e, = or 1 such that (3) ao+6o = €oP+Co, €o+ai+6i = eip+Ci, ei +a2 + ?>2 = €2^4-^2, •■ •• Multiply the first equation by 1, the second by p, the third by p^, etc., and add. Thus A+B = Co-\-Cip+ . . .+Cip'+e,p'+\ "Comptes Rendus Paris, 120, 1895. 408-10. 534-7. "Vpstnik Ceske Ak., 7, 1898, No. 3, 165 (Bohemian). «Messinger Math., 27, 1897-8, 12-15. "Edinburgh Phil. Jour., 1, 1819, 46. "Jour, fur Math., 9, 1832, 73. Proofs by Stern, 12, 1834, 288. "/6ui., 44, 1852, 115-6. Cayley, Math. Quest. Educ. Times, 10, 1868, 88-9. Chap. IX] DIVISIBILITY OF FACTORIALS, MULTINOMIAL COEFFICIENTS. 271 Hence, by Legendre's formula (1), ip-l)N = A-\-B-y-ei-{A-a)-{B-P), a=Sa„ ^ = 26^, 7=Sc,. Insert the value of a+/3 obtained by adding equations (3). Thus A. Genocchi'^^ proved that, if m is the sum of n integers a,h,...,k, each divisible by p — l, and if m<p" — 1, then m\-i- {a\bl. . .k\} is divisible by the prime p. J. Wolstenholme^^ proved that f"ll) = l(mod n^) if n is a prime > 3. H. Anton^ (303-6) proved that if n = vp+a, r = wp+h, where a, h, v, w are all less than the prime p, according as a^ 6 or a < 6. M. Jenkins^^" considered for an odd prime p the sum "^^ ^\mr+k{p-l)J' extended over all the integers k between nr/(p — l) and —mr/{p — l), in- clusive, and proved that (Tr=o'p (mod p) if the g. c. d. of r, p — 1 equals that of p, p-1. E. Catalan^^ noted that C'^_}) = l(mod p), if p is a prime. Ch. Hermite^^ proved by use of roots of unity that the odd prime p divides /2n+l\ , /2n+l\ , /2n+l\ , {p-l) + [2p-2r[sp-3r-' E. Lucas'^^ noted that, if m = pmi-\-ii, n = pni+v, ii<p, v<p, and p is a prime. In general, if fxi, fJL2, ■ ■ ■ denote the residues of m and the integers contained in the fractions m/p, m/p^, . . . , while the v's are the residues of n, [n/p], . . . , e)-t;)t)- '-'^^'- E. Lucas'^'^ proved the preceding results and 0-0. f ;>(-!)". (^:>0(modp), according as n is between and p, and p — l, or 1 and p. "Nouv. Ann. Math., 14, 1855, 241-3. "Quar. Jour. Math., 5, 1862, 35-9. For mod. w^ Math. Quest. Educ. Times, (2), 3, 1903, 33. "'^Math. Quest. Educ. Times, 12, 1869, 29. ^■'Nouv. Corresp. Math., 1, 1874-5, 76. '^Jour. fur Math., 81, 1876. 94. '^Bull. Soc. Math. France, 6, 1877-8, 52. "Amer. Jour. Math., 1, 1878, 229, 230. For the second, anon.« of Ch. Ill (in 1830). 272 History of the Theory of Numbers. [Chap, ix J. Wolstenholme"^ noted that the highest power of 2 dividing i^""^^) isq — p — l, where q is the sum of the digits of 2m — I to base 2, and 2" is the highest power of 2 dividing ?«. J. Petersen"^ proved by Legendre's formula that C^'') equals the product of the powers of all primes p, the exponent of p being (ta+tb — ta+b) -^(p — 1), where ta is the sum of the digits of a to base p. E. Cesaro^° treated Kummer's^^ problem. He stated (Ex. 295) and Van den Broeek^^ proved that the exponent of the highest power of the prime p dividing (-„") is the number of odd integers among [2n/p], [2n/p^], [2n/p'],.... O. Schlomilch^^" stated in effect that („ + i) is divisible by n. E. Catalan^'- proved that if n is odd, p:)+io(t-^) = (modn+2). W. J. C. Sharp^^" noted that {p-\-n)\ — p\n\ is divisible by p^, if p is a prime >n. This follows also from (''t") — 1 (mod p) [Dickson^"]. L. Gegenbauer^^ noted that, if a is any integer, r one of the form 6s or 3s according as n is odd or even, The case n odd, a = 2, r = 3, gives Catalan's result. E. Catalan^ proved Hermite's'^^ theorem. Ch. Hermite^^ stated that (Z) is divisible by m— n-f-1 if w is divisible by n; by (m— n+l)/€ if e is the g. c. d. of m+1 and n; by m/8, if 5 is the g. c. d. of m, n. E. Lucas^^ noted that, ifn^p — 1, p — 2, p — 3, respectively, (^;3)-(-ir(^^±lM)(:nodp), if p is a prime, and proved Hermite's'^ result (p. 506). F. RogeP^ proved Hermite's"^ theorem by use of Fermat's. ^*Jour. de math. 6\6m. et spec., 1877-81, ex. 360. ^»Tidsskrift for Math., (4), 6, 1882, 138-143. soMathesis, 4, 1884, 109-110. 8'7feid., 6, 1886, 179. «>"Zeitschrift Math. Naturw. Unterricht, 17, 1886, 281. <«M6m. Soc. Roy. Sc. de Li6ge, (2), 13, 1886, 237-241 ( = Melanges Math.). Mathesis, 10, 1890. 257-8. 82aMath. Quest. Educ. Times, 49, 1888, 74. s^Sitzunpsber. Ak. Wiss. Wien (Math.), 98, 1889, Ila, 672. wM6m. Soc. Sc. Li^KC, (2), 15, 1888, 253-4 (Melanges Math. III). «*Jour. de math, sp^ciales, problems 257-8. Proofs by Catalan, ibid., 1889, 19-22; 1891, 70; by G. B. Mathews, Math. Quest. Educ. Times, 52, 1890, 63; by H. J. Woodall, 57, 1892, 91. "Th^orie des nombres, 1891, 420. "Archiv Math. Phys., (2), 11, 1892, 81-3. Chap. IX] DIVISIBILITY OF FACTORIALS, MULTINOMIAL COEFFICIENTS. 273 C. Szily^^ noted that no prime >2a divides ?M) • and specified the intervals in which its prime factors occur. F. Morley^^ proved that, if p = 2n+l is a prime, (2„")-(-l)"2*" is divisible by p^ ii p>S. That it is divisible by p^ was stated as an exercise in Mathews' Theory of Numbers, 1892, p. 318, Ex. 16. L. E. Dickson^° extended Rummer's''^ results to a multinomial coefficient M and noted the useful corollary that it is not divisible by a given prime p if and only if the partition of m into nii,..., nit arises by the separate partition of each digit of m written to the base p into the corresponding digits oi TUi, . . . , rrit. In this case he proved that ^= n .1),''" .,, (mod p), m, = ao''Y-{- . . . +a,^''K This also follows from (2) and from (xi+ . . . +xtr= {x,+ ... +x,)"»(xiP+ . . . +x,o"»-i . . . (0^1^''+ . . . +x/'ro (mod p). F. Mertens^^ considered a prime p^n, the highest powers p" and 2" of p and 2 which are ^n, and set n„ = [n/2"]. Then nl-^ {niln2l. . .nj} is divisible by Up'", where p ranges over all the primes p. J. W. L. Glaisher^^ gave Dickson's^" result for the case of binomial coefficients. He considered (349-60) their residues modulo p"', and proved (pp. 361-6) that if {n)r denotes the number of combinations of n things r at a time, 'Z{n)r^(j)k (mod p), where p is any prime, n any integer =j (mod p — l), while the summation extends over all positive integers r, f"^n, r=k (mod p — l), and j, k are any of the integers 1,. . ., p — l. He evaluated S[(?^)r-^p] when r is any number divisible by p — l, and (n)^ is divisible by p, distinguishing three cases to obtain simple results. Dickson^^ generalized Glaisher's^^ theorem to multinomial coefficients: Let k be that one of the numbers 1, 2, . . ., p — l to which m is congruent modulo p — l, and let ki,..., kt be fixed numbers of that set such that ki-\- ■ ■ ■ -\-kt=k (mod p — l). Then if p is a prime, where , . , m (mi, . . . , nit) = J ; The second of the two proofs given is much the simpler. ssNouv. Ann. Math., (3), 12, 1893, Exercices, p. 52.* Proof, (4), 16, 1916, 39-42. s^Annals of Math., 9, 1895, 168-170. ^"Ibid., (1), 11, 1896-7. 75-6: Quart. Jour. Math., 33, 1902, 378-384. siSitzungsber. Ak. Wiss. Wien (Math.), 106, lla, 1897, 255-6. '2Quar. Jour. Math., 30, 1899, 150-6, 349-366. o^Ibid., 33, 1902, 381-4. 274 History of the Theory of Numbers. [Chap, ix Glaisher^^ discussed the residues modulo p^ of binomial coefficients. T. Hayashi^^ proved that if p is a prime and fjL+v = p, (nsr>(-i)'C).».i(-<ip)- according as 0<s^v, v<s<p, or s = 0. T. Hayashi^^ proved that, if Iq is the least positive residue of I modulo p, and if v = p—ii, modulo p. Special cases of the first result had been given by Lucas. *^ A. Cunningham^^ proved that, if p is a prime, (^;^)^(-ir (modp), ^(p^)^! (modp^p>3). B. Ram^^ noted that, if (^), m = l,. . ., n — 1, have a common factor o>l, then a is a prime and n = a''. There is at most one prime <n which does not di\dde n(^) for m = l,. . ., n — 2, and then only when n+l=?a^ where a is a prime and q<a. For m = 0, 1, . . ., n, the number of odd (^) is always a power of 2. P. Bachmann^^ proved that, if h{p — l) is the greatest multiple <A; of p-i, (,!i)+(2(pii))+-+(M/-i>''("^°'^^>' the case k odd being due to Hermite.'^ G. Fonten^ stated and L. Grosschniid^°° proved that (p(pil))^(-l)' (^odp), P = p\ a^O. A. Fleck^^i proved that, if 0^p<p, aH-6=0 (mod p), N. Nielsen^"^ proved Bachmann's^^ result by use of Bernoulli numbers. wQuar. Jour. Math., 31, 1900, 110-124. "Jour, of the Physics School in Tokio, 10, 1901, 391-2; Abh. Geschichte Math. Wias., 28, 1910, 26-28. "Archiv Math. Phys., (3), 5, 1903, 67-9. •'Math. Quest. Educat. Times, (2), 12, 1907, 94-5. "Jour, of the Indian Math. Club, Madras, 1, 1909, 39-43. "Niedere Zahlentheorie, II, 1910, 46. ""•Xouv. Ann. Math., (4), 13, 1913, 521-4. "'Sitzungs. BerUn Math. Gesell., 13, 1913-4, 2-6. Cf. H. Kapferer, Archiv Math. Phys. (3), 23, 1915, 122. "«Annali di mat., (3), 22, 1914, 253. Chap. IX] DIVISIBILITY OF FACTORIALS, MULTINOMIAL COEFFICIENTS. 275 A. Fleck"^ proved that if and only if p is a prime. The case a = 1 is Wilson's theorem. Gu^rin^'^ asked if Wolstenholme's^^ result is new and added that (iLij — ^~1 (modp^), p prime >3. The Congruence 1-2-3. . .(p— 1)/2= ±1 (mod p). J. L. Lagrange^^" noted that p — 1, p— 2, ..., (p+l)/2 are congruent modulo p to —1, —2,..., — (p — 1)/2, respectively, so that Wilson's theorem gives (4) (l-2.3. . .P=iy^(-1)¥ (mod p). For p a prime of the form 4n+3, he noted that (5) l-2-3...^==tl (modp). E. Waring^^^ and an anonymous writer^^^ derived (4) in the same manner. G. L. Dirichlet^^^ noted that, since —1 is a non-residue of p = 4n+3, the sign in (5) is -f or — , according as the left member is a quadratic residue or non-residue of p. Hence if m is the number of quadratic non-residues <p/2ofp, l-2.3...^=(-ir (modp). C. G. J. Jacobi^^^ observed that, for p>3, m is of the same parity as N, where 2N—l = {Q—P)/p, P being the sum of the least positive quadratic residues of p, and Q that of the non-residues. Writing the quadratic residues in the form ^k, l^A;^|(p — 1), let m be the number of negative terms —k, and — T their sum. Since — 1 is a non-residue, m is the number of non-residues < ^p and ^{=i=k)=Sp, P=2{+k)-{-X{p-k)=mp+Sp, Since p = 4n+3, N = n-\-l—m—S. But 7i-|-l and S are of the same parity since p>S+2r = l+2+...+Kp-l)=iy-l) = (2n-H)(n+l). lo^Sitzungs. BerUn Math. GeseU., 15, 1915, 7-8. lo^L'intermgdiaire des math., 23, 1916, 174. ""Nouv. M6m. Ac. BerHn, 2, 1773, ann6e 1771, 125; Oeuvres, 3, 432. "iMeditat. Algebr., 1770, 218; ed. 3, 1782, 380. i"Jour. fur Math., 6, 1830, 105. "'/bid., 3, 1828, 407-8; Werke, 1, 107. Cf. Lucas, Th^orie des nombres, 438; rinterm^diaire des math., 7, 1900, 347. i"/6id., 9, 1832, 189-92; Werke, 6, 240-4. 276 History of the Theory of Numbers. [Chap, ix He stated empirically that N is the number of reduced forms ay^+hyz-\-cz^, 4:ac — b~ = p for b odd, ac — \b~ = p for b even, where b<a, b<c. C. F. Arndt^^^ proved in two ways that the product of all integers relatively prime to M = %r or 2/)", and not exceeding (ilf — 1)/2, is =±1 (mod M), when p is a prime 4fe+3, the sign being + or — according as the number of residues >M/2 of M is even or odd. Again, {l-3-5-7...(p-2)}2=±l (modp), the sign being + or — according as the prime p is of the form 4n+3 or 472+1. In the first case, 1-3. . .(p — 2)=±1 (mod p). L. Kronecker^^^ obtained, for Dirichlet's^^^ exponent m, the result m=j/ (mod 2), where v is the number of positive integers of the form q^^^'^r^ in the set p — 2^, p — 4", p — 6", . . ., and g is a prime not dividing r. Liou- \'ille (p. 267) gave m=k-\-v" (mod 2), when p = 8^+3 and v" is the number of positive integers of the form g'^'+V^ in the set p— 4^, p — 8^, p — 12^, . . .. J. Liou\'ille^" gave the result ?n=cr+r (mod 2), for the case p = 8^'+3, where r is the number of positive integers of the form 2g^'"^^ r^ {q a prime not dividing r) in the set p — 1^, p— 3^, p — 5^, . . . , and a is the number of equal or distinct primes 4gr+l di\'iding b, where p = a^+26" (uniquely). A. Korkine^^^ stated that, if [x] is the greatest integer ^x, _p-3 (p-3)/4 S [Vp^l (modp). 4 J. Franel"^ proved the last result by use of Legendre's symbol and (-i)""'TG> ©=(-^)'' "-TBI (-°'^2)- M. Lerch^^° obtained Jacobi's"* result. H. S. Vandiver^^"" proved Dirichlet's"^ result and that (p-i)/2r.-2-i m= S Y~\ (mod 2). R. D. CarmichaeP^^ noted that (4) holds if and only if p is a prime. E. Malo^^^ considered the residue ±r of 1-2. . .(p — 1)/2 modulo p, where p is a prime 4m+l, and 0<r<p/2. Thus r^= —1. The numbers 2, 3, . . ., (p — 1)/2, with r excluded, may be paired so that the product of the two of a pair is = =•= 1 (mod p) . If this sign is minus for k pairs, 1-2. . .(p-l)/2=(-l)V (mod p). *J. Ouspensky gave a rule to find the sign in (5). Other Congruences Involving Factorials. V. Bouniakowskyi29 noted that (p-l)! = PP', P±P'=0 (mod p) accord- ing as p = 4/j=f1. For, if p is a primitive root of p, we may set P = pp^ I'SArchiv Math. Phys., 2, 1842, 32, 34-35. i^o^Amer. Math. Monthly, 11, 1904, 51-6. "«Jour. de Math., (2), 5, 1860, 127. ^^'IMd., 12, 1905, 106-8. ^^Ubid., 128. '22i;interm6diaire des math., 13, 1906, 131-2 »«L'interm6diaire des math., 1, 1894, 95. »23Bu11. Soc. Phys. Math. Kasan, (2), 21. "»/6id., 2, 1895, 35-37. i"M6m. Ac. Sc. St. P6tersbourg, (6), 1, 1831, 564. '"Prag Sitzungsber. (Math.). 1898, No. 2. Chap. IX] DiVISIBILTY OF FACTORIALS, MULTINOMIAL COEFFICIENTS. 277 ...p\ P' = p'+^..p^-' with t = {v-l)/2, when p = 4A;-l; but P=pp''-^ pV-^ . ., P' = pY-'' pV"". • ., when p = 4A; + l. G. 01tramare^^° gave several algebraic series for the reciprocal of the binomial coefficient C^) and concluded that, if the moduli are primes, !+(-')= -2{(i)%(||)%(i|^J+ . . .} (mod 4^+1), 2=+(-') = -K(iy+(riT+(wy+ ■ • •} ^^'^ *-+3)- V. Bouniakowsky^^^ considered the integers qi,. . ., Qs, each <N and prime to N, arranged in ascending order of magnitude. If X is any chosen integer ^s, multiply q, = N-qi, qs-i = N-q2,..., g,_x+i = iV-gx together and multiply the resulting equation by qi. . . q^^x- Apply the generalized Wilson theorem qi. . .g^+( — 1)^=0 (mod A'"). Hence 9i?2- • •5x-gi?2. . .g.-x-f(-l)'+^=0 (mod N). For N a prime, we have s = N—l and X!(iV-l-X)!+(-l)'=0 (mod N) (l^XSN-l). C. A. Laisant and E. Beaujeux^^^ gave the last result and {'-.'} (-If (mod p), ^ = ^- F. G. Teixeira^^^ proved that if a=-2^''-^p-a, a<2p-l, a{a+l) . . .{a+2p-l)=3^-5\ . .{2p-iyp (mod a+a+l+a+2+...+a+2p-l). Thus, for p = 3, a = 1 , a = 95, 95-96-97-98-99-100=32-52-3 (mod 585 = 95+ •• .+100). M. Vecchi^^^ noted that the final formula by Bouniakowsky^^^ follows by induction. Taking X = (iV— 1)/2, we get Lagrange's formula (4). From the latter, we get {3.5-7. . . (22/-l)}2| (^^^^) \f/2'^^{-l)'^ (mod p). The case y={p — l)/2 gives Arndt's"^ result (6) {3-5-7...(p-2)P=(-l)~ (modp). Vecchi^^^ proved that, if v is the number of odd quadratic non-residues of a prime p = 4n+3, then 1-3-5. . .(p — 2) = ( — 1)" (mod p). If n is the number of non-residues <p/2, 1-3-5. . .{p-2)={-iy+^2^''-^^^^ (mod p). ""M^m. de I'lnstitut Nat. Genevois, 4, 1856, "sjomal de Sciencias Math, e Astr., 3, 1881, 33-6. 105-115. "iBull. Ac. Sc. St. P^tersbourg, 15, 1857, 202-5. i^^Periodico di Mat., 16, 1901, 22-4. "2Nouv. Corresp. Math., 5, 1879, 156 (177). '^Hbid., 22, 1907, 285-8. 278 History of the Theory of Numbers. [Chap. DC R. D. Carmichael^^^ proved that, if a+1 and 2a + 1 are both primes, (a!)* — 1 is di^'isible by (a + l)(2a + l), and conversely. A. Ar^valo^^^ proved (6) and Lucas'" residues of binomial coefficients. N. G. W. H. Beeger"^ proved that [if p is a prime] (p_l)!+l = s-p+l (rnodp^), s = l+2^-'-\- . . .+{p-iy-' = pK.„ where h is a. Bernoulli number defined by the symbolical equation (/i + l)" = /i", hi = l/2. By use of Adams'^"" table of /i„ 2<114, it was verified that p = 5, p = 13 are the only p<114 for which (p — 1)!+1=0 (mod p^). T. E. Mason^^^ and J. M. Child^'^ noted that, if p is a prime >3, inp)\ = nl(piy (modp"+^). N. Nielsen^^'' proved that, if p = 2n+l, P=l-3-5. . . (2n-l), (-l)'^2np2=22'».3.5. . . (4n-l) (mod IGn^). If p is a prime >3, P=(-l)"2^"n! (mod p^). He gave the last result also elsewhere. ^"^^ C. I. Marks"- found the smallest integer x such that 2-4. . .{2n)x is di- visible by 3-5 ... (2n- 1). i»«Revista de la Sociedad Mat. Espanola, 2, "»Math. Quest. Educat. Times, 26, 1914, 19. 1913, 130-1. ""Annali di mat., (3), 22, 1914, 81-2. "'Messenger Math., 43, 1913-4, 83-4. »«K. Danske Vidensk. Selsk. Skrifter, (7), 10 ""a Jour, fiir Math., 85, 1878, 269-72. 1913, 353. "STohoku Math. Jour., 5, 1914, 137. "'Math. Quest. Educ. Times, 21, 1912, 84-6. I CHAPTER X. SUM AND NUMBER OF DIVISORS. The sum of the A;th powers of the divisors of n will be designated crk(n) Often (r(n) will be used for (7i(n),and T{n) for the number ao{n)oi the divisors of n; also, !r(7i)=T(l)+T(2)+...+r(n). The early papers in which occur the formulas for T{n) and a{n) were cited in Chapter II. L. Euler^'^'^ applied to the theory of partitions the formula (1) p{x)='n.{l-x'')=s^l-x-x'+z^+x^-x'^-.... fc=l Euler'' verified for n<300 that (2) a{n)=(T{n-l)+(7{n-2)-a{n-b)-(j{n-7)+(7in-12)+.. ., in which two successive plus signs alternate with two successive minus signs, while the differences of 1, 2, 5, 7, 12, . . . are 1, 3, 2, 5, 3, 7, . . ., the alternate ones being 1, 2, 3, 4, . . . and the others being the successive odd numbers. He stated that (2) can be derived from (1). Euler^ noted that the numbers subtracted from n in (2) are pentagonal numbers (3a;^— a:)/2 for positive and negative integers x, and that if a(n—n) occurs it is to be replaced by n. He was led to the law of the series s by multipljdng together the earlier factors of p{x), but had no proof at that time that p = s. Comparing the derivatives of the logarithms of p and s, he found for —xdp/{pdx) the two expressions equated in ,„. « nx"" x+2x^-bx^-1x^+l2x^^+ . . . {o) 2j ^= n=l \—X S He verified for a few terms that the expansion of the left member is (4) I a;V(n). n=l Multiplying the latter by the series s and equating the product to the numer- ator of the right member of (3), he obtained (2) from the coefficients of x". Euler® proved (1) by induction. To prove (2), multiply the left member of (3) by —dx/x and integrate. He obtained log p{x) and hence log s, and then (3) by differentiation. ^Letter to D. Bernoulli, Jan. 28, 1741, Corresp. Math. Phys. (ed. Fuss), II, 1843, 467. ''Euler, Introductio in Analysin Infinitorum, 1748, I, ch. 16. ^Novi Comm. Ac. Petrop., 3, 1750-1, 125; Comm. Arith., 1, 91. ^Letter to Goldbach, Apr. 1, 1747, Corresp. Math. Phys. (ed. Fuss), I, 1843, 407. »Posth. paper of 1747, Comm. Arith., 2, 639; Opera postuma, 1, 1862, 76-84. Novi Comm. Ac. Petrop., 5, ad annos 1754-5, 59-74; Comm. Arith., 1, 146-154. "Letter to Goldbach, June 9, 1750, Corresp. Math. Phys. (ed. Fuss), I, 1843, 521-4. Novi Comm. Ac. Petrop., 5, 1754-5, 75-83; Acta Ac. Petrop., 41, 1780, 47, 56; Comm. Arith., 1, 234-8; 2, 105. Cf. Bachmann, Die Analytische Zahlentheorie, 1894, 13-29. 279 280 History of the Theory of Numbers. [Chap, x Material on (1) will be given in the chapter on partitions in Vol. II. J. H. Lambert/ by expanding the terms by simple division, obtained n = l 1—X in which the coefficient of x"* is T{n). Similarly, he obtained (4) from the left member of (3). E. Waring^ reproduced Euler's^ proof of (2). E. Waring^ employed the identity n {x''-l)=x''-x'-'-x'-^+x'-^+x'-''- -...=A, k=l the coefficient of x^'", for v^n, being ( — 1)^ if v={3z^^z)/2 and zero if v is not of that form. If m^n, the sum of the mth powers of the roots of A=0 is a{m). Thus (2) follows from Newton's identities between the coefficients and sums of powers of the roots. He deduced m(m-l) ,o^ , m{m-l)im-2) m{m-l){m-2){m-S) \o) I (t{2)-\ a{S) o-(4) , ?n(m-l)(m-2)(m-3) ( .^ + ■■■+ ^ {o-(2)j^-...= c-ml, where c= =•= 1 or is the coefficient of x^~"* in series A. Let U{x''-l)=x''-x'''-'-x''-^+x''-''-\-x''-^- . . . =A', where p ranges over the primes 1, 2, 3, 5, . . ., n. If m^n, the sum of the mth powers of the roots of A' = equals the sum a'{m) of the prime divisors of m. Thus ff'(m) =(r'(w- 1) +o-'(m-2) -t7'(m-4) -o-'(w-8) +(r'(m- 10) +o-'(m- 11) -(T'{m-12)-a'{m-lQ)+. . .. We obtain (5) with a replaced by a', and c by the coefficient of ic^'"*" in series A'. Consider n {x^^-l)=x^-x^-^ -x^-2'+x^-^'+ ... =5, with coefficients as in series A. The sum of the (Zm)th powers of the roots of B = equals the sum (T^^\m) of those divisors of m which are multiples of I. Thus ^ (T'^'\m)=(T'^'^{m-l)W\m-2l)-a^'\m-U)- . . ., with the same laws as (2) . The sum of those divisors of m which are divisible 'Anlage ziir Architectonic, oder Theorie des Ersten und des Einfachen in der phil. und math. Erkenntniss, Riga, 1771, 507. Quoted by Glaisher.'* ^Meditationes Algebraicse, ed. 3, 1782, 345. •Phil. Trans. Roy. Soc. London, 78, 1788, 388-394. Chap. X] SuM AND NuMBER OF DiVISORS. 281 by the relatively prime numbers a,h, c,. . . is Waring noted that o-(a|8) = ao-(/3) + (sum of those divisors of jS which are not divisible by a) . Similarly, <T(a^y . . . ) = aai^y . . . ) + (sum of divisors of 187 . . . not divisible by a) = a(3a{yd . . .) + (sum of divisors of JS7. . . not divisible by a) +a(sum of divisors of 76 . . . not divisible by jS), etc. Again, (r^'^(a/3)=ao-^"(i8) + (sum of divisors of ^ divisible by I but not by a). The generalization is similar to that just given for a. C. G. J. Jacobi^^ proved for the series s in (1) that 00 s^ = l-3x-\-5x^-7x^+...= S (-l)"(2n+l) a;"'"+^)/2. n=0 Jacobi^^ considered the excess E{n) of the number of divisors of the form 4w + l of n over the number of divisors of the form 4m +3 of n. If n = 2^uv, where each prime factor of u is of the form 4m + 1 and each prime factor of V is of the form 4m+3, he stated that E{n) =0 unless y is a square, and then E{n) =t{u). Jacobi^^ proved the identity (6) {l+x-{-x^-{- . . . +a;'(^+i)/24- . . .y = l-\-a(3)x+ . . . +(r(2n+lK+ .... A. M. Legendre^^ proved (1). G. L. Dirichlet^^ noted that the mean (mittlerer Werth) of (T{n) is x^n/6 — 1/2, that of T{n) is log n+2C, where C isEuler's constant 0.57721. . . . He stated the approximations to T{n) and \pin), proved later^'^, without ob- taining the order of magnitude of the error. Dirichlet^^ expressed m in all ways as a product of a square by a com- plementary factor e, denoted by v the number of distinct primes dividing e, and proved that 22" = T(m). Stern^^" proved (2) by expanding the logarithm of (1). If C"„ is the number of all combinations with repetitions with the sum n, (T(n)=nCn-C\ain-l)-C'2(T{n-2)- . . .. Let S{n) be the sum of the even divisors of n. Then, by (1), S{2n)=Si2n-2)-\-S{2n-4:)-S{2n-10)-Si2n-U)-\- . . ., S{0)=2n. "Fundamenta Nova, 1829, § 66, (7); Werke, 1, 237. Jour, fiir Math., 21, 1840, 13; French transl., Jour, de Math, 7, 1842, 85; Werke, 6, 281. Cf. Bachmann,« pp. 31-7. "Zfeid., §40; Werke, 1, 1881, 163. i^Attributed to Jacobi by Bouniakowsky" without reference. See Legendre (1828) and Plana (1863) in the chapter on polygonal numbers, vol. 2. "Th^orie des nombres, ed. 3, 1830, vol. 2, 128. "Jour, fiir Math., 18, 1838, 273; Bericht Berlin Ak., 1838, 13-15; Werke, 1, 373, 351-6. ^'Ibid., 21, 1840, 4. Zahlentheorie, § 124. i5»76id., 177-192. 282 History of the Theory of Numbers. [Chap, x Let S'(n) be the sum of the odd di\'isors of n, and C„ be the number of all combinations without repetitions with the sum n, so that C7 = 5. Then S'in)=nCn-S'{n-l)Ci-S\n-2)C2+ . . ., Z)(n) = -D(n-l)-D(n-3)-D(n-6)-..., D{n)=S'{n)-S{n). A complicated recursion formula for T(n)is derived from \og{{l-x){l-x^y{l-3^)r . .} = - I ^-Tin)x\ n=in Complicated recursion fonnulas are found for the number of integers <m not factors of m, and for the sum of these integers. A recursion formula for the sum Sr{n) of the di\'isors ^r of n is obtained by expanding log {l-x)(l-x2)...(l-a:'-)l = - S -Sr(n)x". n=in Jacobi^® proved (1). Dirichlet^^ obtained approximations to T(n). An integer s^n occurs in as many terms of this sum as there are multiples of s among 1, 2, . . . , n. The number of these multiples is [n/s], the greatest integer ^n/s. Hence ''(")=iG] This sum is approximately the product of n by £i = logn+C+i+.... Hence T{n) is of the same order of magnitude as n log n. Let ju be the least integer ^ y/n and set v = [n/ii]. Then if g{x) is any function and G{x)=g{l)-\-g{2)-\- . . . +^(x), 2 r^i^(s)= -.GGu)+s pi^(s)+s Gjr^ii- »=:LsJ «=iLsJ «=i LLsJJ In particular, if ^(x) = 1, «=iLsJ «=iLsJ Giving to [n/s] the approximation n/s, we see that (7) T(n)=n log,n+(2C-l)n+e, where € is of the same order of magnitude as Vn. Let pin) be the number of distinct prime factors >1 of ti. Then 2"^"^ is the number of ways of factoring n into two relatively prime factors, taking "Jour, fur Math., 32, 1846, 164; 37, 1848, 67, 73. "Abhand. Ak. Wiss. Berlin, 1849, Math., 69-83; Werke, 2, 49-66. French transl., Jour, de Math., (2), 1, 1856, 353-370. 4 Chap. X] SuM AND NuMBER OF DiVISORS. 283 account of the order of the factors. The number of pairs of relatively prime integers ^, 17 for which ^r^^n is therefore y=i For the preceding C and r(n), it is proved that r(n)=S^,/.[P], . t = [V^], ^in)=^(log.n+'-^+2C-l)+m, " C- I H^, IT IT 8=2 S where m is of the order of magnitude of n\ 8>y/2, while 7 is determined by 2)s~^ = l (s = 2 to 00). Moreover, T(n) is the number of pairs of integers X, y for which xy^n. He noted that (7(l)+(r(2) + ...+(r(n)=Ssr^l 8=1 Ls-i and that the difference between this sum and ir^n^/12 is of an order of magni- tude not exceeding n loge n. G. H. Burhenne^^ proved by use of infinite series that r(n)=i2)/"K0), fix)^- "^^ and then expressed the result as a trigonometric series. V. Bouniakowsky^^ changed x into x^ in (6), multiplied the result by x'^ and obtained (x^ +x' +x^ + . . .)* = x*+(r(3)x'2_^ . . . +(r(2w+l)a:^"'+H .... Thus every number 8m+4 is a sum of four odd squares in (r(2w+l) ways. By comparing coefficients in the logarithmic derivative, we get (8) (l2-2m+l)(r(2m+l) + (3^-2m-l)(7(2m-l) + (52-2m-5)(r(2m-5) + ...=0, in which the successive differences of the arguments of <r are 2, 4, 6, 8, ... . For any integer N, (9) {l^-N)a{N) + {S^-N-h2)(r{N-l-2) + i5''-N-2-3)a{N-2'3) + -..=0, where o-(O) , if it occurs, means A^/6. It is proved (p. 269) by use of Jacobi's^° result for s^ that l+x+x'+x'+ . . . =P^= (i+x)a+x'){l+x') . . . {l-x')(l-x'){l-x')..., "Archiv Math. Phys., 19, 1852, 442-9. »M6m. Ac. Sc. St. P^tersbourg (Sc. Math. Phys.), (6), 4, 1850, 259-295 (presented, 1848). Extract in Bulletin, 7, 170 and 15, 1857, 267-9. 284 History of the Theory of Numbers. [Chap, x where the exponents in the series are triangular numbers. Hence if we count the number of ways in which n can be formed as a sum of different terms from 1, 2, 3, . . . together w^ith different terms from 2, 4, 6, . . ., first taking an even number of the latter and second an odd number, the differ- ence of the counts is 1 or according as n is a triangular number or not. It is proved that (10) <r(n) + {(T(2)-4o-(l))(r(n-2)+(r(3)(T(n-4) + {(r(4)-4(r(2))(r(n-6) +(r(5)(7(n-8) + {(r(6)-4(r(3))tr(n-10)+. . . =^(7(n+2). The fact that the second member must be an integer is generaUzed as follows: for n odd, (T(n) is even or odd according as n is not or is a square; for n even, (T{n) is even if n is not a square or the double of a square, odd in the contrary case. Hence squares and their doubles are the only integers whose sums of divisors are odd. V. Bouniakowsky-'^ proved that (r(A^) = 2 (mod 4) only when N = kc^ or 2kc^, where A: is a prime 4Z+1 [corrected by Liouville^°]. V. A. Lebesgue-^ denoted by l-{-AiX+A2X^-\- . . . the expansion of the mth power of p{x), given by (1), and proved, by the method used by Euler for the case m = 1, that a{n)+A,a{n-l)-\-A2<T{n-2)+ . . .+Ar,_,a{l)-\-nAjm = 0. This recursion formula gives . m(m— 3) . —m(m — l)(m — S) A,= -m, A, = — ^^2— ' ^^ = 1:2:3 •••• The expression for Aj, was not found. E. MeisseP2 proved that (c/. Dirichlet^^) (11) T{n) = i^[jj =^i:[j] -'' (^ = [V^])- J. Liouville^^ noted that by taking the derivative of the logarithm of each member of (6) we get the formula, equivalent to (8) : J 5m(m+l) 1 /o , 1 2 N n S^n ^ Ya{2n+l—m—m)=0, summed for m = 0, 1, . . ., the argument of a remaining ^0. J. Liouville^^ stated that it is easily shown that Sd<T(d)=s(|y(r(d), 20M6m. Ac. Sc. St. P^tersbourg, (6), 5, 1853, 303-322. "Nouv. Ann. Math., 12, 1853, 232-4. "Jour, fur Math., 48, 1854, 306. "Jour, de Math., (2), 1, 1856, 349-350 (2, 1857, 412). ^Ibid., (2), 2, 1857, 56; Nouv. Ann. Math., 16, 1857, 181; proof by J. J. Hemming, ibid., (2), 4, 1865, 547. Chap. X] SuM AND NuMBER OF DiVISORS. 285 where d ranges over the divisors of m. He proved (p. 411) that S(-l)'"/'^fi = 2(T(m/2)-(7(m). J. Liouville^^ stated without proof the following formulas, in which d ranges over all the divisors of m, while 5 = m/d : Xaid) =2:5r(d), S0(d)r(5) =(r(m), 20(d)r(6) = [rim)}^, XcT{d)cr{8) =SdT(d)r(5), Sr(d)r(5) =2:|t(^)}' where (}){d) is the number of integers < d and prime to d, 6{d) is the number of decompositions of d into two relatively prime factors, and the accent on S denotes that the summation extends only over the square divisors D^ of m. He gave (p. 184) S0(cf)=r(m2), ^'e{^)i=r{m), the latter being implied in a result due to Dirichlet.^^ Liouville"*' gave the formulas, numbered (a),. . ., {k) by him, in which X(m) = +1 or —1, according as the total number of equal or distinct prime factors of m is even or odd: Sr(d2'')=T(m)r(m''), 2r(d2'')T(5) =ST(d)rOT, S(^(5)(7((i) =mT(m), S5(7(d) =SdT(d), SX(c^) = 1 or 0, ^\{d)d{d)r{b) = 1 or 0, according as m is or is not a square; i:\{d)d{d)r{h^) = l, X\{d)e{d)=\im), SX(d)0(5) = l, l^X{d)d{d)did)=0, SX(5)o-(ci) =mS'-^. The number of square divisors D^ of m is '2\(d)T{8). Liouville^^ gave the formulas, numbered I-XVIII by him: ST(52)(/)(d) =S5^(d), Sdr(52) =S^(5)(r(d), ST(52)X(d) =T(m), 2 {T{8)}Md)d{d) =tW, S0(d)T(5)r(5'') =SdT(62''), ^e{b)T{d)r{d'') =Sr(52)T(d-''), ST(52'')(7(d) =25r(d)T(d''), S'0(i))T(^) =S'Z) ^(^) , SX(5)T(d)TW =2't(^) . ^\id)(T{d) =mX(m)S'- 2)2 '^Jour. de Mathematiques, (2), 2, 1857, 141-4. "Sur quelques fonctiona num^riques," 1st article. Here Sa6c denotes S(a6c). ^^Ihid., 244-8, second article of his series. "76id., 377-384, third article of his seriea 286 History of the Theory of Numbers. [Chap, x S'X(i))r(^2) =^"d(^^' 2{W!'' = T(m^), Sr(0^(5) =S{^(d))''T(52), 2r(OX(5) =2|0(^) j^ where, in 2", e ranges over the biquadrate divisors of m. Liouville^^ gave the formula X{T{d)V={Xr{d)]', which implies that if 2m (m odd) has no factor of the form 4)u+3 and if we find the number of decompositions of each of its even factors as a sum of two odd squares, the sum of the cubes of the numbers of decompositions found will equal thesquare of their sum. Thus, for m = 25, 50=l2+72 = 72+l2 = 52+5^ 10 = 32+12 = 12+32, 2 = 1 + 1, whence 3H2Hl' = 62. Liouville2^ stated that, if a, 6, . . . are relatively prime in pairs, a^iah. . .)=o'n(oVn(?>)- • •, while if p, 9, . . . are distinct primes, He stated the formulas 2(r^((i)</)(5) =m<T,_,{m), 2ciV,(5) =2d''(T^(5), 2X(d)r(d2)(r^(5) =2d''r(5)X(5), 2dV^(6) =2c/''r(d), 2dX(d) =252v,(d), 2dV3,(5) =2c^V2,(d), 2dX+,(d)(7,(5) =2(iV,+,(d)(r,(5), 2X(d)(T,(5) =S'(^)' 2T(d2'')(r,(5) =2^^(5)7(5"), 2{^(rf)} V,(5) =2d''r(52'), and various special cases of them. To the seventh of these Liouville^" later gave several forms, one being the case p = of 2d''-V.+X^)(r,+,(5)=2d''-X+,((i)(r,+,(5), and proved (p. 84) the known theorem that a{m) is odd if and only if m is a square or the double of a square [cf. Bouniakowsky,^^ end]. He proved that (t{N) = 2 (mod 4) if and only if N is the product of a prime 4X + 1, raised to the power 4Z + 1 (Z^O), by a square or by the double of a square not divis- "Jour. de Math6matiques, (2), 2, 1857, 393-6; Comptes Rendus Paris, 44, 1857, 753, ^^Ibid., 425-432, fourth article of his series, "/bid., (2), 3, 1858, 63. Chap. X] SuM AND NuMBER OF DiVISORS. 287 ible by the prime 4X+1. The condition given by Bouniakowsky^° is neces- sary, but not sufficient. Also, o-3(m) = S <T{2j-l)a{2m-2j+l) {m odd). J. Liouville's series of 18 articles, "Sur quelques formules . . .utiles dans la th^orie des nombres," in Jour, de Math., 1858-1865, involve the function (r„, but will be reported on in volume II of this History in connection with sums of squares. A paper of 1860 by Kronecker will be considered in connection with one by Hermite.'^'^ C. Traub^^ investigated the number {N; M, t) of divisors T oi N which are = t (mod M) , where M is prime to t and N. Let a,h,. . .,lhe the integers < M and prime to M ; let them belong modulo M to the respective exponents a', h',. . ., V; let m be a common multiple of the latter. Since any prime factor of N is of the form Mx+k, where k = a,. . .,1, any T is congruent to a^6^. . .Z^=« (mod M), O^A<a',. . ., O^KV. Let A',. . ., L' he one of the n sets of exponents satisfying these conditions. If P is a primitive mth root of unity, the function 1/' = ^7-^SP^ e = {A-A')am/a'+ . . .+{L-L')\m/V, summed for all sets 0^a<a', . . ., O^X<r, has the property that i^ = l if A = A'(mod a'),. . ., L=L'(mod V) simultaneously, while i/' = in all other cases. Thus {N] M, t) =SSt/', where one summation refers to the n sets mentioned, while the other refers to the various divisors T of N. This double sum is simplified. [The properties found (pp. 278-294) for the set of residues modulo M of the products of powers oi a,. . ., I may be deduced more simply from the modern theory of commutative groups.] V. Bouniakowsky^^ considered the series n=l'c. n=l "' By forming the product of xl/ix)""'^ by \{/{x) , he proved that z„, 2 is the number No{n)=T{n) of the divisors of n, and Zn,m equals where (and below) d ranges over the divisors of n. Also, \p(x)\l/{x-l)= 2) ——• n=l '«' From \l/{xYxl/(x-iy for (i, j) = (2, 1), (2, 2), (1, 2), he derived the first and fourth formulas of Liouville's^^ first article and the fourth of his^^ second article. He extended these three formulas to sums of powers of the divisors ^lArchiv Math. Phys., 37, 1861, 277-345. 32M^m. Ac. Sc. St. P^tersbourg, (7), 4, 1862, No. 2, 35 pp. 288 History of the Theory of Numbers. [Chap, x and proved the second formula in Liouville's first article and the first two summation formulas of Liouville.-^ He proved i.(2.-l)=2.-l+z[^-^], .= [^], where 77 = 1 or according as 2o-— 1 is divisible by 3 or not. The last two were later generaUzed by Gegenbauer.^^ E. Lionnet^ proved the first two formulas of Liouville.^^ J. Liou^ille^ noted that, if q is divisible by the prime a, (r,(a5)+a''a-M^|j = (a'' + l)(r,(g). C. Sardi^^ denoted by A„ the coefficient of x" in Jacobi's^'' series for s^, so that An = unless n is a triangular number. From that series he got S(-l)P(2p+l)(7{7i-p(p + l)/2)=(-l)^'+^'/W3orO {t = Vl+8n), p according as n is or is not a triangular number, and |.4„+A„_,cr(l)+...+Ai(r(n-l)+Ao(r(n)=0. This recursion formula determines A„ in terms of the c's, or (T{n) in terms of the A's. In each case the values are expressed by means of determinants of order n. IM. A. Andreievsk}^^ wrote N^h^^i for the number of the divisors of the form 4/i± 1 of n = a'^h^ . . . , where a, b,. . . are distinct primes. We have where d ranges over all the di\'isors of n and the symbols are Legendre's. Evidently „ /-i\a' S (— ^) = a + l if a = 4Z + l, a'=o\ a } = or 1 if a = 4Z-l, according as a is odd or even. Hence, if any prime factor 4Z — 1 of n occurs to an odd power, we have iV4A+i=iV4A_i. Next, let *| where each p, is a prime of the form 4Z + 1 , each g, of the form 4? — 1 . Then iV4A+i-iV4.-i = (ai + l)(ao + l). . . =r(^), D = q,\^\ . .. ^'Souv. Ann. Math., (2), 7, 1868, 68-72. "Jour, de math., (2), 14, 1869, 263-4. "Giomale di Mat., 7, 1869, 112-5. 3%Iat. Sbomik (Math. Soc. Moscow), 6, 1872-3, 97-106 (Russian). Chap. X] SUM AND NUMBER OF DiVISORS. 289 The sum of the N's is T{n) ^riD^Mn/D^). Hence N^^^rm + l which is never an integer other than 1 or 2 when n is odd. If it be 2, t(D^) = 3 requires that D be a prime. Similarly, for Legendre's symbol (2/a), is zero if any prime factor 8^± 3 of n occurs to an odd power, but is 11 (a^H- 1) if in n each p, is a prime 8Z±1 and each Qi a prime 8Z±3. For n odd, Ngh^i/Nsh^s can not be an integer other than 1 or 2; if 2, D is a prime. F. Mertens" proved (11). He considered the number v{n) of divisors of n which are not divisible by a square > 1. Evidently v{n) =2", where p is the number of distinct prime factors of n. If ju(n) is zero when n has a square factor > 1 and is + 1 or — 1 according as n is a product of an even or odd number of distinct primes, v{n) =XiJL^{d), where d ranges over the divisors of n. Also, fc=i k=i \/c / He obtained Dirichlet's^'^ expression \l/{n) for this sum, finding for m a limit depending on C and n, of the order of magnitude of \/n log^ n. E. Catalan^'^" noted that So-(i)o-(i) =80-3(72) where i-{-j = 4n. Also, if i is odd, €r{i) equals the sum of the products two at a time of the E's of the odd numbers whose sum is 2i, where E denotes the excess of the number of divisors 4/i+l over the number of divisors 4/^ — 1. H. J. S. Smith^^ proved that, if m = pi''ip2"2. . ., ..W-2..(^)+S.,(^)-. ..=.•- For, if P=l+p'+...+p", P' = l+p'+...+p'"-"', then c.(m) = P.P. ... , <.. (^) = P/P, . . . , a. (^J = P/P/Pa .... and the initial sum equals (Pi — Pi){P2 — P2) ■ . .=m\ J. W. L. Glaisher^^ stated that the excess of the sum of the reciprocals of the odd divisors of a number over that for the even divisors is equal to the sum of the reciprocals of the divisors whose complementary divisors are odd. The excess of the sum of the divisors whose complementary divisors are odd over that when they are even equals the sum of the odd divisors. G. Halphen^° obtained the recursion formula (T(n)=3(r(n-l)-5(r(n-3)+. . . -(-l)"(2a;+l)Jn-^^^^|+. . ., "Jour, flir Math., 77, 1874, 291-4. ^'"Recherches sur quelques produits indefinis, M^m. Ac. Roy. Belgique, 40, 1873, 61-191. Extract in Nouv. Ann. Math., (2), 13, 1874, 518-523. asProc. London Math. Soc, 7, 1875-6, 211. '^Messenger Math., 5, 1876, 52. ^oBuU. Soc. Math. France, 5, 1877, 158. 290 History of the Theory of Numbers. [Chap.X where, if n is of the form x{x+l)/2, (t(0) is to be taken to be n/3 [Glai- sher^"]. The proof follows from the logarithmic derivative of Jacobi's^" expression for s^, as in Euler's^ proof of (2). Halphen"*^ formed for an odd function /(z) the sum of s.i pc! (-1)7 (t*«)' n- ■ +^n-l — "~o-^n> X taking all integral values between the two square roots of a, and y ranging over all positive odd divisors of a—x^. This sum is if a is a square, zero if a is not a square. Taking /(s) =z, we get a recursion formula for the sum of those di\dsors d oi x for which x/d is odd [see the topic Sums of Squares in Vol. II of this History]. Taking f{z)=a^ —oT', we get a recursion formula for the number of odd di\dsors <a/m of a. A generalization of (2) gives a recursion formula for the sima of the divisors of the forms 2nk, n{2k-\-\)^m, wdth fixed n, m. E. Catalan^'- denoted the square of (1) by l+LiX+ . . . +L„x'*H- . . .. Thus o-(n) +IiO-(7i- 1) +L20-(n-2) + I'n-I>n-l-I>.-2+I^n-54-Z>„_7- . . . = Or (2X + 1)(-1)\ according as n is not or is of the form X(X + l)/2. In \dew of the equality of (3) and (4) and the fact that l/p=2;/'(n)x", where yp{n) is the number of partitions of n into equal or distinct positive integers, he concluded that (7(n)=;//(n-l)+2,/'(n-2)-5;/'(n-5)-7i/^(n-7) + 12^(n-12)+. . .". J. W. L. Glaisher^^ noted that, if B{n) is the excess of the sum of the odd di\'isors of n over the sum of the even dii'isors, e{n) +<9(n - 1) +d{n - 3) +d{n - 6) -f . . . = 0, where 1, 3, 6, . . . are the triangular numbers, and B{n—n) = —n. E. Cesaro^ denoted bj^ s„ the sum of the residues obtained by dividing n by each integer <n, and stated that s„+(7(l)+(7(2)+...+(7(n)=n2. E. Catalan^^ proved the equivalent result that the sum of the divisors of 1, . . . , n equals the sum of the greatest multiples, not >/?, of these numbers. Catalan'*^ stated that, if <^(a, n) is the greatest multiple ^^ of a, n a{n)= 2 {</)(a, n)—4>{a, n — 1)). I "Bull. Soc. Math. France, 6, 1877-S, 119-120, 173-188. "Assoc, frang. avanc. sc, 6, 1877, 127-8. Cf. Catalan.^^" «Messenger Math., 7, 1877-8, 66-7. "Nouv. Corresp. Math., 4, 1878, 329; 5, 1879, 22; Nouv. Ann. Math., (3), 2, 1883, 289; 4, 1885, 473. «/Wd., 5, 1879, 296-8; stated, 4, 1879, ex. 447. *mid., 6, 1880, 192. Chap. X] SuM AND NuMBER OF DiVISORS. 291 Radicke (p. 280) gave an easy proof and noted that if we take n = 1, . . . , m and add, we get the result by E. Lucas^^ o-(l) + . . . +o-(m) =0(1, m) + . . . +0(m, m). J. W. L. Glaisher^^ stated that if f{n) is the sum of the odd divisors of n and if g{n) is the sum of the even divisors of n, and /(O) =0, g{0) =n, then /(n)+/(n-l)+/(n-3)+/(n-6)+/(n-10) + ... = g{n)+g{n-l)+gin-S)+ . . .. Chr.Zeller^^ proved (11). R. Lipschitz^o wrote G(t) for <t{1)-\- . . .-\-a{t), D{t) for {t''+t)/2, and $(0 for (^(1) + . . . +(f>{t), using Euler's (f>{t). Then if 2, 3, 5, 6, . . . are the integers not divisible by a square > 1 , ^w--[l]--[l]--G] + . . . =n, G(n)-2G D(n)-D the sign depending on the number of prime factors of the denominator. He discussed (pp. 985-7) Dirichlet's^'^ results on the mean of T{n), o-(n), (f>{n). A. Berger^^ proved by use of gamma functions that the mean of the sum of the divisors d of n is ir^n/Q, that of S d/2'^ is 1, that of Sl/d! is ir^/Q, G. Cantor^^" gave the second formula of Liouville^^ and his^^ third. A. Piltz^^ considered the, number Tk{n) of sets of positive integral solu- tions of Ui. . .Uk = n, where differently arranged u's give different sets. Thus T'i(n) = 1, T'2(n) =T{n). If a- is the real part of the complex number s, and n* denotes e^ '°^ " for the real value of the logarithm, he proved that n=l "' m = where l = \—(T — \/k, and the 6's are constants, 6^ = for s ?^ 1 ; while 0(/) is^'^ of the order of magnitude of /. Taking s = 0, we obtain the number 'SiTkin) of sets of positive integral solutions of t^i . . .u^'^x. H. Ahlborn^Hreated (11). E. Cesaro^^ noted that the mean of the difference between the number of odd and number of even divisors of any integer is log 2 ; the limit for ^^Nouv. Corresp. Math., 5, 1879, 296. 48NOUV. Corresp. Math., 5, 1879, 176. "Gottingen Nachrichten, 1879, 265. ^oComptes Rendus Paris, 89, 1879, 948-50. Cf. Bachmann^" of Ch. XIX. "Nova Acta Soc. Sc. Upsal., (3), 11, 1883, No. 1 (1880). Extract by Catalan in Nouv. Corresp. Math., 6, 1880, 551-2. Cf. Gram.«^« "«G6ttmgen Nachr., 1880, 161; Math. Ann., 16, 1880, 586. "Ueber das Gesetz, nach welchem die mittlere Darstellbarkeit der natiirlichen Zahlen ala Produkte einer gegebenen Anzahl Faktoren mit der Grosse der Zahlen wachst. Diss., Berlin, 1881. "Progr., Hamburg, 1881. "Mathesis, 1, 1881, 99-102. Nouv. Ann. Math., (3), 1, 1882, 240; 2, 1883, 239, 240. Also Ces^ro," 113-123, 133. 292 History of the Theory of Numbers. [Chap, x 7i= 00 of r(r2)/(n log r?) is l;cf. (7); themean of 2(d+p)-' is (1 + 1/2+... + l/p)/P- -'^s generalizations of Berger's^^ results, the mean of H.d/'p^ is l/(p — 1); the mean of the sum of the rth powers of the divisors of n is ^r ^(/--f 1) and that of the inverses of their rth powers is f(r+l), where (12) f(s)=ilM n=l J. W. L. Glaisher^^ proved the last formula of Catalan^^ and (r(n)-(r(n-4)-(r(n-8)+(T(n-20)+o-(n-28)-... = Q(n-l)+3Q(n-3)-6Q(n-6)-10Q(n-10)+..., where Q{n) is the number of partitions of n without repetitions, and 4, 8, 20, . . . are the quadruples of the pentagonal numbers. He gave another formula of the latter tj-pe. R. Lipschitz,^^ using his notations,^" proved that 7'(n)-.r0+.r[^]-...=n+z[|], G(n)-SaGg]+2«6G[^] - . . . =n+Sp[|], D(.) -SD [2] +XD [^] - . . . =$(n) +2*[|] , where P ranges over those numbers ^ n which are composed exclusivelj'' of primes other than given primes a,h,. . ., each ^ n. Ch. Hermite"^ proved (11) very simply. R. Lipschitz^^ considered the number T^it) of those divisors of t which are exact sth powers of integers and proved that where p' is the largest sth power ^rz, and v = [n/ii']. The last expression, found by taking /i = [n^^"^*^" ], gives a generahzation of (11). T. J. Stieltjes^^ proved (7) by use of definite integrals. E. Cesaro^° proved (7) arithmetically and (11). E. Cesaro^^ proved that, if d ranges over the divisors of n, and 5 over those of X, (13) 2G(d)/Q)=2^(d)FQ), F{x)^i:fi8), G{x)^Xg{8). Taking g(x) = l,f{x) =x, 4>{x), 1/x, we get the first two formulas of Liouville^^ "Messenger Math., 12, 1882-3, 16&-170. "Comptes Rendus Paris, 96, 1883, 327-9. "Acta Math., 2, 1883, 299-300. "/&id., 301-4. 5»Cdmpte9 Rendus Paris, 96, 1883, 764-6. ^Hhid., 1029. "Mdm. Soc. Sc. Li^e, (2), 10, 1883, Mem. 6, pp. 26-34. Chap. X] SuM AND NuMBER OF DiVISORS. 293 and the fourth of Liouville.^^ Taking g = x, f=(t>, we get the third for- mula of Liouville.^^ For g=l/x,f=(j), we get S#(d)(7 0) ^ =Sdl For g=(f) or x'',f=x% we get the first two of Liouville's^^ summation formulas. If ir(x) is the product of the negatives of the prime factors 5^ 1 of a:, Sx(d)*(d)<70)i = T(»), ST(d)<#,(d)J3 = ^,2#(d). Further specializations of (13) and of the generalization (p. 47) 2G(d)/(^) =i:F(d)g(^fj, F{x)^l:^|^{^)f(^^y G(a;) ^2,^(5)^ (^), led Cesaro (pp. 36-59) to various formulas of Liouville^^"^^ and many- similar ones. It is shown (p. 64) that n=l fl' n=l "• for f and F as in (12), (13). For /(n) =(^(n), we have the result quoted under Cesaro^^ in Ch. V. For/(n) = 1 and n'', m — k>l, ^—^=r{m), S-— - = ^(m)f(m-A;). n=l '«' n '«' If (n, j) is the g. c. d. of n, j, then (pp. 77-86) .ST^-jr = 2S(7(d)-l, nT(n)=i:a{n,j), <j{n)=^T{n, 3), S (Tk{n, j)=n(Tk-i(n), ^j(T{n, j) =-^lnT{n)-\-a{n)}. y=i ^ If in the second formula of Liouville^^ we take m = l,. . .,n and add, we get s0(i)rr?i=s<T(i). Similarly (pp. 97-112) we may derive a relation in [x] from any given relation involving all the divisors of x, or any set of numbers defined by x, such as the numbers a, h,. . . for which x — a^, x — W,. . . are all squares. Formula (7) is proved (pp. 124-8). It is shown (pp. 135-143) that the mean of the sum of the inverses of divisors of n which are multiples of k is 7rV(6A;^) ; the excess of the number of divisors 4)U+1 over the number of divisors 4^i+3 is in mean 7r/4, and that for 4/x+2 and 4ju is ^ log 2; the mean of the sum of the inverses of the odd divisors of any integer is ttYS ; the mean is found of various functions of the divisors. The mean (p. 172) of the number of divisors of an integer which are mth powers is f(^)j and hence is 7rV6 if 294 History of the Theory of Numbers. [Chap, x m = 2. The mean (pp. 216-9) of the number of divisors of the form aix+r of n is, for r>0, i+ijlog«/a+2C-/;ij^dx} (cf. pp. 341-2 and, for a = 4, 6, pp. 136-8), while several proofs (also, p. 134) are given of the known result that the number of divisors of n which are multiples of a is in mean -(log7i/a+2C). a If (pp. 291-2) a ranges over the integers for which [2n/d] is odd, the number (sum) of the a's is the excess of the number (sum) of the divisors of n + 1, n+2, . . . , 2n over that of 1, . . . , n; the means are n log 4 and 7rW/6. If (pp. 294-9) k ranges over the integers for which [n/k] is odd, the number of the A:'s is the excess of the number of odd divisors of 1, . . ., n over the number of their even divisors, and the sum of the A;'s is the sum of the odd divisors of 1, . . . , w; also S*W=9^ 9=[^]' Several asymptotic evaluations by Cesaro are erroneous. For instance, for the functions \{n) and At(n), defined by Liouville^^ and Mertens,^^ Cesaro (p. 307, p. 157) gave as the mean values 6/7r^ and 36/7r*, whereas each is zero.^^ J. W. L. Glaisher^^ considered the sum A(n) of the odd divisors of n. If n = 2^m {m odd), A(n) =(j{m). The following theorems were proved by use of series for elliptic functions : A(l)A(2n-l)+A(3)A(2n-3)+A(5)A(2n-5)+...+A(2n-l)A(l) equals the sum of the cubes of those divisors of n whose complementary divisors are odd. The sum of the cubes of all divisors of 2n+l is A(2n+l) + 12{A(l)A(2n)+A(2)A(2n-l)+. . . +A(2n)A(l)). If A, £, C are the sums of the cubes of those divisors of 2n which are respec- tively even, odd, with odd complementary divisor, 2A(2n)+24JA(2)A(2n-2)+A(4)A(2n-4)+. . .+A(2n-2)A(2)) = i(2A-2J5-C)=i(3-23^-10)5 o 7 if 2n = 2'"m (ttz odd). Halphen's formula^*' is stated on p. 220. Next, n(r(2n+l) + (n-5)(r(2n-l) + (n-15)(T(2n-5) + (n-30)(7(2n-ll)+. . . =0, "H. V. Mangoldt, Sitzungsber. Ak. Wiss. Berlin, 1897, 849, 852; E. Landau, Sitzungsber, Ak. Wiss. Wien, 112, II a, 1903, 537. «Quar. Jour. Math., 19, 1883, 216-223. Chap. X] SuM AND NuMBER OF DiVISORS. 295 in which the differences between the arguments of a in the successive terms are 2, 4, 6, 8, ... , and those between the coefficients are 5, 10, 15, ... , while o-(O) =0. Finally, there is a similar recursion formula for A(n). Glaisher^^ proved his^^ recursion formula for Q{n), gave a more compli- cated one and the following for (j{n) : o-(n)-2{o-(n-l)+o-(n-2)) +3{(7(n-3)+(r(n-4)+(7(n-5)! - . . . + (-irV{...+(r(l)) = (-l)V-s)/6, where s = r unless ra-(l) is the last term of a group, in which case, s = r+l. He proved Jacobi's^^ statement and concluded from the same proof that E{n) =JlE{ni) if n=nn„ the n's being relatively prime. It is evident that E{p')=r-\-l if p is a prime 4m+l, while £'(pO = l or if p is a prime 4m+3, according as r is even or odd. Also £'(2'') = 1. Hence we can at once evaluate E{n). He gave a table of the values oi E{n),n = \,. . ., 1000. By use of elliptic functions he found the recursion formulae E{n)-2E{n-^)+2E{n-\io)-2E{n-m)+ . . . =0 or (-l)'^-^^/^^, for n odd, according as n is not or is a square; for any n. E{n)-E{n-l)-E{n-S)+E{n-6)-{-E{n-10)- . . . = or (-ir{(-l)('-i)/2^-l}/4, ^^ Vs^^fl, according as n is not or is a triangular number 1, 3, 6, 10, . . .. He gave recursion formulae for S{2n) =E(2)+E(4)+ . . . +E{2n), S{2n-l)=E{l)-\-E{S)+ . . .+Ei2n-1). The functions E, S, 6, a are expressed as determinants. J. P. Gram^^" deduced results of Berger^^ and Cesaro.^'* Ch. Hermite^^ expressed (T(l)+<r (3) + . • - +o-(2n-l), (t(3) +o-(7) + . . . +o-(4n — 1) and o-(1)+<j(5)+ . . . +(7(4n+l) as sums of functions E,{x)=^{[xf-^[x\]/2. Chr. Zeller^^ gave the final formula of Catalan.^^ J. W. L. Glaisher®^ noted that, if in Halphen's^" formula, n is a triangular number, (T{n—n) is to be given the value n/3; if, however, we suppress the undefined term (7(0), the formula is (T(n)-3(j(n-l)+5(7(n-3)- . . . =0 or {-lY-\l''+2''+ . . .+r''), according as n is not a triangular number or is the triangular number r(r+ 1)/2. He reproduced two of his^^'^"*'^^ own recursion formulas for <T{n) (with yp for <j in two) and added o-(n)-{(7(n-2)+o-(n-3)+(r(n-4)j + !(7(n-7)-f(r(n-8)+(7(n-9) +o-(n-10)+(7(n-ll)[-{(T(n-15)+...) + ...=A-B, «^Proc. London Math. Soc, 15, 1883-4, 104-122. "°Det K. Danske Vidensk. Selskabs Skrifter, (6), 2 1881-6 (1884), 215-220 296. "^Amer. Jour. Math., 6, 1884, 173-4. 6«Acta Math., 4, 1884, 415-6. 6Troc. Cambr. Phil. Soc, 5, 1884, 108-120. 296 History of the Theory of Numbers. [Chap, x where A and B denote the number of positive and negative terms respec- tively, not counting cr{0) =n as a term; n<T(n)+2{(n-2)(r(n-2) + (n-4)(r(n-4)l +3{(n-6)(r(n-6) + (n-8)(7(n-8) + (n-10)tr(n-10)} + ... = a{n) + {V-+3')\a{n-2)+a{n-4:)] + (l-+3- + 5') {(r(n-6)+(r(n-8)+<r(n-10)l + ... (n odd). He reproduced his^ formulas for d{n) and E{n). He announced {ibid., p. 86) the completion of tables of the values of ^(n), T{n), (T{n) up to n = 3000, and inverse tables. Mobius^^ obtained certain results on the reversion of series which were combined by J. W. L. Glaisher^^ into the general theorem: Let a,h, ... be distinct primes; in terms of the undefined quantities e^, %,..., let e„ = e^V/ ... if n = a'^lP . . . , and let ei = 1. Then, if F(a:)=SeJ(a:"), where n ranges over all products of powers of a, 6, . . . , we have /(x)=S(-l)^6^(a;0, where v ranges over the numbers wdthout square factors and divisible by no prime other than a, 6, ... , while r is the number of the prime factors of v. Taking Glaisher obtained the formula of H. J. S. Smith^^ and aM -2aV,(^) +2a'fcV.(^^^ - . . . = 1. Using the same/, but taking €2 = 0, Cp = p\ when p is an odd prime, he proved that, if Ar{n) is the sum of the rth powers of the odd divisors of n, A,(„)-.A,©-f.A,(|) = or rf, according as n is even or odd. In the latter case, it reduces to Smith's. If A'r(^) is the sum of the rth powers of those di\'isors of n whose com- plementary divisors are odd, while Er{n) [or E'r{n)] is the excess of the sum of the rth powers of those divisors of n which [whose complementary divisors] are of the form 47?i + 1 over the sum of the rth powers of those divisors which [whose complementary divisors] are of the form 4772+3, A',(n)-2a'A',(^)+2a'6'A',(^^) - , . , =;, = ijl-(-l)-), A',(„)-2A',Q+2A-,(^)-...=n-, "Jour, fiir Math., 9, 1832, 105-123; Werke, 4, 591. "London, Ed. Dublin Phil. Mag., (5), 18, 1884, 518-540. Chap. X] SuM AND NUMBER OF DiVISORS. 297 EM -s^.(^) +s^.(£) - . . . = (-i)^"-^>/v., E'Xn) -^a^E\(^^ +^a%^E',{^^ -... =^{-\r-'^'\ E\{n)-U-ir-'"'E\{^ +S(-l)(^«-i)/2^;(^JL^ - . . . =n^ where A,B,. . . are the odd prime factors of n. Note that ^ = or 1 according as n is even or odd. By means of these equations, each of the five functions (Tr{i^),- ■ -J E'r{n) is expressed in two or more ways as a determinant of order n. Ch. Hermite^° quoted five formulas obtained by L. Kronecker'^^ from the expansions of elHptic functions and involving as coefficients the functions $(n)=o-(n), the sum Z(n)_of the odd divisors of n, the excess ^(n) of the sum of the divisors >\/n of n over the sum of those <\/n, the excess $'(n) of the sum of the divisors of the form 8k=^l of n over the sum of the divisors of the form 8k^S, and the excess ^'(n) of the sum of the divisors 8A;± 1 exceeding Vn and the divisors 8A;± 3 less than y/n over the sum of the divisors Sk=i=l less than \/n and the divisors 8A;± 3 exceeding \/n. Hermite found the expansions into series of the right-hand members of the five formulas, employing the notations Ei{x) = [a:+i] - [x], E^ix) = [x\[x+\]/2, a = l, 3, 5,...; 6 = 2,4,6,...; c = l, 2, 3,..., and A for a number of type a, etc. He obtained Z(l)+X(3)+. . .+X(A)=SE2(^), (r(l) +(7(2) + . . . + (7 (C) =SE2(C/c), ^(l)+^(2)+. . .+^(0=2^2 (^'), X(2)+Z(4)+ . . . +Z(B) =is|a[^] +&^i[|]| «l>'(l)+*'(3)+. . .+$'(A)=S(-l)^"^-^)/«a[^], ^'(l)+^'(3)+ . . . +^'{A) =S(-l)('^^+^>/«a| l^ ^+^^-^' j "BuU. Ac. Sc. St. Petersbourg, 29, 1884, 340-3; Acta Math., 5, 1884-5, 315-9. "Jour, fiir Math., 57, 1860, bottom p. 252 and top p. 253. 298 History of the Theory of Numbers. [Chap, x The first three had been found and proved purely arithmetically by Lipschitz and communicated to Hermite. Hermite proved (11) by use of series. Also, i F{a)=i r^l/(a), F(n)^2/(d), a = l a = l LuJ where d ranges over the di\'isors of n. When f{d) = I, F{n) becomes T{n) and the formula becomes the first one by Dirichlet.^^ L. Gegenbauer"'- considered the sum p^. , (n) of the A:th powers of those divisors d, of n whose complementary divisors are exact ^th powers, as well as Jordan's function Jk{n) [see Ch. V]. By means of the f -function, (12), he proved that 2 (Tk{m)po, 2(n) =2po. 2t{d)pk, t K j ' where d ranges over the divisors of r, and 7n, n over all pairs of integers for which mv} = r', 2 Jtk{n)p,, t{m) = rV^jt. <(r) , 2 <T,_k{m)T{n)m'' ='Epk, t{d)p,, t \j) ' the latter for t = l being Liouville's^® seventh formula ioT v = 0; 2dV,(^) =2dV..(0, 2/.(d)dV.(0 =P.+.<(r), the latter f or ^ = t- = 1 , A- = 0, being the second formula of Liouville"^, while for < = 1 it is the final formula by Cesaro^^° of Ch. V; 2X(d)dV.,2.(0 =2X(d)p,.,((i)p*.,^0 =0 or P2k,t{\^), according as r is not or is a square; 2X(n)p,. M =p,,2t{r), 2X(d)T(d2) =Hr)T{r), ^r\d)J,Q =r^^, 2dV(d2)(7,(0 =2dV(d), 2 \l\r{x') =2 At), 2 r^1x(x)(7,(x) =2 p^r). 2=iLa;J r=l i=lL3;J r=l By changing the sign of the first subscript of p, we obtain formulas for the sum Pk,i{n)=n''p_i,j{7i) of the A:th powers of those divisors of 7i which are tth powers. By taking the second subscript of p to be unity, we get formulas for (Tkin). There are given many formulas invohnng also the number fain) of solutions of nin2. . .71^ = 71, and the number co(«) of ways n can be expressed as a product of two relatively prime factors. Two special cases [(107), (128)] of these are the first formula of Liou\ille-^ and the ninth summation formula of Liouville,^^ a fact not observed by Gegenbauer. He proved that, if p^n, 2 B{x) = - 2 Cix)-\-Bn-Ap, i=p+l x = A + l "Sitzungsber. Ak. Wiss. Wien (Math.), 89, II, 1884, 47-73, 76-79. Chap. X] SuM AND Number of Divisors. 299 where and B = B{n), A=B{p+l); also that 2 D{x)= S F{x)+Dn-Ep, z=p+l x=D+l where and D = D(n), E = D(p-{-l). It is stated that special cases of these two formulas (here reported with greater compactness) were given by Dirichlet, Zeller, Berger and Cesaro. In the second, take t = 1, p = 0, and choose the integers a, j8, b, n so that hn''^-^>a>h{n-iy-\-^, whence D = 0. If Xr is the number of divisors of r which are of the form bx'+jS, we get Change n to n + 1 and set i3 = 0, 6 = o- = l, whence a = n [also set p = [Vn]]; we get Meissel's^^ formula (11). Other speciaHzations give the last one of the formulas by Lipschitz,^^ and where v = [-\/n\, k{r) is the number of odd divisors of r, while ^ = or 1 according as [n/v — ^]>v — \ or =v — \. L. Gegenbauer'^^ proved by use of ^-functions many formulas involving his'^^ functions p, / and divisors d<. Among the simplest formulas, special cases of the more general ones, are 2(r,(d)d^=S(r,+x(0rf'=2(r^(0d^+\ Xn\d,) =SX(A), mh)tx\d,) =Xfx\h), i:r{h')}x\d,) =Zd{h), 2m'(^)^(0 ='r(r'), Xrid^Uh) =e{r), Xi/{d)J,(^^ ^^dMh), summed for d, c?2, d^, where h = Vr/dg. Other special cases are the fourth and sixth formulas of Liouville,^^ the first, third and last of Liouville.^^ Beginning with p. 414, the formulas involve also oi,{n)^n'Ti {l + \/vt), n=ilp:\ 1=1 1=1 "Sitzungsber. Ak. Wien (Math.), 90, II, 1884, 395-459. The functions used are not defined in the paper. For his \pf,, ^, u, we write 0-^, r, e, where e is the notation of Liouville." n" 300 History of the Theory of Numbers. [Chap, x Beginning with p. 425 and p. 430 there enter the two functions ■.?,(i)-{© -'■'-}• in which (A/p) is Legendre's symbol, with the value 1 or —1. J. W. L. Glaisher^'* investigated the excess tr{n) of the sum of the rth powers of the odd di\'isors of n over the sum of the rth powers of the even divisors, the sum A'r(n) of the rth powers of those divisors of n whose complementary divisors are odd, wrote f for f i, and A' for A'l, and proved A'3(n)=nA'(n)+4A'(l)A'(n-l)+4A'(2)A'(n-2)+. . .+4A'(n-l)A'(l), r3(n) = (2n-l)r(n)-4r(l)r(n-l)-4r(2)r(n-2)-...-4r(n-l)f(l), nA'(n)=A'(l)A'(2n-l)-A'(2)A'(2n-2) + . . .+A'(2n-1)A'(1), (-l)''-H(n)=A'(n)+8r(l)A'(n-2)+8f(2)A'(n-4) + . . ., A'3(n)=7zA'(n)+A'(2)A'(2n-2)+A'(4)A'(2n-4) + . . .+A'(2n-2)A'(2), -f3(7i)=3A(n)+4{A(l)A(n-l)+9A(2)A(n-2)+A(3)A(n-3) +9A(4)A(n-4)+ . . . +A(n-1)A(1)) {n even), 2^-W2.+i(n) _ [l,2r-l] [3, 2r-3] [2r-l, 1] (2r)! l!(2r-l)!"^3!(2r-3)!'^""'^(2r-l)!l!' where b,?]=(7p(lK(2n-l)+(rp(3K(2n-3) + . . .+(Tp(2n-lK(l). For n odd, f (n) =A'(n) =a{n) and the fourth formula gives (/i-lM7i)=8{(7(lMn-2)H-r(2M7i-4)+(7(3Mn-6)+r(4Mn-8) + . . .). Glaisher'''^ proved that 5o'3(n) — 6w(7(n) -\-(j{n) = 12{(7(l)(7(7i-l)+o-(2)(r(n-2)+...+(7(n-l)(T(l)), (r(l)(r(2n-l)+(r(3)(r(2n-3) + . . .+or(2n-l)(T(l) =A'3(n)=|{cr3(2ii)-(T3(n)}. The latter includes the first theorem in his^^ earUer paper. Glaisher'^^ proved for Jacobi's^^ E{n) that cr(2m + l)=E(l)E(4m + l)+£;(5)^(4w-3)+E(9)E(4w-7) + . . . +£(4m + l)E(l), E(0-2^(^-4)+2E(<-16)-2E(«-36) + . . . =0 (« = 8n+5), (7(y)-2(r(y-4)+2(7(y-16)-2(7(y-36) + . . . =0 (y = 8n+7), <t(w) +(t(?/- 8) H-(r(i/- 24) +(r(w- 48) + . . . =4{(r(w)+2(T(m-4) +2(r(rn-16)+2(7(m-36)+. . .) (m = 2n + l, w = 8n+3), and three formulas analogous to the last (pp. 125, 129). He repeated (p. 158) his^'* expressions for A'3(n). ^^Messenger Math., 14, 1884r-5, 102-8. "•Hbid., 156-163. '•Quar. Jour. Math., 20, 1885, 109, 116, 121, 118. r,,.,(n)=SM.(^|);'' Chap. X] SuM AND NuMBER OF DiVISORS. 301 L. Gegenbauer^^ considered the number Ti(k) of the divisors ^[\/n] of ^ and the number T2ik) of the remaining divisors and proved that STi(/c)=5(log,n+2C)+0(V^), ^r^ik) =|(log,n+2C-2)+0(V^), 0(s) being ^° of the order of magnitude of s. He proved (p. 55) that the mean of the sum of the reciprocals of the square divisors of any integer is 7rV90; that (p. 64) of the reciprocals of the odd divisors is ttVS; the mean (p. 65) of the cubes of the reciprocals of the odd divisors of any integer is 7r^/96, that of their fifth powers is 7r^/960. The mean (p. 68) of Jacobi's^^ E{n) is 7r/4. G. L. Dirichlet^^ noted that in (7), p. 282 above, we may take e to be of lower [unstated] order of magnitude than his former -\/n- L. Gegenbauer^^ considered the sum r^ k,s (n) of the kth powers of those divisors of n which are rth powers and 'are divisible by no (sr) th power except 1 ; also the number Qa{b) of integers ^ b which are divisible by no ath power except 1. It follows at once that, if /Xg(^) =0 if m is divisible by an sth power >1, but =1 otherwise, where the summation extends over all the divisors dr of n whose com- plementary divisors are rth powers, and that (14) ST..,,(a;)= S -; hr^V.(^), v = Wn]. From.the known formula Qr{n) =S[n/a;'^]/z(x), x = 1, . . ., j^, is deduced the right member reducing to n for A; = and thus giving a result due to Bougaief. From this special result and (14) is derived From these results he derived various expressions for the mean value of Tr,-k,s{^) and of the sum t^.aj.X^) of the A;th powers of those divisors of n which are rth powers and are divisible by at least one (sr) th power other than 1. He obtained theorems of the type: The mean value of the number of square divisors not divisible by a biquadrate is 15/x^; the mean value of the excess of the number of divisors of one of the forms 4rjLi+y(j = l, 3, . . ., 2r — 1) over the number of the remaining odd divisors is 1 i cot(2LdV. 4ri^i 4r "Denkschr. Akad. Wien (Math.), 49, I, 1885, 24. 78G6ttingen Nachrichten, 1885, 379; Werke, 2, 407; letter to Kronecker, July 23, 1858. "Sitzungsberichte Ak. Wiss. Wien (Math.), 91, II, 1885, 600-^21. 302 History of the Theory of Numbers. [Chap, x L. Gegenbauer^" considered the number Ao{a) of those di\'isors of a which are congruent modulo k and have a complementary divisor =1 (mod k). He proved that, if p<k, If we replace (t by <t — 1 and subtract, we obtain expressions for Ao{k(T—p). The above formulas give, for k = 2, p = l, and formulas of Bouniakowsky.^^ The same developments show that an odd number a is a prime if L2(2a:+1) ^2j L2(2x+l)^2j for x^[(a — Z)/2]; likewise for a = 6fc=*=l if the same equality holds when x^[(a — 5)/Q], with similar tests for a = 3n — 1, or 4n — 1. C. Runge^^ proved that T{n)/n* has the limit zero as n increases indefi- nitely, for every e>0. E. Catalan^- noted that, if x^p is the number of ways of decomposing a product of n distinct primes into p factors >1, order being immaterial, x„p = px„_ip+x„_i,_i = jp'-^-(PTi)(p-ir-^+(^-^)(p-2r-^-...±i} -^{(p-l!). E. Cesaro^ considered the number F„ (x) of integers ^x which are not divisible by mth powers, and the number T^ (x) of those di\'isors of x which are mth. powers, evaluated sums involving these and other functions, and determined mean values and probabilities relating to the greatest square divisor of an arbitrary integer. R. Lipschitz^ considered the sum k{m) of the odd di\'isors of m increased by half the sum of the even di\'isors, and the function l(m) obtained by interchanging the words "even," "odd." He proved that k{m)-2k{m-l)+2k{m-9)- . . . =(-1)"-'^ or 0, according as m is a square or is not; l{m)+l{m-l)+l{77i-S)+l{m-6)+. . . = -m or 0, according as m is a triangular number or is not ; XW=A'(1)+A'(2)+...+A-W = H+[|]+3[|]+2[^] + ...+m[^], L(m) = /(l)+Z(2)+ . . . +l{m) = -[m]-^2[f\ -^[f\+^[f\ -■■■> "Sitzunpsberichte Ak. Wien. (Math.), 91, II, 1885, 1194-1201. "Acta Math., 7, 1885, 181-3. "M^m. 80C. roy. sc. Lifege, (2), 12, 1885, 18-20; Melanges Math., 1868, 18. "Annali di Mat., (2), 13, 1885, 251-268. Reprint "Excursions arith. k Tinfini," 17-34. "Comptes Rendus Paris, 100, 1885, 845. Cf. Glaisher"«, also Fergola" of Ch. XI, Vol. II. Chap. X] SuM AND NuMBER OF DiVISORS. 303 where fx = m or m/2 according as m is odd or even. Cf . Hacks.^^ M. A. Stern^^ noted that Zeller's^^ formula follows from B=pA, where 1 00 n 00 = A= Si/'(n)a;", -^=S= S o-(n)a:'*-\ p = l+2a:-5x*-7x^+ pix) n=o V{x) where p(rc) is defined by (1), yp{n) is the number of partitions of n, and the second equation follows from the equality of (3) and (4) after remov- ing the factor x. Next, if N{n) denotes the number of combinations of 1, 2, . . ., n without repetitions producing the sum n, X N(n)x''= {l+x){l+x^) . . . = 2, (l-x')il-x') rZi ^ ' ' ' '' '^■■- {l-x){l-x')...' then by the second equation above, B{\-x^-x'^x^''+x^^- . . .)=pSA^(n)x~, (r(n)-o-(n-2)-o'(7i-4)+(r(n-10)+(7(n-14)- . . . =i\r(n-l)+2i\r(n-2)-5iV(n-5)-7iV(n-7)+..., where (T{n — n) =0, N{n—n) = 1. S. Roberts^^ noted that Euler's^ formula (2) is identical with Newton's relation S^n = S-n+i+S^n+2— ■ ■ ■ for obtaining the sum aS_„ of the (— n)th powers of the roots of s = 0, where s and p are defined by (2). In p, the sum of the ( — n)th powers of the roots of 1 —a:^ = is A; or according as k is or is not a divisor of n. Hence the like sum for p is (r{n). [Cf. Waring^.] The process can be applied to products of factors 1 —f(k)x^. His further results may be given the following simpler form. Let 0„ be the sum of the even divisors of n, and xpn the sum of the odd divisors, and set s„=0„+2i^„ if n is even, s„= — 2i^„ if n is odd. By elliptic function expansions, S2n + 8{s2n_itAi+3S2H-2'/'2+S2n-3'A3 + 3S2n-4^4+ • • • -hSiXf/ 2n-l] +12ni/^2n = 0, S2n+l+8{s2„lAi+3S2„_l^2+ ■ • ■ +3SilA2n) +(4n + 2)l/'2n+l=0» the coefficients being 1 and 3 alternately. He indicated a process for finding a recursion formula involving the sums of the cubes of the even divisors and the sums of the cubes of the odd divisors, but did not give the formula. N. V. Bougaief^®" obtained, as special cases of a summation formula, ^{Sx+5-5i2u-iy}(Ti2x + l-u'' + u) = 0, S{n -3(7(t/)}P{n -o-(w)} = 0, where P{n) is the number of solutions u, v of a{u) +(t(v) =n. L. Gegenbauer^®'' proved that the number of odd divisors of 1, 2, . . ., n equals the sum of the greatest integers in (n+l)/2, (n+2)/4, (nH-3)/6, . . ., (2n)/(2n). The number of divisors of the form Bx—'yoil,...,nis ex- pressed as a sum of greatest integers; etc. J. W. L. Glaisher^^ considered the sum A^n) of the sth powers of the odd divisors of n, the Hke sum Dsin) for the even divisors, the sum D',{n) of the s^Acta Mathematica, 6, 1885, 327-8. 8«Quar. Jour. Math., 20, 1885, 370-8. 8««Comptes Rendus Paris, 100, 1885, 1125, 1160. sebDenkschr. Akad. Wiss Wien (Math.), 49, II, 1885, 111. 8'Messenger Math., 15, 1885-6, 1-20. 304 History of the Theory of Numbers. [Chap, x sth powers of the divisors of n whose complementary divisors are even, the excess f ',(«) of the sum of the sth powers of the divisors whose com- plementary'' di\'isors are odd over that when they are even, and the similar functions'^ A'„ f „ a,. The seven functions can be expressed in terms of any two: where the arguments are all n. Since D\{2k) =(T,(k), we may express all the functions in terms of a^n) and (TXn/2), provided the latter be defined to be zero when n is odd. Employ the abbreviation 'EfF='S,Ff for /(l)F(n-l)+/(2)F(n-2)+/(3)F(n-3) + . . .+/(n-l)F(l). This sum is evaluated when/ and F are any two of the above seven functions w^th s = 1 (the subscript 1 is dropped) . If f{n)=aa(n)+^D'in), F(n) =aV(n)+/3'Z)'(n), then 2/F = aa'2(T(r+(ai8'+a'/3)S(Ti)'H-/3iS'2i)'Z)'. By using the first formula in each of two earUer papers,'^' '^ we get 12'Z(ja = 5(T2{n) —6na{n) -\-a{n) , 122D'D' = 5Ds'(n)-dnD'{n)+D'in), 242o-D' = 2(r3(n) + (l-3n)(7(7i) + (l-6n)D'(n)+8Z)3'(n). Hence all 21 functions can now be expressed at once linearly in terms of 0-3, Ds', (T and D'. The resulting expressions are tabulated; they give the coefficients in the products of any two of the series 2f/(n)x", where/ is any one of our seven functions ■vsithout subscript. Glaisher^^ gave the values of l^a^ai for 2 = 3, 5, 9 and ^<r^(TT, where the notation is that of the preceding paper. Also, if p = 7i—r, 12 S rpa{f)a{fi) ^n^a^in) -nV(n), S rj{r)F{p) =^/F. r=l r=l A L. Gegenbauer^^ gave purely arithmetical proofs of generalizations of theorems obtained by Hermite'" by use of elliptic function expansions. Let 5,(r) =2/, (7=J^5,([^]) -v&,{y), v^\yM' Then (p. 1059), The left member is knowTi to equal the sum of the ^th powers of all the divisors of 1, 2, . . ., n. The first sum on the right is the sum of the A-th powers of the divisors ^ y/n of 1, . . . , n. Hence if A^fx) is the excess of the "Messenp;er Math., 15, 188.5-6, p. 36. '•Sitzungsberichte Ak. Wien (Math.), 92, II, 1886, 1055-78. Chap. X] SuM AND NuMBER OF DiVISORS. 305 sum t/'fc'(a;) of the kth powers of the divisors > \/x of x over the sum of the A;th powers of the remaining divisors, it follows at once that Also n V ['fi~\ x=l x=lL-CJ ^J,\x) =£'S^([3) +^''+M - (^+i)5,(.), with a similar formula for ^^^(a;), where "^k(^) is the excess of xpki^) over the sum of the A;th powers of the divisors < y/x of x. For k — l, the last formula reduces to the third one of Hermite's. Let Xk{^) be the sum of the kth powers of the odd divisors of x; Xk\^) that for the odd divisors > \/x; Xk"{x) the excess of the latter sum over the sum of the A;th powers of the odd divisors < ^/x of x; Xk"'{'^) the excess of the sum of the kih. powers of the divisors 8s±l>\/x of x over the sum of the kih. powers of the divisors 8s='=3<\/^ of ^- For y = 2x and y = 2x — l, the sum from x = l to a: = n of Xkiv), Xk'iv), X/iy) and Xk"{y) are expressed as complicated sums involving the functions Sk and [x\. E. Pfeiffer^° attempted to prove a formula like (7) of Dirichlet/^ where now e is 0{'n}^^^^) for every k>0. Here Og{T) means a function whose quotient hy g{T) remains numerically less than a fixed finite value for all sufficiently large values of 7". E. Landau^ ^ noted that the final step in the proof fails from lack of uniform convergence and reconstructed the proof to secure this convergence. L. Gegenbauer,^^ in continuation of his^° paper, gave similar but longer expressions for S r{y), S (Tkiy) (2/ = 4a;+l, 6a;+l, 8a:+3, 8a:+5, 8a:+7) and deduced similar tests for the primality of y. Gegenbauer^^" found the mean of the number of divisors \x-\-a of a number of s digits with a complementary divisor iiy-\-^; also for divisors ax'^+hy'^. Gegenbauer^^'' evaluated A(l) + . . . +A(n) where A{x) is the sum of the pth powers of the crth roots of those divisors d oi x which are exact o-th powers and whose complementary divisors exceed kdJ/'^. A special case gives (11), p. 284 above. Gegenbauer^^" gave a formula involving the sum of the A;th powers of those divisors of 1, . . . , m whose complementary divisors are divisible by no rth power >1. '"Ueber die Periodicitat in der Teilbarkeit . . . , Jahresbericht der Pf eiffer'schen Lehr- und Erzieh- ungs-Anstalt zu Jena, 1885-6, 1-21. "Sitzungsber. Ak. Wiss. Wien (Math.), 121, Ila, 1912, 2195-2332; 124, Ila, 1915, 469-550. Landau.^^^ »276id., 93, II, 1886, 447-454. "-^Sitzungsber. Ak. Wiss. Wien (Math.), 93, 1886, II, 90-105. 92676id., 94, 1886, II, 35-40. "^c/feid., 757-762. 306 History of the Theory of Numbers. [Chap, x 1 C^' Ch. Hermite^' proved that if F{N) is the number of odd divisors of N, n = l and then that F(l)+F(2)+ . . . +F{N) =^N log iV+ (c-^N, $(l)+$(2)+ . . . +$(iV) =^N log N/k+ (c-^N, asATnptotically, where ^(N) is the number of decompositions of N into two factors d, d', such that d'>kd. E. Catalan^^" noted that, if n = i+i' = 2i"d, 'La{i)a{i')=2(P, i:{aii)a{2n-i)} =Si:{<j{i)<x{n-i)]. E. Ces^ro^ proved Lambert's^ result that T{n) is the coefficient of x" in 2xV(l — a;*). Let T,{n) be the number of sets of positive integral solutions of and s,{n) the sum of the values taken by ^,. Then sM = TM + TXn-v) + TXn -2v)+..., T{n) =Si(n) -Szin) +83(71) Let aa^)=S(-l)'*+^r,(x-n), summed for the divisors d of n. Then Tin)=tM+t2{n)+ . . .+TM-T2{n) + Ts{n)- . . .. E. Busche^^ employed two complementary di\isors 5^ and 8 J of m, an arbitrary function/, and a function y=^(x) increasing with x whose inverse function is x = ?/)/' (7/). Then 2 limx)], X) -/(O, x) } =2 {/(5'^, 5 J -/(5'^- 1, 5 J ) , x=l where in the second member the summation extends over all divisors of all positive integers, and $(w)^6;„^a. In particular, 2 /(x)[tA(x)] =2/(5J, 2 [rA(x)] = number of 5„, 1=1 2=1 subject to the same inequalities. In the last equation take \l/(x)=x, a = [\/n]', we get (11). J. Hacks^® proved that, if 7?i is odd, ^W^T(l)+r(3)+T(5) + . . . +r(7n) =2[^], wjour. fur Math., 99, 1886, 324-8. •"^Mdm. .Soc. R. Sc. Li^ge, (2), 13, 1886, 318 (Melanges Math., II). •*Jomal de sciencias math, e astr., 7, 1886, 3-6. «Jour. fiir Math., 100, 1887, 459-464. Cf. Busche.»" "Acta Math., 9, 1887, 177-181. Corrections, Hacks, »^ p. 6, footnote. Chap. X] SuM AND NUMBER OF DiVISORS. 307 @(m)=or(l)+(7(3)+(r(5) + . . .+(7(m)=S^[^^], where t ranges over the odd integers ^ m. For the K and L of Lipschitz^ and G{m) =(r(l)+o-(2) + . . . -\-<T(m), it is shown that LW^(?(m)^[Vm] + [^|], !r(m)^[v^] (mod 2). J. Hacks^^ gave a geometrical proof of (11) and of Dirichlet's^^ expression for T{n), just preceding (7). He proved that the smn of all the divisors, which are exact ath powers, of 1, 2, . . . , m is m S{1^+2«+...+[a/^H. 3 = 1 He gave (pp. 13-15) several expressions for his^^ i^M, &(m), K{m). L. Gegenbauer^^" gave simple proofs of the congruences of Hacks. ^^ M. Lerch^^ considered the number \p{a, h) of divisors >6 of a and proved that [n/2] n (15) X \}/{n—p, p) =71, Si/'(n4-p, p)=2n. p=0 p=0 A. Strnad^^ considered the same formulas (15). M. Lerch^°° considered the number x(«j b) of the divisors ^6 of a and proved that [(m-l)/a] S {\J/{m—aa,k+a)—xi'm—aa,a)] <7 = k + S {\J/{m+\a, X-l)-x(m+Xa, a)) =0. x=i This reduces to his (15) for a = l, k = l orm + 1. Let (k, n; m) denote the g. c. d. {k, n) of k, n or zero, according as {k, n) is or is not a divisor of m. Then a— 1 a (16) S {i/'(m4-an, a)— ;/'(m+an, a)] = S (A;, n; m). In case m and n are relatively prime, the right member equals the number 0(a, n) of integers^ a which are prime to n. Finally, it is stated that (17) S 1/^(772 — an, a) = S x(^ — ctn, n), c= . a = a = L n J Gegenbauer,^^ Ch. VIII, proved (16) and the formula preceding it. "Acta Math., 10, 1887, 9-11. "oSitzungsber. Ak. Wiss. Wien (Math.), 95, 1887, II, 297-8. "Prag Sitzungsberichte (Math.), 1887, 683-8. "Casopis mat. fys., 18, 1888, 204. ""Compt. Rend. Paris, 106, 1888, 186. Bull, des sc. math, et astr., (2), 12, 1, 1888, 100-108, 121-6. 308 History of the Theory of Numbers. [Chap, x C. A. Laisant^"^ considered the number nk{N) of ways N can be expressed as a product of k factors (including factors unity), counting PQ . . . and QP . . . as distinct decompositions. Then n,{N) =n,_,{N)u(l-\-^y N=Upr. E. Ces^ro^"^ proved Gauss' result that the number of di\'isors, not squares, of n is asymptotic to Gtt"^ logn. Hence T{n~) is asymptotic to Stt"^ log^n. The number of decompositions of n into two factors whose g. c. d. has a certain property is asymptotic to the product of log n by the probability that the g. c. d. of two numbers taken at random has the same property. E. Busche^''^ gave a geometric proof of his^^ formula. But if we take $(x) to be a continuous function decreasing as x increases, with $(0)>0, then the number of positive divisors of y which are ^4^iy) is S[$(a:)/x], summed for x = 1, 2, . . . , with $(x) ^ 0. This result is extended to give the number of non-associated di\'isors of y+zi whose absolute value is ^4>{y, z). J. W. L. Glaisher^^ considered the excess H{n) of the number of divisors = 1 (mod 3) of n over the number of divisors =2 (mod 3), proved that H{pq) =H{p)H{q) if p, q are relatively prime, and discussed the relation of H{n) to Jacobi's^i E{n). Glaisher^^^ gave recursion formulae for H{n) and a table of its values for n = l,..., 100. L. Gegenbauer^°^ found the mean value of the number of divisors of an integer which are relatively prime to given primes pi, . . . , Pa, and are also (pr) th powers and have a complementary divisor which is di\'isible by no rth powers. Also the mean of the sum of the reciprocals of the A;th powers of those di\'isors of an integer which are prime to pi, . . . , p„ and are rth powers. Also many similar theorems. Gegenbauer^o^" expressed S^=S i^(4.T+l) and SF(4a;+3) in terms of Jacobi's symbols (A/?/) and greatest integers [ij] when F{x) is the sum of the A;th powers of those divisors ^ -x/x of x which are prime to D, or are divisible by no rih. power > 1, etc.; and gave asymptotic evaluations of these sums. J. P. Gram^°^ considered the number D„(m) of di\dsors ^m of n, the number iV"2, 3 ..(^) of integers ^n which are products of powers of the primes 2, 3, . . , and the sum -Lo. 3. . . (n) of the values of \(k) whose arguments k are the preceding N numbers, where X(2"3^ . . . ) = ( ~ 1)"+'^+ •. If p = Pi°'P2°'. . ., where the p, are distinct primes, Dj,{n) = Nin) -'LN(n/p,''^+') +SiV(n/pi"'+ ^2'^'^') - • • • • ""Bull. Soc. Math. France, 16, 1888, 150. i^Atti R. Accad. Lincei, Rendiconti, 4, 1888, I, 452-7. >MJour. fur Math., 104, 1889, 32-37. "xProc. London Math. Soc, 21, 1889-90, 198-201, 209. ^°Hbid., 395-402. See Glaisher.'" '»«Denkschi-iften Ak. Wiss. Wien (Math.), 57, 1890, 497-530. '""'Sitzungsber. Ak. Wiss. Wien (Math.), 99, 1890, Ila, 390-9. '"Det K. Danske Videnskab. Selskaba Skrifter (natur. og math.), (6), 7, 1890, 1-28, with r6- 8um6 in French, 29-34. i Chap. X] Sum and Number of Divisors. 309 In particular, if the pi include all the primes in order, we may replace N{x) by [x], the greatest integer ^x. Since there are as many divisors >a of n as there are divisors <n/a, D»+i),(^^^=€+n(a,+l), where € = 1 or according as n is or is not a divisor of p. These two formulas serve as recursion formulas for the computation of N{n). For the case of two primes pi = 2, p2 = 3, The functions L satisfy similar formulas and are computed similarly. J. W. L. Glaisher^"^ stated a theorem, which reduces for m = l to Halphen's,4° /S=o-^(n)-3o-^(n-l)+5o-^(n-3)-7(r„(n-6)+9(7^(n-10)-. . . = 2s(^'^^{cr^_,(n-l)-(l^+2V^_,(n-3)+(lH2H3V^_,(n-6)-. . . j provided m is odd, where k ranges over the even numbers 2, 4, . . ., ?n — 1, while 6 = or 6 = 1 according as n is not or is of the form g'(^ + l)/2. As in Glaisher^^ for m = l, the series are stopped before any term (Ti{n — n) is reached; but, if we retain such terms, we must set 6 = for every n and define o-i(O) by m+2 m ©''^^'^^l^-^'+i^^'K <r(0) = ©.;(o)= m+2 B. where Bi, B2, . . . are the Bernoullian numbers. Glaisher^°^ stated the simpler generalization of Halphen'*": 'S+S 2Hk^(!k) f^-'^^'') -3^+V^_,(n-l)+5^+V_,(n-3) - . . . } where the summation index k ranges over the even numbers 2, 4, ... , m — 1, and m is odd. If we include the terms <T2,_i(0) = ( — !)' 5r/(4r) in the left member, the right member is to be replaced by 5{-iy 2'"+2(m+2) "^Messenger Math., 20, 1890-1, 129-135. ^o^Ibid., 177-181. 310 History of the Theory of Numbers. [Chap, x Glaisher"" considered the set Gn[\l/(d), x(^>- •• ) of the values of \l/(d), x(cO, • • • when d ranges over all the di\isors of n, and wrote —G{\J/, x,-) for G{—\f/, — X, • •)• By use of the ^-function (12), he proved (p. 377) that the numbers given by GM-Gn-i(d, d=^l)+Gn-z{d, d^l, d=^2)-G,.M d^l, ^±2, ^±3) + . . . all cancel if n is not a triangular number, but reduce to one 1, two 2's, three 3's, . . ., g g's, each taken with the sign ( — )*~\ if n is the ^th tri- angular number ^(gr+l)/2. For example, if n = 6, whence g = S, {1, 2, 3, 6) - {1, 5; 2, 6; 0, 4} + {l, 3; 2, 4; 0, 2; 3, 5; -1, 1} = {1,2,2,3,3,3). Let \l/{d) be an odd function, so that \f/{ — d)=—\l/{d), and let 2r/(c?) denote the sum of the values of f(d) when d ranges over the divisors of r. Then the above theorem implies that 2„iA(d) -2„-i iHd) +4^{d=^ 1) ) +2„_i_2 [rPid) +rl^{d^ 1) -{-^p{d^ 2) ) -2„_i_2-3 {rpid) +yp{d^ 1) +iA(d± 2) +^(d± 3) 1 + . . . = 5( - 1)''-^ {,^(1) +2^(2) +3V^(3) + . . . +g^l^{g) } , where 5 = or 1 according as n is not or is of the form ^(^+l)/2, and where \l/{d^i) is to be replaced by \}/{d+i)-{-yl/{d — i). Taking \l/{d) =d"', where m is odd, we obtain Glaisher's^°^ recursion formula for (Tm{n), other forms of which are derived. For the function^^ ^3, we derive Un)+Un-1)+Un-S) + . . .+Q{nn-l)-{l'-2'mn-3) + (l2-22+3^)r(n-6)-...) = (-l)''-^(l^-2H3'- . . . +(-l)'-y) or 0, according as n is of the form ^(^+l)/2 or not. Next he proved a companion theorem to the first : 2d-\-7 \, -[2d-7];+ ^/ 2d+l \ ^ / 2d+3 V^ / 2d+5 \ r f all cancel if n is not a triangular number, but reduce to 1, 3, 5,. .., 2^ — 1, each taken with the sign ( — )", together with ( — l)''+^(2^+l) taken g times, if n is the gth triangular number ^(^+l)/2. For example, if n = 6, {-tt J-;?}-{-t"M-3;-"}={-'-'-- ^- ^- ^}- Hence if x(c^) be any even function, so that x(—d) = x{d), 2„{x(2d+l)-x(2d-l)l-2„_i{x(2ci+3)-x(2d-3)l+S„_3-.... = 5(-l)''-M^x(2^ + l)-x(l)-x(3)- . . . -x(2^-l)l. Taking ^(A;) = fc'""*"\ where k and m are odd, we get Glaisher's^°^ formula. "oProc. London Math. Soc, 22, 1890-1, 359-410. Results stated in London, Edinb., Dublin Phil. Mag., (5), 33, 1892, 54-61. I Chap. X] SUM AND NuMBER OF DiVISORS. 311 He proved two theorems relating to the divisors of 1, 2, . . ., w: + (G^n-3 + ^n-4 + G^n-5)( _y_3l ) — • • • all cancel with the exception of —2, — 4, . . ., — (p— 2), each taken twice, p taken p — 1 times and — 0, if p be even; but with the exception of 1, 3, ... , p — 2, each taken twice, and —p taken p — 1 times, if p be odd, where p(p+l)/2 is the triangular number next >n; all cancel with the exception of k taken k times, for A; = 1, 3, 5, . . . , p — 1, if p be even; and of —A; taken k times, for A; = 2, 4, 6, . . . , p — 1, if p be odd; here zeros are ignored. The last two theorems yield (as before) corresponding relations for any- even function x aiid any odd function xp. Applying them to x{d+l) = (d+l)"' and \l/{d)=d"*, where m is odd, and in the first case dividing by 2(m+l), and modifying the right members, we get for r=o-Jn)-2{(T^(n-l)+(r^(n-2)}+3{Mn-3) +(rm{n-4:)+(Tm{n-5)]-. .. the respective relations +3*+^ (next three) - . . . } -^-^M2(m^"^^^ +3 V2J^P -5V4/3"^ +...=^2 -p^, where s = (m+l)/2 and 0-^(0) terms are suppressed; !r=S2(^^{(r^_,(n-l)+(7^_fc(n-2)-2^ (next three) + ( 1^3') (next four) _(2^+4*^) (next five) + (lH3*+5'=) (next six) - (2H4H6') (next seven) + . . . } r ^m+i_^^m+i_^^m+i_^ _{_ (^ _ l)-+i^ if p bg eveu, i-|_ 2^+1 _ 4^+1 _gm+i_ _(p_i)m+i^ if p be odd, where, in each, A: takes the values 2, 4, . . . , m — 1. These sums of like powers of odd or even numbers are expressed by the same function of Bernoullian numbers. For m = l, the first formula becomes that by Glaisher,^^ repub- Ushed.^^ Three further (t„ formulas are given, but not applied to o-„. J. Hammond"^ wrote (n; m) = l or according as n/m is integral or fractional, also t{x) =a{x)=0 if x is fractional, and stated that 00 00 T(n/m) = S (n; jm), a (n/m) = 2 j(n; jm). y=i }=i "^Messenger Math., 20, 1890-1, 158-163. 312 History of the Theory of Numbers. [Chap, x From the sum of Euler's (f}(d) for the di\'isors d of n, he obtained a{n)=iT(jj<t>iJ), nr{n)=i<T(tjct>{j). E. Lucas^^- proved the last formulas, the result of Cesaro/* and the related one o-(n)+s„ = s„_i+2n — 1. A. Berger"^ considered the mean of the number of decompositions of 1,2,. . ., X into three or more factors, and gave long expressions for \l/{l)-\- .. .+^(7J), where i/'(A-) =2d'c?i*', summed for the solutions of ddi = k. He gave (pp. 116-125) complicated results on the mean value of (Tk{n). D. N. Sokolov and D. T. Egorov^^^" proved, by use of Bougaief's formu- las for sums extending over all the divisors of a number, the formulas in Liou\'ille's-°"'-^ series of four articles. J. W. L. Glaisher^^"* gave Zeller's^^ formula and P(n - 1) +2-P(n - 2) - 5-P(n - 5) - 7'P{n - 7) + . . . = ^|5(r3(n)-(18n-lMn)l, where 1, 2, 5, . . . are pentagonal numbers (3r=i=r)/2 and P(0) = 1. Glaisher"^ proved formulae which are greatly shortened by setting a./n)=n''(rXn)-3(n-l)V/n-l)+5(n-3)VXn-3)-7(n-6)'(T,(n-6)+.... Write Qij for ay(n). Besides the formula [of Halphen^°] aoi = 0, he gave 40 ao3-2an = 0, ao5-10ai3+ya2i = 0, 126 ,756 ^_ . 007 5~^15H H~^23 — 105031 = 0, Oo9 - 50017+720025 - 336O033 +336O041 = 0, with the agreement that o-(O) =n/3 and — t--\-l f — 1 — ^'*+l t^—1 ^3(0) = -24o-' <^M = -^' ^7(0)— ^^> <^9i0)=-^, where t = Sn+l, but did not find the general formula of this type. Next, he gave five formulas of another set, the first one being that of his earlier paper,^ the second involving the same function of 0-3 with added terms in ra{r). Finally, denoting Euler's formula (2) by Ea{n) =0, it is shown that 5Eas{n)-lSE{n(T{n)\ =0. Glaisher^^^ showed that his"^ third formula holds for all odd numbers v not expressible as a sum of three squares and hence in particular for the "T'htorie des nombres, 1891, 403-6, 374, 388. '"Nova Acta Soc. Upsal., (3), 14, 1891 (1886), No. 2, p. 63. >"^Math. Soc. Moscow, 16, 1891, 89-112, 236-255. "♦Messenger Math., 21, 1891-92, 47-8. ^''Ibid., 4^-69. ^*Ibid., 122, 126. The further results are quoted in the chapter on sums of three squares. Chap. X] SUM AND NuMBER OF DiVISORS. 313 former case v=7 (mod 8). Also the left member of the third formula equals 4jE(y-l)-3^(y-9)+5^(y-25)- . . .) when V is odd, provided E{0) = 1/4. If A'(n) denotes the sum of those divi- sors of n whose complementary divisors are odd, A'(7i)-2A'(n-l)+2A'(n-4)-2A'(n-9)+. . . =0 or (-l)'^-^ according as n is not or is a square. [Cf. Lipschitz.^^] Since A'(n) =a{n) for n odd, we deduce a formula involving c's and A"s. M. Lerch"'' proved (11) and if F{n) =2/(d), d ranging over the divisors of n. K. Th. Vahlen^^^ proved Liouville's^^ formula and Jacobi's^" result. A. P. Minin"^ proved that 2, 8, 9, 12, 18, 8g and 12p (where g is a prime >2, p a prime >3) are the only numbers such that each is divisible by the number of its divisors and the quotient is a prime. Minin^^° found that 1, 3, 8, 10, 18, 24 and 30 are the only numbers N for which the number of divisors equals the number of integers < A^ and prime to A^. M. Lerch^^^ considered the number x{a, b) and sum X{a, h) of the divisors ^b oi a, proved his^°° final formula (17) and c c a, X{m — a.n, a) = 2a{x(m — an, n)—yp{m — an, a)}, o=l a=l (18) S \l/[m-an, ^) = 2 xim-an, rn), c=\ ^^-— • a=o \ ^/ a=o L n J If 6 ranges over the divisors of n, i sV{(a-am, n)} =2^^^^, | sV|(a-am, n)\ =2(5, m) a), 't a=0 n a=0 S (am, n) =nS^-(5, (m, n)). a=l TO— 1 He quoted (p. 8) from a letter to him from Chr. Zeller that 2 a\p{m — a, a) a=l equals the sum of the remainders obtained on dividing m by the integers <m. M. Lerch^^^ proved that 'Zxf/im+p—crn, a) =2x(w+p— o-n, n) — 2 p \, i:\pim-p-pn, a)=Xx{m-p-(rn, ^) -2[^:j:YjLn+fJ' i^Casopis, Prag, 21, 1892, 90-95, 185-190 (in Bohemian). Cf. Jahrbuch Fortschritte Math., 24, 1892, 186-7. "sjour. fur Math., 112, 1893, 29. i^Math. Soc. Moscow, 17, 1893, 240-253. i2o/6id., 17, 1894, 537-544. "iPrag Sitzungsberichte (Math.), 1894, No. 11. ^Ubid., No. 32. 314 History of the Theory of Numbers. [Chap, x summed for p, o- = 0, 1, . . ., with p^<r. Also, "s (-l)V(w-a,a)=2 2 (-l)"e'(/n-a) + (-l)'"m, a-O a=0 m— 1 m PtwH 2^'(7n-a,a)=2(-l)*-Mf , a-O t = l L^J 2 Um-a, 2a) =m+^-ii- ^+2 2 (-1)' "^ , «=o ^ v=i L if J where 6' (A") is the number of odd divisors of k; yp'{n, a) is the number of div'isors >a of n whose complementary di\isors are odd; while \po{k, n) is the number of even di\'isors >/i of k. In No. 33, he expressed in terms of greatest integer functions X{\p{7n—p—an, k-jr(T)—xi'm—p — (7n,n)}} 2{^(w — a, k-\-a) — {k-]-a)\l/{m — a, k-\-a)}- a E. Busche^^^ gave a geometrical proof of Meissel's^^ (11). J. Schroder^^ obtained (11) and the first relation (15) of Lerch^^ as special cases of the theorem that 0,1,2,... m m 2 \J/r,+sin-ri: ipi, 2p,) Pi.--.Pm= » = 1 » = 1 equals the coefficient of x" in the expansion of m-l 1- n (l-a:"+0 1=0 li::ri ' (l-x^"*) n(l-x"+0 t=0 where ypr,+>{o-j ^) is the number of di\isors of a which are >^ and have a complementary divisor of the form rv-{-s{v = 0,l,. . .). He obtained 2 \f/r,+i{n-rp, p) = y J • Schroder^^ determined the mentioned coefficient of x". Schroder ^^^ proved the generaUzation of (11): p=iLpJ p=iLpJ p=2 LpJ For (j{\)-\- . . . +(r(n), Dirichlet," end, he gave the value E. Busche^^^ proved that if X = 4)(m) is an increasing (or decreasing) function whose inverse function is m=<l>(X), the divisors of the natural i»Mittheilungen Math. GeseU. Hamburg, 3, 1894, 167-172. "*Ibid., 177-188. ^Ibid., 3, 1897, 302-8. "•/feid., 3, 1895, 219-223. "Ubid., 3, 1896, 239-40. Chap. X] SuM AND NuMBEK OF DiVISORS. 315 numbers between <^(m) and a, including the limits, are the numbers x from 1 to o (or those ^a) each taken ^= [$(x)/x] times, and the numbers within the limits which are multiples of x are x,2x,. . ., ^x. For example, if a = 3, </)(m)=900/m^, then <I>(a;)=30/Vx and it is a question of the divisors of 3. . ., 17; for x = 'd, ^ = 5 and 3 is a divisor of 3, 6, 9, 12, 15. For«I>(x)=n, a = l, the theorem states that among the divisors of 1,. . ., n any one x occurs [n/x] times and that these divisors are l,...,7i;l,..., [n/2]; 1,. . ., [w/3]; etc. Hence the sum of the divisors of 1, . . . , n is and their product is n3.[n/x]=n[nA]!. x=l x=\ He proved (pp. 244-6) that the number of divisors =r (mod s) of 1, 2, . . . , n equals A-\-B, where A is the number of integers [n/x] for x = \,. . ., n which have one of the residues r, r+l,...,s — 1 (mod s), and B is the number of all divisors of 1, 2, . . . , [n/s\. The number of the divisors b of m, such that \ n n and such that 5" divides m/8, equals the number of divisors of 1, 2, . . ., n. The number of primes among n, [n/2], . . ., [n/n] equals the number of those divisors of 1, . . ., n which are primes decreased by the number of divisors which exceed by unity a prime. P. Bachmann^^^ gave an exposition of the work of Dirichlet,^^' ^^ Mer- tens,^'^ Hermite,^^ Lipschitz,^^ Ces^ro,^'' Gegenbauer,'^'^ Busche,^^^' ^" Schr6der.i24. 126 N. V. Bougaiefi29 stated that where d ranges over the divisors > 1 of n, and v = [Vn] ; -i^HMM' where d ranges over the divisors of n for which d <n. If 6 is any function, nZ-^d{d)= Xi:d{d), a y=i d where, on the left, d ranges over all the divisors of n; on the right, only over those ^ [nVj]. For 6{d) = l, this gives „a(n) = Sx(n,[y]). "8Die Analytische Zahlentheorie, 1894, 401-422, 431-6, 490-3. i^'Comptes Rendus Paris, 120, 1895, 432-4. He used $ (a, 6), ^(a, b) with the same meaning as xib, a), X{b, a) of Lerch,"^ and fi(n) for (r{n). 316 History of the Theory of Numbers. [Chap, x i>^ M. Lerch"° proved relations of the type The number of solutions of [n/ x] = [n/ (x-\-l)], x<n, is 2i/^(n-r, r)+2 x{n-p, p) (A:= -§ + Vn + 1/4 ). F. NachtikaP^^ gave an elementary proof of (15). M. Lerch^^' proved that 2 \yp{vi—aa,-)-]r'4^{m—(Ta,ra)\ remains unaltered if we interchange r and s. He proved (18) and showed that it also follows from the special case (17) . From (17) f orn = 2 he derived L. Gegenbauer^^^" proved a formula which includes as special cases four of the five general formulas by Bougaief .^^^ When x ranges over a given set S of n positive integers, the sum2/(x)[x(a:)] is expressed as sums of expressions $(p) and <J>i(p), where p takes values depending upon x, while $(2) is the sum of the values of /(x) for x in *S and x^z, and $1(2) is the analogous sum with X'^Z. F. RogeP^^ differentiated repeatedly the relation |x|<l, 00 n(i- -xr''-'=e-^, then set a: = and found that 22 - (-1)'' \<T^{2)\^ Jcr .(r)l r'=ss( -!)■/ the summations extending over all sets of a's for which CI1+CI2+ . . . +ar = i, ai+2a2+ . . . -\-ra.r = r. Starting with the reciprocals of the members of the initial relation, he obtained similarly a second formula; subtracting it from the former result, he obtained .„(r)=."+|22{nf^+^;^-i)-(-i)'np} ai!. . .a,_3!;=2l j J ""Casopis, Prag, 24, 1895, 25-34, 118-124; 25, 1896, 228-30. "i/Wd., 25, 1896, 344-6. »«Jornal de Scienciaa Math, e Astr. (TeLxeira), 12, 1896, 129-136. "^'Monatshefte Math. Phys., 7, 1896, 26. i»Sitzung8ber. Geaell. Wiss. (Math.), Prag, 1897, No. 7, 9 pp. 'y-'.J Chap. X] SuM AND NuMBER OF DiVISORS. 317 where i = 3, 5, 7,. . . in S', while the a's range over the solutions of ai+. . .+ar-3 = i, ai+2a2+. . . +(r-3)a,_3 = r. The case n = leads to relations for T{r). J. de Vries^^^" proved the first formula of Lereh's.^^'^ A. Berger"^ considered the excess \l/{k) of the sum of the odd divisors of k over the sum of the even divisors and proved that xl/{n)+x{/in-l)+\l/{n-3)+x(/{n-Q)+xl/in-10)+ . . .=0 orn, according as n is not or is a triangular number; also Euler's (2). J. FraneP^^ employed two arbitrary functions /, g and set d{n) ^Xmg( fj, F{n) = ifU), G{n) = S gU), where d ranges over the divisors of n. Then id{j) = 2 /(r)G[^] +^j(r)F\j] -F{v)G{v), where v = ['\^n]. The case f{x)=g(x) = l gives Meissel's^" (11). Next, he evaluated St?(j), where ??(n) =Xf{x)g{y)h(z), sunmied for the sets of positive integral solutions of xyz = n. In particular, ■d(n) is the number of such sets if f=g = h = l. Using Dirichlet's series, it is shown (p. 386) that 2^(i) = Si(logn+3C-l)2-3C2+6Ci+l}+e, where e is of the order of magnitude of nP^^ log n, C is Euler's constant and Ci = 0.0728 . . . [Piltz,^2 Landau^"]. FraneP^^ proved that 2^ = 1 log' P+2C log p+e+Ao, r=i r where Aq is a coefficient in a certain expansion, and ep^^^ remains in absolute value inferior to a fixed number for every p. E. Landau ^^'^ gave an immediate proof of (11) and of S7^3(^)=2T(.)r-l y = l u=l Li'J where T^iv) is the number of decompositions of v into three factors. He obtained by elementary methods a formula yielding the final result of R. D. von Sterneck^^^" proved Jacobi's^^ formula for s^. i33aK. Akad. Wetenschappen te Amsterdam, Verslagen, 5, 1897, 223. i3*Nova Acta Soc. Sc. UpsaUensis, (3), 17, 1898, No. 3, p. 26. "^Math. Annalen, 51, 1899, 369-387. i3676id., 52, 1899, 536-8. i"/6ici., 54, 1901, 592-601. i^^Sitzungsber. Ak. Wiss. Wien (Math.), 109, Ila, 1900, 31-33. 318 History of the Theory of Numbers. [Chap, x J. Franel"^ stated that, if /(n) is the number of positive integral solutions of x''y^ = n, where a, h are distinct positive integers, 2/(r) =^i^l)n^+^{^n'+o{n'-^ +6J, where'" 0{s) is of the order of magnitude of s. Taking a = l, 6 = 2, we see that /(n) is the number of di\isors of q, where <f is the greatest square divid- ing n, and that the mean of /(n) is tt"/ 6. E. Landau^^^ proved the preceding formula of Franel's. EUiott^^ of Ch, V gave formulas involving a{n) and rin). L. Kronecker^'*'^ proved that the sum of the odd divisors of a number equals the algebraic sum of all its di\isors taken positive or negative according as the complementary^ di\'isor is odd or even (attributed to Euler^); proved (pp. 267-8) the result of Dirichlet'^ and (p. 345) proved (7) and found the median value (Mittelwert) of T{n) to be log^ n+2C with an error of the order of magnitude of n~^^^when the number of values employed is of the order of n^''^. Calling a di\isor of n a smaller or greater di\dsor according as it is less than or greater than \/n, he found (pp. 343-369) the mean and median value of the sum of all smaller (or greater) di\'isors of 1, 2, . . . , A^ [cf. Gegenbauer^^], the sum of their reciprocals, and the sum of their logarithms. The mean of Jacobi's^^ E{n) is x/4 (p. 374). J. W. L. Glaisher^^^ tabulated for n = l,..., 1000 the values of the function^'^ H{n) and of the excess J{n) of the number of divisors of n which are of the form 8A- + 1 or 8A;+3 over the number of divisors of the form 8A:+5 or 8A:+7. WTien n is odd, 2J{n) is the number of representations of n by x'-\-2f. J. W. L. Glaisher"^ derived from Dirichlet's^^ formula, and also inde- pendently, the simpler formula 2 0.(s) = -pG(p)+f[5],(.)+£(?{g]}, where p = {\/n]. The case g{s) = l gives Meissel's^- formula (11), which is applied to find asymptotic formulae involving n/s — [n/s]. The error of the approximation (7) is discussed at length (pp. 38-75, 180-2). The first formula above is applied (pp. 183-229) to find exact and asymptotic formu- las for 2/(s), when/(n) is Jacobi's" E{n), Glaisher's^^^ H{n) or J{n), or the excess D{n) of the number of odd divisors of n over the number of even di\'isors, or more general functions (p. 215, p. 223) involving the number of di\isors with specified residues modulo r. G. Voronoi^^^ proved a formula like Dirichlet's^^ (7), but with e now of the same order of magnitude as -^/n log^ n. •"L'intermddiaire des math., 6, 1899, 53; 18, 1911, 52-3. "»/6ui., 20, 1913, 155. •"Vorlesungen iiber Zahlentheorie, I, 1901, 54-55. "'Messenger Math., 31, 1901-2, 64-72, 82-91. >«Quar. Jour. Math., 33, 1902, 1-75, 180-229. '«Jour. fiir Math., 126, 1903, 241-282. Chap. X] SuM AND NUMBER OF DiVISORS. 319 H. Mellin"^ obtained asymptotic expressions for2T(n), So-(n). I. Giulini^^^ noted that, if m and h are given integers, and /3(r) is the sum of the divisors d = mk-\-h of r, then i8(l) + . . .+/3(n)=2:4n/d], A; = 0, 1,. . ., [{n-h)/m], k The number and sum of the divisors d='mk-\-h oil,. . ., n are [(n-w«]r^-i [n/im+h)] /^-hA , ,%'^\-n-sh , ,-| S hj ' ^ 2 Eal ]+hi:\ +1 h fc=o LttJ r=i \ mr / s=iL ms J respectively, where ^2(2^) = W[a^+l]/2. G. Voronoi^^^" gave for T{x) the precise analytic expression x(logx+2(7-l)+i+Ma^)-2| g{t)dt^ {g{-x+ti)-g(-x-U)}idt, and (p. 515) approximations to these integrals, where ,(x)= -i log .-iC-!2i^+2^ i^x(») ,og5(^+-i-). He discussed at length the function g{x) and (pp. 467, 480-514) the asymp- totic value of Sr(n)(x — 7i)V^!- J. Schroder^^® proved that the sum of the I'th powers of 1, . . . , n is S pf-1 =n(T._i(n)+ "S p'+ r pT ^1, p=i LpJ p=t+i P=i LPJ where t = [n/2], and the accent on the last S denotes that the summation extends only over the values ^ ^ of p which are not divisors of n. E. Busche^*^ proved that, if we multiply each divisor of m by each divisor of n, the number of times we obtain a given divisor a of mn is Tiixv/a), where jjL is the g. c. d. of m,a, and v is that of n, a. A like theorem is proved for th^ divisors of mnp .... He stated (p. 233; cf. Bachmann^^^) that (Th{m)(Th{n) =SdV;,f ^j, where d ranges over the common divisors of m, n. C. Hansen ^^^ denoted by Ti{n) and T^{n) the number of divisors of n of the respective forms 4/c — 1 and 4/^ — 3, and set A„=r3(4n-3)-ri(4n-3). By use of Jacobi's B^{v, s) for ^ = 1/4, he proved that „=i '^^ ~Zx^ ^ l-s'^"-^ l-2s^+2s^«+... i«Acta Math., 28, 1904, 49. i«Giornale di mat., 42, 1904, 103-8. i^'^Annales sc. I'ecole norm, sup., (3), 21, 1904, 213-6, 245-9, 258-267, 472-480. Cf. Hardy.i^" "«Mitt. Math. GeseU. Hamburg, 4, 1906, 256-8. "V6id., 4, 1906, 229. "^Oversigt K. Danske Videnskabemes Selskabs Forhandlinger, 1906, 19-30 (in French). 320 History of the Theory of Numbers. [Chap, x and hence deduced the law of a recursion formula for A„. The law of a recursion formula for B„ = 4{T2{7i) — Ti{n)\ is obtained from S ^y S s''"+'^' cos(2n+l)^= i; (2n+l)s<2n+i)'sijj(2n+l)7, n=0 n=0 4 n=0 4 with Bo=l, which was found by use of Jacobi's d{\, s). Next, is shown to satisfy the functional equation $(ts)=lj$(s)-$(-s)}-$(s2)+2$(s*). If a convergent series liens'" is a solution $(s) of the latter, the coefficients are uniquely determined by the C4i_3(/j = 1, 2, . . . ), which are arbitrary. Hence the function 5„ is determined for all values of n by its values forn = 4/c— 3 (A: = l, 2,...). S. Wigert^^^ proved that, for sufficiently large values of n, r(n)<2', where f = (l + e) log n^-log log n, for every e >0; while there exist certain values of n above any limit for which riri) >2', s = (1 — e) log n -i-log log n. J. V. Pexider^^° proved that, if a, n are positive, a an integer, by the method used, for the case in which n is an integral multiple of a, by E. Cesaro.^° Taking a = [Vn], we have the second equation (11). Proof is given of the first equation (11) and S.[g=2.W, 2[2][!^]=S<?-.W, where d ranges over the divisors of [n]. 0. Meissner^" noted that, if m =pi". . .p/", where pi is the least of the distinct primes pi, . . . , 7?„, then ,=iPj — 1 m ,=2Pt — 1 w log m where G is finite and independent of m. If /v> 1, (Tk{m)/rn!' is bounded. W. Sierpinski^^^ proved that the mean of the number of integers whose squares divide n, of their sum, and of the greatest of them, are x^ 1, .3^ 3 , , 9C , 36 - logs — , -logn+^C, -:2lognH — rA — -^Z ^^, respectively, where C is Euler's constant. J. W. L. Glaisher^^^ derived formulas differing from his^^° earlier ones only in the replacement of d by { — lY~^d, i. e., by changing the sign of each "»Arkiv for mat., ast., fys., 3, 1906-7, No. 18, 9 pp. »"Rendiconti Circolo Mat. Palermo, 24, 1907, 58-63. "'Archiv Math. Phys., (3), 12, 1907, 199. "^prawozdania Tow. Nank. (Proc. Sc. Soc. Warsaw), 1, 1908, 215-226 (Polish). i"Proc. London Math. Soc, (2), 6, 1908, 424-^67. Chap. X] SuM AND NuMBER OF DiVISORS. 321 even divisor d. In the case of the theorems on the cancellation of actual divisors, the results follow at once from the earlier ones. But the recursion formulae for o-„ and f „ are new and too numerous to quote. Cancellation formulas (pp. 449-467) are proved for the divisors whose complementary- divisors are odd, and applied to obtain recursion formulae for the related function A/(n) of Glaisher.''^' " E. Landau ^^^ proved that log 2 is the superior limit for x= oo of log T(x)-log log a;-^log X. M. Fekete^^^ employed the determinant RkX obtained by deleting the last t rows and last t columns of Sylvester's eliminant of x'^— 1 = and a;"-l = 0. Set, for A;^n, Then 6„(A;) = 1 or according as k is or is not a divisor of n; while c„(i, k) = l if ik = n and i is relatively prime to k, but = in the contrary cases. Thus T(n)=S6„(fc), (T{n)=ikK{k), k = \ A: = l while the number and sum of those divisors d of n, which are relatively prime to the complementary divisors n/d, equal, respectively, n 1 " S Cn{i, k), - i: (i+k) c^{i, k). i, k = \ ^ i,k = l J. Schroder^" deduced from his^^"* final equation the results The final sum equals XIZI i//(s, [s/{r+l)]). P. Bachmann^^^ gave an exposition of the work of Euler,^'^ Glaisher,^^' ^^ Zeller,66 stern,^^ Glaisher,iio Liouville.^^ E. Landau ^^^ proved that the number of positive integers^ a; which have exactly n positive integral divisors is asymptotic to Aa;^/^^-^^(log log rr)'"-Vlog x, where p is the least prime factor of n, and p occurs exactly w times in n, while A depends only on n. K. Knopp^^" obtained, by enumerations of lattice points, n n w w i:Mq,k)= i:hik,q)= i:f,iq,k)+i:f2{k,q)-F{w,w), k=l k=l k=l k=l where q = [n/k] and /i(r, k)=k fij, k), h{k, s)=i f{k, j), F(r, s)=i Mr, j). y=i y=i i=i i66Handbuch. . .Verteilung der Primzahlen, I, 1909, 219-222. i^^Math. 6s Phys. Lapok (Math. phys. soc), Budapest, 18, 1909, 349-370. German transl., Math. Naturwiss. Berichte aus Ungarn, 26, 1913 (1908), 196-211, >"Mitt. Math. Gesell. Hamburg, 4, 1910, 467-470. issNiedere Zahlentheorie, II, 1910, 268-273, 284-304, 375. i^Annaes So. Acad. Polyt. do Porto, Coimbra, 6, 1911, 129-137. ""Sitzungsber. Berlin Math. Gesell., 11, 1912, 32-9; with Archiv Math. Phys. 322 History of the Theory of Numbers. [Chap, x Taking /(/j, k) = l, we obtain Meissel's" (11), a direct proof of which is also given. Taking /(/i, k)=f{h)g{hk), we get S S/0>(j/c)=S/(/c)S^(iA:), -m- I special cases of which yield niany known formulas involving Mobius's func- tion ju(n) or Euler's function (f>{n). E. Landau^^^ proved the result due to PfeifTer^°, and a theorem more effective than that by Piltz^^, having the terms replaced by 0{x°-), where, for every e>0, k-1 . E. Landau^^- extended the theorem of Piltz^^ to an arbitrary algebraic domain, defining Tk{n) to be the number of representations of n as the norm of a product of k ideals of the domain. J. W. L. Glaisher^^, generalizing his^^^ formula, proved that Sf[^]^(s)=Sf[^]^(s) + 2g[^]/(s)-F(p)G(p), where F(s) =/(!)+ . . . +/(s), G{s)=g{l)+. . . -\-g{s), p = [v^]. A similar generalization of another formula by Dirichlet^^ is proved, also analogous theorems involving only odd arguments. Glaisher^^ applied the formulas just mentioned to obtain theorems on the number and sum of powers of divisors, which include all or only the even or only the odd divisors. Among the results are (11) and those of Hacks.^®'^^ The larger part of the paper relates to asymptotic formulas for the functions mentioned, and the theorems are too numerous to be cited here. E. Landau^^ gave another proof of the result by Voronoi^^^. He proved (p. 2223) that T(n)< 471^/^ J. W. L. Glaisher^^^ stated again many of his^^ results, but without determining the limits of the errors of the asymptotic formulas. S. Minetola^^^ proved that the number of ways a product of m distinct primes can be expressed as a product of n factors is iy{»"-G)("-')"+(2)(»-2)"--(„:ii)4 T. H. GronwalP^^ noted that the superior limits for a:= oo of aM/x" (a>l), (r{x)/{x\oglogx) are the zeta function f (a) and e^, respectively, C being Euler's constant. "'Gottingen Nachrichten, 1912, 687-690, 716-731. "»Tran8. Amer. Math. Soc, 13, 1912, 1-21. »«Quar. Jour. Math., 43, 1912, 123-132. ^**Ibid., 315-377. Summary in Glaisher.i« '"Messenger Math., 42, 1912-13, 1-12. i««Il Boll, di Matematica Gior. Sc.-Didat., Roma, 11, 1912, 43-46; cf. Giomale di Mat., 45, 1907, 344-5; 47, 1909, 173, §1, No. 7. "Trans. Amer. Math. Soc, 14, 1913, 113-122. Chap. X] SuM AND NuMBEE OF DiVISORS. 323 P. Bachmann^^^ proved the final formula of Busche.^^' K. Knopp^^^ studied the convergence of 26„a:V(l — 2:"), including the series of Lambert^, and proved that the function defined in the unit circle by Euler's^ product (1) can not be continued beyond that circle. E. T. BelP'° proved that, if P is the product of all the distinct prime factors of m, and X is their number, and d ranges over all divisors of m, 6^Sr(d)T(^^ =r{m)T{Pm)T{P''m). J. F. Steffensen^'^^ proved that,^° if Ix denotes log x, S. Wigert^^^ proved, for the sum n's{n) of the divisors of n, (1 — €)e^ log log n< s{n) < {l-\-e)e^ log log n, ^s{n) = '^x-^l^{x), rP{x) = x^ 1 + 2 Ip(^), for €> and p{x)=x — [x]. For x sufficiently large, (i-e) log x<xP{x)<(l+e) log x. Besides results on Ss(a^)(x— n)*, lls{n) log x/n, he proved that X ns{n)=^+xlh\ogx-rP{x)}+0{x). E. Landau^^^ gave corrections and simplifications in the proofs by Wigert."2 E. T. Bell^^^ introduced a function including as special cases the functions treated by Liouville,^^"-^ restated his theorems and gave others. J. G. van der Corput^^^ proved, for ix(d) as in Chapter XIX, Sd'')u(d)So-„(A;)=x. S. Ramanujan^'^® proved that t{N) is always less than 2* and 2', where^" ^ = lWV+« { (IsSpf '=^*-('°^ ^)+^f'°« iVe— .-!, for Li(x) as in Ch. XVIII, and for a a constant. Also, t(N) exceeds 2*'and 2' for an infinitude of values of N. A highly composite number N is one for which TiN)>T{n) when N>n', if Ar = 2''^3"». . .p"p, then aa^as^ag^ "SArchiv Math. Phys., (3), 21, 1913, 91. "9Jour. flir Math., 142, 1913, 283-315; minor errata, 143, 1913, 50. ""Amer. Math. Monthly, 21, 1914, 130-1. i"Acta Math., 37, 1914, 107. Extract from his Danish Diss., "Analytiske Studier med Anven- delser paa Taltheorien," Kopenhagen, 1912. "HUd., 113-140. "^Gottingsche gelehrte Anzeigen, 177, 1915, 377-414. "<Univ. of Washington PubUcations Math. Phys., 1, 1915, 6-8, 38-44. '"Wiskundige Opgaven, 12, 1915, 182-4. "oProc. London Math. Soc, (2), 14, 1915, 347-409. 324 History of the Theory of Numbers. [Chap. X . . . ^flp, while ap= 1 except when A'' = 4 or 36. The value of X for which a2>ai> . . . >ax is investigated at length. The ratio of two consecutive highly composite numbers A'' tends to unity. There is a table of A"s up to t{N) = 10080. An N is called a superior highly composite number k there exists a positive number e such that N.^ N' = Ni^ for all values of A^ and No such that A'2> A'> A^i. Properties of t{N) are found for (superior) highly composite numbers. Ramanujan^" gave for the zeta function (12) the formula and found asjTnptotic formulae for j=i S r'(j), =1 Sr(jr+c), S(7„(j>i(i), A(n), for a = or 1 , where A(n) = S^r(i.) =SM(d)r (0Z),(^), summed for the di\isors d of v. If 5 is a common di\isor of u, v, xM=iMW.g)rg)=2.W.@x(0. E. Landau^'^^'' gave another asjinptotic formula for the number of de- compositions of the numbers ^ x into k factors, A' ^ 2. Ramanujan^'* wrote c^O) =^^("5) and proved that 2,,(,>.(n-,) ^^Pm^ ■ ^^l^lV J=0 r(r+s + 2) f(r+s+2) ^^+*+^ r(i-r)+r(i-^) (n) /Z(r,+,_i(n)+0(n2'^+«+^^/^), for positive odd integers r, s. Also that there is no error term in the right member if r=l, s = 1, 3, 5, 7, 11; r = 3, s=3, 5, 9; r = 5, s = 7. J. G. van der Corput^"^ wrote s for the g. c. d. of the exponents ai, a-z,... in m='n.pi''i and expressed in terms of zeta function f(i), i=2, . . ., k-\-l, 2 {a,{s)-l]/m m=2 if A' > 1 ; the sum being 1 — CifA=— 1, where C is Euler's constant. "'Messenger Math., 45, 1915-6, 81-84. "'"Sitzungsber. Ak. Wiss. Miinchen, 1915, 317-28. i^Trans. Cambr. Phil. Soc, 22, 1916, 159-173. "•Wiskundige Opgaven, 12, 1916, 116-7. Chap. X] SuM AND NUMBEK OF DiVISORS. 325 G. H. Hardy^^° proved that for Dirichlet's^^ formula (7) there exists a constant K such that e > Kn^^'^, e < — Kn^^'^, for an infinitude of values of n surpassing all limit. In Piltz's^^ formula S Tk{n) = x{akii\og xy-'-\- . . . -^-a^k} +e„ n=l ek>Kx\ ik<—Kx\ where t={k — \)/{2k). He gave two proofs of an equivalent to Voronoi's^^^'' explicit expression for T{x). Hardy^^^ wrote A(n) for Dirichlet's e in (7) and proved that,^° for every positive e, /l{n) = 0{n'^^''^) on the average, i. e., iJiA(0M«=O(n'+"*). G. H. Hardy and S. Ramanujan^^^ employed the phrase "almost all numbers have a specified property" to mean that the number of the num- bers ^ X having this property is asymptotic to a: as a; increases indefinitely, and proved that if / is a function of n which tends steadily to infinity with n, then almost all numbers have between a — 6 and a-\-h different prime factors, where a = log log n, h=f-\/d. The same result holds also for the total number of prime factors, not necessarily distinct. Also a is the normal order of the number of distinct prime factors of n or of the total number of its prime factors, where the normal order of g{n) is defined to mean f{n) if , for every positive e, (1— €)/(n)<gr(n)<(l+e)/(n) for almost all values of n. S. Wigert^^^ gave an asymptotic representation for l!,n^j:r{n){x — n)^. E. T. BelP^ gave results bearing on this chapter. F. RogeP^^ expressed the sum of the rth powers of the divisors ^g* of m as an infinite series involving Bernoullian functions. A. Cunningham^^^ found the primes p< lO'* (or 10^) for which the number of divisors of p — 1 is a maximum 64 (or 120). Hammond^^ of Ch. XI and RogeP^^ of Ch. XVIII gave formulas involv- ing (J and r. Bougaief^^' ^^ of Ch. XIX treated the number of divisors ^ m of n. Gegenbauer^° of Ch. XIX treated the sum of the pth powers of the divisors ^ m of n. i^oProc. London Math. Soc, (2), 15, 1916, 1-25. i"/&id., 192-213. i82Quar. Jour. Math., 48, 1917, 76-92. i83Acta Math., 41, 1917, 197-218. is^Annals of Math., 19, 1918, 210-6. i85Math. Quest. Educ. Times, 72, 1900, 125-6. i86Math. Quest, and Solutions, 3, 1917, 65. I CHAPTER XL MISCELLANEOUS THEOREMS ON DIVISIBILITY, GREATEST COMMON DIVISOR. LEAST COMMON MULTIPLE. Theorems on Divisibility. An anonymous author^ noted that for n a prime the sum of 1 , 2, . . . , n — 1 taken by twos (as 1+2, 1+3,. . .), by fours, by sixes, etc., when divided by n give equally often the residues 1, 2,..., n — 1, and once oftener the residue 0. The sum by threes, fives, . . . , give equally often the residues 1,. . ., n — 1 and once fewer the residue 0. J. Dienger^ noted that if w''-+'±l and (m^'-+2_ 1)7(^2 _i) are divisible by the prime p, then the sum of any 2r+l consecutive terms of the set 1, m^", m^'^", m^'^", . . . is divisible by p. The case m = 2, r = l, p = 3 or 7 was noted by Stifel (Arith. Integra). G. L. Dirichlet^ proved that when n is divided by 1, 2, . . ., n in turn the number of cases in which the remainder is less than half the divisor bears to n a ratio which, as n increases, has the limit 2 — log 4 = 0.6137 . . . ; the sum of the quotients of the n remainders by the corresponding divisors bears to n a ratio with the limit 0.423 .... Dirichlet^ generalized his preceding result. The number h of those divisors 1,2,. . . , p (p^ ti), which yield a remainder whose ratio to the divisor is less than a given proper fraction a, is -liH-B-"]} Assuming that pVn increases indefinitely with n, the limit of /i/p is a if n/p increases indefinitely with n, but if n/p remains finite is J. J. Sylvester^ noted that 2"""^^ is a factor of the integral part of /c^"*"*"^ and of the integer just exceeding h^"^, where ^ = l + \/3- N. V. Bougaief^ called a number primitive if divisible by no square >1, secondary if divisible by no cube. The number of primitive numbers ^ n is H,{n)=i:q{u)+iq{u)+.. ., <i = [VnA'], 1 1 ' where q{u) is zero if u is not primitive, but is +1 or —1 for a primitive u, according as ?/ is a product of an even or odd number of prime factors. iJour. fiir Math., 6, 1830, 100-4. ^Archiv Math. Phys., 12, 1849, 425-9. 3Abh. Ak. Wiss. BerMn, 1849, 75-6; Werke, 2, 57-58. Cf. Sylvester, Amer. Jour. Math., 5, 1882, 298-303; CoU. Math. Papers, IV, 49-54. <Jour. fur. Math., 47, 1854, 151-4. Berlin Berichte, 1851, 20-25; Werke, 2, 97'-104; French transl. by O. Terquem, Nouv. Ann. Math., 13, 1854, 396. ^uar. Joum. Math., 1, 1857, 185. Lady's and Gentleman's Diary, London, 1857, 60-1. "Comptes Rendus Paris, 74, 1872, 449-450. BuU. Sc. Math. Astr., 10, I, 1876, 24. Math. Sbornik (Math. Soc. Moscow), 6, 1872-3, I, 317-9, 323-331. 327 328 History of the Theory of Numbers. [Chap, xi To obtain the number Hoin) of secondary numbers ^n, replace square roots by cube roots in the /,. We have ffi(n)+Hi([|:,])+/fi([|]) + . . . =n, H2{n)+Ho{]^^ + . . . =n, and similarly for Hi,_i{n) given by (2) below. J. Grolous" considered the probability R^ that a number be divisible by at least one of the integers Qi,. . ., Qk, relatively prime by twos, and showed that Chr. Zeller''" modified Dirichlet's^ expression for h. The sums ,=iLs J ,=iLs + aJ are equal. The sum of the terms of the second with s>fjL = [\/p] equals the excess of the sum of the first n terms of the first over fx^ or ju^ — 1 , the latter in the case of numbers between fi^ and m^+m- Hence we may abbre- \iate the computation of h. E. Cesaro^ obtained Dirichlet's^'^ results and similar ones. The mean (p. 174) of the number of decompositions of A^ into two factors having p as their g. c. d. is 6(log N)/(p~Tr^). The mean (p. 230) of the number of di\isors common to two positive integers n, n' is 7rV6, that of the sum of their common di\isors is ilog, nn'+2C-Y^+i, where C = 0.57721 .... The sum of the inverses of the nth powers of two posi- tive integers is in mean ^^+2) where s" is defined by (12) of Ch. X. E. Cesaro^ proved the preceding results on mean values; showed that the number of couples of integers whose 1. c. m. is n is the number of divisors of n", if (a, b) and (6, a) are both counted when a^^b; found the mean of the 1. c. m. of two numbers; found the probability that in a random division the quotient is odd, and the mean of the first or last digit of the quotient; the probability that the g. c.d. of several numbers shall have specified properties. Cesaro^" noted that the probability that an integer has no divisor > 1 which is an exact rth power is l/f(r). L. Gegenbauer^° proved that the number of integers ^ x and divisible by no square is asymptotic to Gx/tt", with an error of order inferior to \/x- He proved the final formulas of Bougaief.^ 'Bull. Sc. Soc. Philomatique de Paris, 1872, 11(>-128. '"Nachrichten Gesell. Wiss. Gottingen, 1879, 265-8. 8M6m. Soc. R. Sc. de Li^ge, (2), 10, 1883, No. 6, 175-191, 219-220 (corrections, p. 343). •Annali di mat., (2), 13, 1885, 235-351, "Excursions arith. 4 I'lnfini." •"Nouv. Ann. Math., (3), 4, 1885, 421. I'Denkschr. Akad. Wien (Math.), 49, 1, 1885, 47-8. Sitzungsber. Akad. Wien, 112, II a, 1903, 562; 115, II a, 1906, 589. Cf. A. Berger, Nova Acta Soc. Upsal., (3), 14, 1891, M6m. 2, p. 110; E. Landau, Bull. Soc. Math. France, 33, 1905, 241. See Gegenbauer,".'" Ch. X. Chap. XI] MISCELLANEOUS THEOREMS ON DIVISIBILITY. 329 Gegenbauer^°" proved that the arithmetical mean of the greatest integers contained in k times the remainders on the division of n by 1, 2, . . ., n approaches k—l k\ogk-{-k-l-ki:i/x as n increases. The case A; = 2 is due to Dirichlet. Gegenbauer^^ gave formulas involving the greatest divisor t^n), not divisible by a, of the integer n. In particular, he gave the mean value of the greatest divisor not divisible by an ath power. L. Gegenbauer,^^ employing Merten's function ix (Ch. XIX) and R{a)=a — \a\, gave the three general formulas 2 sV(^V(2/) = sV(A;) - i: m - i m, Xi j/=i \y / k=i A=i it=i where X2 ranges over the divisors >n of (r — l)n+l, (r — l)n+2, . . ., rn, while Xi ranges over all positive integers for which r-\-n ~ g r n \ ' g' ' ' gj where g is the g. c. d. of r, n. Take f{x) = 1 or according as x is an sth power or not. Then the functions (1) 2 /(A;), 2/x(^)/(2/) k = \ y = \ \y/ become [-^m] and \{^), with the value if the exponent of any prime factor of X is ^0, 1 (mod s), otherwise the value ( — 1)", where a is the number of primes occurring in x to the power /cs+1. Thus 2x,(x2) = \y^ - \</V^r?^ - [i/i\ • If j{x) = or 1 according as x is divisible by an sth power or not, the func- tions (1) become Qs(w) and ix{\/x)j the former being the number of integers ^ m divisible by no sth power. If J{x) = 1 or according as x is prime or not, the functions (1) become the number of primes ^m and a simple func- tion a(x) ; then the third formula shows that the mean density of the primes loiDenkschr. Akad. Wien (Math.), 49, II, 1885, 108. "Sitzungsber. Akad. Wiss. Wien (Math.), 94, 1886, II, 714. i276id., 97, 1888, Ila, 420-6. 330 History of the Theory of Numbers. [Chap, xi If /(x)=log X, the second function (1) becomes v{x), ha\ing the value* log p when x is a power of the prime p, otherwise the value 0. Besides the resulting formulas, others are found by taking J{x) = v{x), Jacobi's symbol (A/x) in the theor>' of quadratic residues, and finally the number of repre- sentations of X by the system of quadratic forms of discriminant A. L. Saint-Loup^^ represented graphically the divisors of a number. Write the first 300 odd numbers in a horizontal line; the 300 following numbers are represented by points above the first, etc. Take any prime as 17 and mark all its multiples; we get a rectilinear distribution of these mul- tiples, which are at the points of intersection of two sets of parallel lines. J. Hacks^^ proved that the number of integers ^m which are divisible by an nth power >1 is p„(m) =S g„] -2 [^„] +S [jtTi^.] - . . . , where the A;'s range over the primes >1 [Bougaief^]. Then yp2{fn) = m—p2{'m) is the number of integers ^m not divisible by a square >1, and ^.w+^.(f)+^.(f)+...+^.([-^.) = m. A like formula holds for \p3 = 7n — p3(m), using quotients of m by cubes. L. Gegenbauer"" found the mean of the sum of the reciprocals of the A:th powers of those divisors of a term of an unlimited arithmetical progres- sion which are rth powers ; also the probabiUty that a term be divisible by no rth power; and many such results. L. Gegenbauer^^ noted that the number of integers 1, . . . , n not divisible by a Xth power is (2) Qx(n)= S^[5J/x(x). Ch. de la Valine Poussin^® proved that, if x is divided by each positive number ky-\-b^x, the mean of the fractional parts of the quotients has for x= 00 the limit 1 — C; if x is divided by the primes ^x, the mean of the fractional parts of the quotients has for x = co the limit 1 — C. Here C is Euler's constant.^ L. Gegenbauer^^ proved, concerning Dirichlet's^ quotients Q of the remainders (found on di\'iding n by 1 , 2, . . . , n in turn) by the corresponding divisors, that the number of Q's between and 1/3 exceeds the number of Q's between 2/3 and 1 by approximately 0.179n, and similar theorems. ♦Cf. Bougaief 1" of Ch. XIX. "Comptes Rendus Paris, 107. 1888, 24; ficole Norm. Sup., 7, 1890, 89. "Acta Math., 14, 1890-1, 329-336. ""Sitzungsber. Ak. Wien (Math.), 100, Ila, 1891, 1018-1053. ^Ibid., 100, 1891, Ila, 1054. Denkschr. Akad. Wien (Math.), 49 I, II, 1885; 50 I, 1885. Cf. Gegenbauer" of Ch. X. "Annale.^ de la soc. ac. Bruxellea, 22, 1898, 84-90. "Sitzungsberichte Ak. Wiaa. Wien (Math.), 110, 1901, Ila, 148-161. Chap. XI] MISCELLANEOUS ThEOEEMS ON DIVISIBILITY. 331 He investigated the related problem of Dirichlet.* Finally, he used as divisors all the sth powers ^ n and found the ratio of the number of remain- ders less than half of the corresponding divisors to the number of the others. L. E. Dickson^^" and H. S. Vandiver proved that 2">2(7i+l)(n' + l) . . ., if 1, n, n', . • • are the divisors of an odd number n> 3. R. Birkeland^^ considered the sum Sg of the qth. powers of the roots Oi, . . ., flm of z'^+Aiz"'~^-\- . . . +^m = 0. If Si, . . . , s^ are divisible by the power a^ of a prime a, then A^ is divisible by a" unless q is divisible by a. If g is divisible by a, and a^' is the highest power of a dividing q, then Afi is divisible by a^~^\ Then (n+aai) . . . (n+aa^) —n"" is divisible by a^. In particular, the product of m consecutive odd integers is of the form 1+2^^ if m is divisible by 2". E. Landau^^ reproduced Poussin's^^ proof of the final theorem and added a simplification. He then proved a theorem which includes as special cases the two of Poussin and the final one by Dirichlet^. Given an infinite class of positive numbers q without a finite limit point and such that the number of g's ^a; is asymptotic to x/w{x), where w{x) is a non-decreasing posi- tive function having x=oo w{x) then if x is divided by all the q's ^ x, the mean of the fractional parts of the quotients has for x = «> the limit 1 — C. St. GuzeP° wrote 5(n) for the greatest odd divisor of n and proved in an elementary way the asymptotic formulas [X] rfi \X\ U^\ S 5(n) =|-+0(x), S ^^ =f:r+0(l), n=l O n=l n for as in Pfeiffer^", Ch. X. A. Axer^^ considered the x'''''(^) decompositions of n into such a pair of factors that always the first factor is not divisible by a Xth power and the second factor not by a z^th power, X^2, v'^2. Then S^iix'"'" (n) is given asymptotically by a compHcated formula involving the zeta function. F. RogeP^ wrote Rx,n for the algebraic sum of the partial remainders <— [i] in (2), with n replaced by 2, and obtained Qx(2)=2P,,„-|-i2x.n, Px.n= n (l-:;^x)' where p„ is the nth prime and Pn''^ 2<p„+i. He gave relations between the values of Qx{z) for various 2's and treated sums of such values, and tabu- lated the values of ^2(2) and jB2,n for 2^288. He^^" gave many relations I'^Amer. Math. Monthly, 10, 1903, 272; 11, 1904, 38-9. "Archiv Math, og Natur., Kristiania, 26, 1904, No. 10. "Bull. Acad. Roy. Belgique, 1911, 443-472. "Wiadomoaci mat., Warsaw, 14, 1910, 171-180. "Prace mat. fiz., 22, 1911, 73-99 (Polish), 99-102 (German). Review in Bull, des sc. math., (2), 38, II, 1914, 11-13. ^Sitzungsber. Ak. Wiss. Wien (Math.), 121, Ila, 1912, 2419-52. "«/6id., 122, Ila, 1913, 669-700. See RogeP« of Ch. XVIII. 332 History of the Theory of Numbers. [Chap, xi between the QAz), relations involving the number A{z) of primes ^z, and relations involving both Q's and A's. A. Rothe'*^^ called b a maximal divisor of a if no larger divisor of a con- tains 6 as a factor. Then a/b is called the index of b with respect to a. 1| If also c is a maximal divisor of b, etc., a,b, c, . . ., I are said to form a series of composition of a. In all series of composition of a, the sets of indices are the same apart from order [a corollary of Jordan's theorem on finite groups applied to the case of a cyclic group of order a]. *Weitbrecht-^ noted tricks on the divisibility of numbers. *E. Moschietti-^ discussed the product of the divisors of a number. Each-^ of the consecutive numbers 242, 243, 244, 245 has a square factor > 1 ; likewise for the sets of three consecutive numbers beginning with 48 or 98 or 124. C. Avery and N. Verson^^ noted that the consecutive numbers 1375, 1376, 1377 are divisible by 5^ 2\ 3^ respectively. J. G. van derCorput^^ evaluated the sum of thenth powersof all integers, not divisible by a square >1, which are ^x and are formed of r prime factors of m. Greatest Common Divisor, Least Common Multiple. On the number of divisions in finding the g. c. d. of two integers, see Lame^^ et seq. in Ch. XVII; also Binet^^ and Dupre^. V.A.Lebesgue^^notedthatthel.c.m.of a, . . .,A;is(p]P3P5. . ■)/{v2ViP&- ■ ■) if pi is the product of a, ... , k, while p2 is the product of their g. c. d.'s two at a time, and ps the product of their g. c. d.'s three at a time, etc. If a, 6, c have no common divisor, there exist an infinitude of numbers ax-^b rela- tively prime to c. V. Bouniakowsky^^ determined the g. c. d. N of all integers represented by a polynomial /(x) with integral coefficients without a common factor. Since A^ divides the constant term of f{x), it remains to find the highest power p" of a prime p which divides J{x) identically, i. e., for x = 1, 2, . . . , p". Divide /(x) by Xp={x — 1). . .{x — p) and call the quotient Q and remain- der R. Then must R^O (mod p") for x = l,. . ., p, so that each coefficient of R is divisible by p", and iu = Mu vvhere p"' is the highest power of p divid- ing the coefficients of i?. If /ii = l, wehaveju= 1. Next, let /ii>l. Divide ^^Zeitschrift Math.-Xaturw. Unterricht, 44, 1913, 317-320. "Vom Zahlenkunststiick zur Zahlentheorie, Korrcspondenz-Blatt d. Schulen Wiirttembergs, Stuttgart, 20, 1913, 200-6. "Suppl. al Periodico di Mat., 17, 1914, 115-6. i^Math. Quest. Educ. Times, 36, 1881, 48. 2'Math. Miscellany, Flushing, N. Y., 1, 1836, 370-1. "Nieuw Archief voor Wi.skunde, (2), 12, 1918, 213-27. "Jour, de Math., (1), 6, 1841, 453. »Ibid., (1), 11, 1846, 41. '*Nouv. Ann. Math., 8, 1849, 350; Introduction k la th6orie des nombres, 1862, 51-53; Exercises d'analyse num^rique, 1859, 31-32, 118-9. "M<5m. acad. sc. St. P^tersbourg, (6), ac. math, et phys. 6 (so. math. phys. et nat. 8), 1857 305-329 (read 1854); extract in Bulletin. 13, 149. Chap. XI] GREATEST COMMON DiVISOR. 333 Q by (x — p — l)... {x — 2p) and call the quotient Q' and remainder R'. Then must X^R'+XopQ'^O and hence XpR'=0 (mod p"). Thus if 1^2 is the exponent of the highest power of p dividing the coefficients of R', we have At^M2 + l- In general, if [j.^ and X^-i are the exponents of the highest powers of p dividing the coefficients of the remainder i?'^~^^ and X^k-Dp identically, then fi^ iJLk+\k-i' Finally, if l = [m/p], /x^X^. The extension to several variables is said to present difficulties. [For simpler methods, see Hensel^^ and Borel.^^] It is noted (p. 323) that are identically divisible by p". It is conjectured (p. 328) that/(a;)/iV repre- sents an infinitude of primes when f{x) is irreducible. E. Cesaro" and J. J. Sylvester^^ proved that the probabihty that two numbers taken at random from 1 , . . . , n be relatively prime is Q/tt^ asymp- totically. L. Gegenbauer^^ gave 16 sums involving the g. c. d. of several integers and deduced 37 asymptotic theorems such as the fact that the square of the g. c. d. of four integers has the mean value IS/tt^. He gave the mean of the kth. power of the g. c. d. of r integers. J. Neuberg^^" noted that, if two numbers be selected at random from 1, . . .,N, the probability that their sum is prime to N is k=cf){N) 0Tk/{N—l) according as N is odd or even. T. J. Stieltjes,^^ starting with a set of n integers, replaced two of them by their g. c. d. and 1. c. m., repeated the same operation on the new set, etc. Finally, we get a set such that one number of every pair divides the other. Such a reduced set is unique. The 1. c. m. of a, . . . , ? can be expressed (pp. 14-16) as a product a'. . J' of relatively prime factors divi- ding a,...,l, respectively. The 1. c. m. (or g. c. d.) oi a,h,. . .,1 equals the quotient oi P = ab. . .Ihy the g. c. d. (or 1. c. m.) of P/a, P/b, . . . , P/l. E. Lucas^^ gave theorems on g. c. d. and 1. c. m. L. Gegenbauer^^" considered in connection with the theory of primes, the g. c. d. of r numbers with specified properties. J. Hacks^^ expressed the g. c. d. of m and n in the forms ql^]-.n^.^n, 2'|;[f]+2'}:g]-2[|][|]-^. where € = or 1 according as m, n are both or not both even. J. Hammond^^ considered arbitrary functions / and F oi p and a, such "Mathesis, 1, 1881, 184; Johns Hopkins Univ. Circ, 2, 1882-3, 85. "Johns Hopkins Univ. Circ, 2, 1883, 45; Comptes Rendus Paris, 96, 1883, 409; Coll. Papers, 3, 675; 4, 86. "Sitzungsberichte Ak. Wiss. Wien (Math.) 92, 1885, II, 1290-1306. 39«Math. Quest. Educ. Times, 50, 1889, 113-4. ^''Sur la theorie des nombres, Annales de la fac. des sciences de Toulouse, 4, 1890, final paper. ^iTheorie des nombres, 1891, 345-6; 369, exs. 4, 5. ""Monatshefte Math. Phys., 3, 1892, 319-335. *2Acta Math., 17, 1893, 208. ^Messenger Math. 24 1894^5 17-19. 334 History of the Theory of Numbers. [Chap, xi that /(p, 0) = 1 , F{p, 0)=0, and any two integers m=np", n='n.p^, where the p's are distinct primes and, for any p, a ^ 0, /3 ^ 0. Set rP{7n)=Uf{p,a), $=2F(p, a). By the usual proof that mn equals the product of the g. c. d. M of m and n by their 1. c. m. n, we get yPim)4/{n)=^P{M)xl/(jx), <I>(w)+$(n) =$(M)+$(iu). In particular, if m and n are relatively prime, yl/{m)\p{n) =\p{vin), $(w)+<l>(n) =<l>(mn). These hold if i/' is Euler's 0-function, the sum o-(m) of the divisors of m or the number T{m) of divisors of ?n ; also, if ^{m) is the number of prime factors of vi or the sum of the exponents a in m = Iip''. K. Hensel^ proved that the g. c. d. of all numbers represented by a polynomial F{u) of degree n with integral coefficients equals the g. c. d. of the values of F{u) for any n+1 consecutive arguments. For a polynomial of degree ni in Ui, 712 in ^2, • ■ • we have only to use ni + 1 consecutive values of ui, 712+ 1 consecutive values of U2, etc. F. Klein"*^ discussed geometrically Euclid's g. c. d. process. F. ^Vlertens^^ calls a set of numbers primitive if their g. c. d. is unity. If 7719^0, k>\, and ai,. . ., o^, m is a primitive set, we can find integers Xi,. . ., Xk so that ai-\-mxx,. . ., ak+mxk is a primitive set. Let d be the g. c. d. of fli, . . ., Oi- and find 5, ji so that db-\-vi^ = \. Take integral solu- tions a of OittiH-. . .+akak = d and primitive solutions ^i not all zero of aij3i+ . . . +aii3/; = 0. Then 7i=/3,+6a,('i = l,. . ., k) is a primitive set. Determine integers ^ so that 71^1+. . .+7*^^ = 1 and set a:,=/i^<. Then Ci+TTix, form a primitive set. R. Dedekind^^ employed the g. c. d. d oi a,h, c; the g. c. d. (6, c) =Oi, (c, a) = 61, (a, h) = Ci. Then a' = ajd, h' = hi/d, c' = C]/d are relatively prime in pairs. Then cf6'c' is the 1. c. m. of 61, Ci, and hence is a divisor of a. Thus a = dh'c'a", h = dc'a'b", c = da'h'c". The 7 numbers a', . . .,a" ,. . .,d are called the " Kerne" of a, h, c. The generalization from 3 to n numbers is given. E. Borel'*^ considered the highest power of a prime p which di\'ides a polynomial P{x, y,. . .) with integral coefficients for all integral values of X, y,. . .. If each exponent is less than p, we have only to find the highest power of p dividing all the coefficients. In the contrary case, reduce all exponents below p by use of x^ = x-\-pxi,Xi' = Xi -\-px2,. . . and proceed as above with the new polynomial in x, Xi, X2,...,y,yi,.... Then to find all arithmetical divisors of a polynomial P, take as p in turn each prime less than the highest exponent appearing in P. L. Kronecker^^ found the number of pairs of integers i, k having t as their g. c. d., where l^i^m, l^k^n. The quotient of this number by «Jour. fur Math., 116, 1896, 350-6. "Ausgewahlte Kapitel der Zahlentheorie, I, 1896. *«Sitzung8berichte Ak. Wiss. Wien (Math.), 106, 1897, II a, 132-3. *^Ueber Zerlegungen von Zahlen durch d. grossten gemeinsamen Teller, Braunschweig, 1897. "BuU. Sc. Math. Astr., (2), 24 I, 1900, 75-80. Cf. Borel and Drach'^ of Ch. III. "Vorlesungen uber Zahlentheorie, I, 1901, 306-312. i Chap. XI] GREATEST COMMON DiVISOR. 335 mn is the mean. When m and n increase indefinitely, the mean becomes Q/iirH"^). The case ^=1 gives the probability that two arbitrarily chosen integers are relatively prime; the proof in Dirichlet's Zahlentheorie fails to establish the existence of the probability. E. DintzP° proved that the g. c. d. A(a, . . ., e) is a linear function of a, . . . , e, and reproduced the proof of Lebesgue's^^ formula as given in Merten's Vorlesungen iiber Zahlentheorie and by de Jough.^^ A. Pichler,^°" given the 1. c. m. or g. c. d. of two numbers and one of them, found values of the other number. J. C. Kluyver^^ constructed several functions z (involving infinite series or definite integrals) which for positive integral values of the two real variables equals their g. c. d. He gave to Stern's^^ function the somewhat different form [;r] / \ W. Sierpinski^^ stated that the probability that two integers ^n are relatively prime is . „ p -,2 contrary to Bachmann, Analyt. Zahlentheorie, 1894, 430. G. Darbi^^ noted that if a = (a, N) is the g. c. d. of a, N, (iV,abc...)=a(6,^)(c,^(^_^/J and gave a method of finding the g. c. d. and 1. c. m. of rational fractions without bringing them to a common denominator. E. Gelin^® noted that the product of n numbers equals ah, where a is the 1. c. m. of their products r at a time, and h is the g. c. d of their products n — r at a time. B. F. Yanney^'^ considered the greatest common divisors Di, D2, ... of tti, . . . , a„ in sets of k, and their 1. c. m.'s Li, L2, . . . . Then HA Lt' ^ (ai . . . «n)^ ^ n D ^-^L„ 5 = (^y c = (^~ I) . The limits coincide ii k = 2. The products have a single term iik = n. P. Bachmann^^ showed how to find the number N obtained by ridding a given number n of its multiple prime factors. Let d be the g. c. d. of n and 0(n). If d = n/d occurs to the rth power, but not to the (r+l)th power in n, set ni = n/5^ From rii build di as before, etc. Then N = 86182 .... "Zeitschrift fiir das Realschulwesen, Wien, 27. 1902, 654-9, 722. 6o«76id., 26, 1901, 331-8. "Nieuw Archief voor Wiskunde, (2), 5, 1901, 262-7. ^^K. Ak. Wetenschappen Amsterdam, Proceedings of the Section of Sciences, 5, II 1903, 658- 662. (Versl. Ak. Wet., 11, 1903, 782-6.) "Jour, flir Math., 102, 1888, 9-19. "Wiadomosci Mat., Warsaw, 11, 1907, 77-80. ^^Giornale di Mat., 46, 1908, 20-30. S6I1 Pitagora, Palermo, 16, 1909-10, 26-27. "Amer. Math. Monthly, 19, 1912, 4-6. 6«Archiv Math. Phys., (3), 19, 1912, 283-5. 336 History of the Theory of Numbers. [Chap. XI Erroneous remarks^^ have been made on the g. c. d. of 2"" — 1, 3"" — 1. ]\I. Lecat^° noted that, if a,j is the 1. c. m. of i and j, the determinant loyl was evaluated by L. Gegenbauer,^^ who, however, used a law of multi- pHcation of determinants valid only when the factors are both of odd class. J. Barinaga®^" proved that, if 5 is prime to iV = nk, the sum of those terms of the progression A'', N-\-d, iV+25, . . . , which are between nk and n{k-\-hd) and which have with n = mp the g. c. d. p, is ^n<l){n/p){2k-\-hd)h. R. P. Willaert®- noted that, if P{n) is a polynomial in n of degree p with integral coefficients, f{n)=aA'"'-\-P{n) is divisible by D for every integral value of 7} if and only if the difference A''f{0) of the Ath order is di\"isible by D for k = 0, 1,. . ., p-\-l. Thus, if p = l, the conditions are that /(O), /(l),/(2) be divisible by D. *H. Verhagen^^ gave theorems on the g. c. d. and 1. c. m. H. H. ]\Iitchell^ determined the number of pairs of residues a, b modulo X whose g. c. d. is prime to X, such that ka, kb is regarded as the same pair as a, b when k is prime to X, and such that X and ax + by have a given g. c. d. W. A. Wijthoff^^ compared the values of the sums S (-l)'"-WF{(w, a)}, "s m'F{{m,a)}, s = l, 2, m=l m=l where {m, a) is the g. c. d. of m, a, while F is any arithmetical function. F. G. W. Brown and C. M. Ross^^ wTote h, U, ...,ln for the 1. c. m the pau-s Ai, A^; A^, Az; . . . ; A„, Ai, and gi, g^, ■ ■ ., gn for the g these pairs, respectively. If L, G are the 1. c, m. and g. c. d. of Ai, A„, then gm . . .gn = G'', . c ^2, of d. of 9i92 C. de Polignac^^ obtained for the g. c. d (a\btJi)={a,by{\,fx).(--\ \{a, b) (X, m) Sylvester^* and others considered the g. c. d 9n G' (a, 6) of a, 6 results like fi \ / b X J \(a, 6)' (X, mV 6)' (X, m)> of Z)„ and Z)„+i where D^ is the n-rowed determinant whose diagonal elements are 1, 3, 5, 7, . . ., and having 1, 2, 3, 4, ... in the line parallel to that diagonal and just above it, and units in the parallel just below it, and zeros elsewhere. On the g. c. d., see papers 33-88, 215-6, 223 of Ch. V, Cesaro" of Ch. X, Cesaro^' ' of Ch. XI, and Kronecker^^ of Ch. XIX. "L'interm6diaire des math., 20, 1913, 112, 183-4, 228; 21, 1914, 36-7. ^''Ibid., 21, 1914, 91-2. •'Sitzungs. Ak. Wiss. Wien (Math.), 101, 1892, II a, 425-494. ""Annaes Sc. Acad. Polyt. do Porto, 8, 1913, 248-253. ^Mathesis, (4), 4, 1914, 57. "Nieuw Tijdschria voor Wiskunde, 2, 1915, 143-9. "Annals of Math., (2), 18, 1917, 121-5. "Wiskundige Opgaven, 12, 1917, 249-251. "Math. Quest, and Solutions, 5, 1918, 17-18. «'Xouv. Corresp. Math., 4, 1878, 181-3. 6»Math. Que.st. Educ. Times, 36, 1881, 97-8; correction, 117-8. CHAPTER XII. CRITERIA FOR DIVISIBILITY BY A GIVEN NUMBER. In the Talmud^ lOOa+6 is stated to be divisible by 7 if 2a+b is divis- ible by 7. Hippolytos^", in the third century, examined the remainder on the division of certain sums of digits by 7 or 9, but made no appHcation to checking numerical computation. Avicenna or Ibn Sina (980-1037) is said to have been the discoverer of the familiar rule for casting out of nines (cf . Fontes^^) ; but it seems to have been of Indian origin.-^'' Alkarkhi^^ (about 1015) tested by 9 and 11. Ibn Musa Alchwarizmi^'* (first quarter of the ninth century) tested by 9. Leonardo Pisano^^ gave in his Liber Abbaci, 1202, a proof of the' test for 9, and indicated tests for 7, 11. Ibn Albanna^-'^ (born about 1252), an Arab, gave tests for 7, 8, 9. In the fifteenth century, the Arab Sibt el-Maridini^'' tested addition by casting out multiples of 7 or 8. Nicolas Chuquet^^ in 1484 checked the four operations by casting out 9's. J. Widmann^'' tested by 7 and 9. Luca Paciuolo^ tested by 7, as well as by 9, the fundamental operations, but gave no rule to calculate rapidly the remainder on division by 7. Petrus Apianus^" tested by 6, 7, 8, 9. Robert Recorde^'' tested by 9. Pierre ForcadeP noted that to test by 7 = 10 — 3 we multiply the first digit by 3, subtract multiples of 7, add the residue to the next digit, then multiply the sum by 3, etc. Blaise Pascal^ stated and proved a criterion for the divisibility of any number N by any number A. Let ri, r2, 7*3, . . . , be the remainders obtained when 10, lOfi, lOrg, ... are divided by A. Then iV = a+ 106 + 100c+ ... is divisible by A if and only if a-\-rib-\-r2C+ . . . is divisible by A. 'Babylonian Talmud, Wilna edition by Romm, Book Aboda Sara, p. 96. i«M. Cantor, Geschichte der Math., ed. 3, I, 1907, 461. ^^Ibid., 511, 611, 756-7, 763-6. i^Cf. Carra de Vaux, Bibliotheca Math., (2), 13, 1899, 33-4. i<^M. Cantor, Geschichte der Math., ed. 3, I, 1907, 717. i^Scritti, 1, 1857, 8, 20, 39, 45; Cantor, Geschichte, 2, 1892, 8-10. '/Le TaUfhys d'Ibn Albanna public et traduit par A. Marre, Atti Accad. Pont. Nuovi Lincei, 17, 1863-4, 297. Cf. M. Cantor, Geschichte Math., I, ed. 2, 757, 759; ed. 3, 805-8. iffLe Triparty en la science de nombres. Bull. Bibl. St. Sc. Math., 13, 1880, 602-3. ^''Behede vnd hubsche Rechnung . . . , Leipzig, 1489. ^Summa de arithmetica geometria proportion! et proportionalita, Venice, 1494, f. 22, r. 2"Ein newe. . .Kauffmans Rechnung, Ingolstadt, 1527, etc. ^^The Grovnd of Artes, London, c. 1542, etc. ^L'Arithmeticqve de P. Forcadel de Beziers, Paris, 1556, 59-60. *De numeris multiphcibus, presented to the Acad^mie Parisienne, in 1654, first published in 1665; Oeuvres de Pascal, 3, Paris, 1908, 311-339; 5, 1779, 123-134. 337 338 History of the Theory of Numbers. [Chap, xii D'Alembert^ noted that if N = A-10'"+B-W-\-. ..+E is divisible by 10-6, then Ab"'+Bb"-\- . ..+E is divisible by 10-6; if A' is divisible by 10+6, then A(-6)'"+B(-6)"+ . . . +^ is divisible by 10+6. The case 6 = 1 gives the test for divisibiUty by 9 or 11. By separating A'' into parts each with an even number of digits, N = A-10"'+ . . . +E, where m, . . .are even; then if A^ is di\'isible by 100-6, Ah"*^^ -\- . . . + E is divisible by 100-6. De Fontenelle^ gave a test for divisibility by 7 which is equivalent to the case 6 = 3 of D'Alembert; to test 3976 multiply the first digit by 3 and add to the second digit; it remains to test 1876. For proof see F. Sanvitali, Hist. Literariae Italiae, vol. 6, and Castelvetri.^ G. W. Kraft^ gave the same test as Pascal for the factor 7. J. A. A. Castelvetri^ gave the test for 99: Separate the digits in pairs, add the two-digit components, and see if the sum is a multiple of 99. For 999 use triples of digits. Castelvetri^ tested 1375, for example, for the factor 11 by noting that 13+75 = 88 is divisible by 11. If the resulting sum be composed of more than two digits, pair them, add and repeat. To test for the factor 111, separate the digits into triples and add. The proof follows from the fact that lO-*" has the remainder 1 when divided by 11. J. L. Lagrange^° modified the method of Pascal by using the least residue modulo A (between — .4/2 andyl/2) in place of the positive residue. He noted that if a number is written to any base a its remainder on division by a — 1 is the same as for the sum of its digits. J. D. Gergonne^^ noted that on di\dding iV = Ao+Ai6"*+A26^'"+ . . ., written to base 6, by a di\'isor of 6'" — 1, the remainder is the same as on dividing the sum A0+A1+A2+ ... of its sets of m digits. Similarly for 6'"+l and A0-A1+A2-A3+ • • •• C. J. D. HilP^ gave rules for abbre\dating the testing for a prime factor p, for p<300 and certain larger primes. C. F. Liljevalch^^a ^^^^^ ^^^^^ jf lO^a-/? is di\^sible by p then a- 10^6 will be a multiple of p if and only if aa — /36 is a multiple of p. J. ]\I. Argardh" used Hill's symbols, treating divisors 7, 17, 27, 1429. F. D. Herter^^ noted that a + 106+100c+ ... is divisible by 10n±l if 'Manuscript R. 240* 6 (8°), Bibl. Inst. France, 21, ff. 316-330, Sur une propri^t^ des nombres. •Histoire Acad. Paris, ann^e 1728, 51-3. 'Comm. Ac. Sc. Petrop, 7, ad annos 1734-5, p. 41. »De Bononiensi Scientiarum et Artium Institute atque Academia Comm., 4, 1757; commen- tarii, 113-139; opuscula, 242-260. •De Bononiensi Scientiarum et Artium Institute atque Academia Comm., vol. 5, 1767, part 1, pp. 134-144; part 2, 108-119. "Lemons 6\6m. sur les math, donn^es k I'^cole normale en 1795, Jour, de I'^cole polytechnique, vols. 7, 8, 1812, 194-9; OemTes, 7, pp. 203-8. "Annales de math, (ed., Gergonne), 5, 1814-5, 170-2. "Jour, fur Math., 11, 1834, 251-261; 12, 1834, 355. Also, De factoribua numerorum com- positonim dignoscendis, Lund, 1838. "<»De factoribus numerorum compositorura dignoscendis, Lund, 1838. "De residuis ex divisione. . ., Diss. Lund, 1839. "Ueber die Kennzeichen der Theiler einer Zahl, Progr. Berlin, 1844. Chap. XII] CRITERIA FOR DIVISIBILITY. 339 a=F&/nH-c/n^=F . . . is divisible by 10n±l, with a like test for 10n±3 (replacing 1/n by 3/n), and deduced the usual tests for 9, 11, 7, 13, etc. A. L. Crelle^^ noted that to test XmA""-^ . . . +XiA-i-Xo for the divisor s we may select any integer n prime to s, take r=nA (mod s), and test for the divisor s. For example, if A = 10, s = 7, 10^=— 1 (mod 7), so that Xo — Xi-\-X2—. . . ±0:^ is to be tested for the divisor 7, where Xq, . . .are the three-digit components of the proposed number from right to left. Simi- larly for s=9, 11, 13, 17, 19. A. Transon^^ gave a test for the divisibility of a number by any divisor of 10"-n±l. A. Niegemann^^ noted that 354578385 is divisible by 7 since 35457 -f 2X8385 is divisible by 7. In general if the number formed by the last m digits of A^ is multiplied by k, and the product is added to the number de- rived from N by suppressing those digits, then N is divisible by d if the resulting sum is divisible by d. Here k{0<k<d) is chosen so that 10'"/: — 1 is divisible by d. Thus k = 2 if m = 4, d = 7. Many of the subsequent papers are listed at the end of the chapter. H. Wilbraham^^ considered the exponent p to which 10 belongs modulo m, where m is not divisible by 2 or 5. Then the decimal for 1/m has a period of p digits. If any number N be marked off into periods of p- digits each, beginning with units, so that A^ = ai + 10^a2+10^^a34- • • •, then ai-\-a2-]- ■ ■ ■ = N (mod m), and N is divisible by m if and only if «i+<J2+ • • ■ is divisible by m. E. B. Elliott^^ let 10'' = MD+r^,. Thus iV = 10%-h . . . +10ni+no is divisible by D if N ='ZfnjMD-{-'Znjrj is divisible by D. The values of the r's are tabulated for D = S, 7, S, 9, 11, 13, 17. A. Zbikowski^° noted that N = a-\-10kis divisible by 7 if k — 2a is divis- ible by 7. If 8 is of the form lOn+1, N = a-\-10k is divisible by 5 if A; — na is divisible by d ; this holds also if 5 is replaced by a divisor of a number 10n+ 1 . V. ZeipeP^ tests for a divisor h by use of nh = 10d-\-l. Then 10a2+ai is divisible by 6 if a2 — aid is divisible by b. J. C. Dupain^^ noted, for use when division by p — 1 is easy, that N={p — 1)Q+R is divisible by p if R — Q is divisible by p. F. Folie^^ proved that if a, c are such that ak'^ck = mp then AB-\-C is divisible by the prime p = aB-{-c if Ak'=^Ck = m'p, provided a, c, k, k' are isjour. fur Math., 27, 1844, 125-136. 16N0UV. Ann. Math., 4, 1845, 173-4 (cf. 81-82 by O. R.). i^Entwickelung u. Begrlindung neuer Gesetze iiber die Theilbarkeit der Zahlen. Jahresber. Kath. Gym. Koln, 1847-8. i^Cambridge and Dublin Math. Jour., 6, 1851, 32. "The Math. Monthly (ed. Runkle), 1, 1859, 45-49. "oBull. ac. sc. St. Petersbourg, (3), 3, 1861, 151-3; Melanges math. astr. ac. St. P^tersbourg, 3, 1859-66, 312. 2iOfversigt finska vetensk. forhandl., Stockhohn, 18, 1861, 425-432. «Nouv. Ann. Math., (2), 6, 1867, 368-9. 23M6m. Soc. Sc. Liege, (2), 3, 1873, 85-96. 340 History of the Theory of Numbers. [Chap, xii not multiples of p. Application is made to the primes p^37. Again, if p is a prime and aB-+cB+d = ak"-\-ck'-\-dk = Ak"-{-Ck'-\-Dk = mp, where k, k', k" are prime to p, then AB~-\-CB-[-D is divisible by p provided k'^ — kk" is a multiple of p. C. F. IMoller and C. Holten-^ would test the divisibility of n by a given prime p by seeking a such that ap= =*= 1 (mod 10) and subtracting from n such a multiple of ap that the difference ends with zero. L. L. Hommel"-^ made remarks on the preceding method. V. SchlegeP^ noted that if the di\isor to be tested ends with 1, 3, 7 or 9, its product by 1, 7, 3 or 9 is of the form (i = lOX+1. Then a, with the final digit u, is divisible by d\i ai = {a — ud)/\0 is. Then treat Oj aswe did a, etc. P. Otto"^ would test Z for a given prime factor p by seeking a number n such that if the product by n of the number formed by the last s digits of Z be subtracted from the number represented by the remaining digits, the remainder is di\'isible by p if and only if Z is. ^Material is tabulated for the application of the method when p<100. N. V. Bougaief-^" noted that a^. . .Ci to base B is divisible by D if fli . . .a„ to base d is divisible by D, where dB= 1 (mod D). For jB = 10 and Z) = 10/1+9, 1, 3, 7, we may take c? = n + l, 9?i+l, 3nH-l, 7n+5, respec- tively. Again, kB--\-aB-\'h is di\'isible by D if kB-\-a-\-hd is divisible. W. Mantel and G. A. Oskamp'^ proved that, to test the di\isibility of a number to any base by a prime, the value of the coefficient required to eliminate one, two, . . . digits on subtraction is periodic. Also the number of terms of the period equals the length of the period of the periodic fraction arising on division by the same prime. G. Dostor-^'' noted that \{)t-\-u is divisible by any divisor a of 10A± 1 if t=^Au is di\dsible by a. [A case of Liljevalch^-''.] Hocevar^^ noted that if N, wTitten to base a, is separated into groups Gi, (x2, . . . each of q digits, N is di\'isible by a factor of a'+l if Gi — G2+G3 - ... is divisible. Thus, for a = 2, g = 4, A'' = 104533, or 11001100001010101 to base 2 is divisible by 17 since 0101-0101 + 1000-1001 + 1 = 0. J. Delboeuf^° stated that if p, q are such that pa-\-qh is a multiple of Z) and if N = Aa.-\-B^ is a multiple of Z) = aa + 6/3, then pA+qB is a multiple of Z). E. Catalan {ibid., p. 508) stated and proved the preceding test in the following form: If a, h and also a', h' are relatively prime, and iV = aa'+66', Nx = Aa-\-Bh, Nx' = A'a'+B'h', then AA'-\-BB' is a multiple of A^ (and a sum of 2 squares if N is). "Tidsskrift for Math., (3^, 5, 1875, 177-180. «*Tidsskrift for Math., (3), 6, 1876, 15-19. "Zeitschrift Math. Phys., 21, 1876, 365-6. »'Zeitschrift Math. Phys., 21, 1876, 366-370. ^"''Mat. Sbornik (Math. Soc. Moscow), 8, 1876, I, 501-5. "Nieuw Archief voor Wiskunde. Amsterdam, 4, 1878, 57-9, 83-94. "-^.Ajchiv Math. Phys., 63, 1879, 221-4. "Zur Lehre von der Teilbarkeit. . ., Prog. Imisbruck, 1881. "La Revue Scientifique de France, (3), 38, 1886, 377-8. Chap. XII] CRITERIA FOR DIVISIBILITY. 341 Noel (ibid., 378-9) gave tests for divisors 11, 13, 17,. . ., 43. Bougon {ibid., 508) gave several tests for the divisor 7. For example, a number is divisible by 7 if the quadruple of the number of its tens dimin- ished by the units digit is divisible by 7, as 1883 since 188-4 — 3 = 749 is divisible by 7. J. Heilmann (ibid., 187) gave a test for the divisor 7. P. Breton and Schobbens {ibid., 444-5) gave tests for the divisor 13. S. Dickstein^^ gave a rule to reduce the question of the divisibility of a number to any base by another to that for a smaller number. A. Loir^2 gave a rule to test the divisibihty of N, having the units digit a, by a prime P. From {N — a)/10, subtract the product of a by the number, say (mP — 1)/10, of tens in such a multiple mP of P that the units digit is 1. To the difference obtained apply the same operation, etc., until we exhaust N. If the final difference be P or 0, N is divisible by P. R. Tucker^^ started with a number N, say 5443, cut off the last digit 3 and defined ^2 = 544 — 2-3 = 538, ^3 = 53 — 2-8, etc. If any one of the ^^'s is divisible by 7, N is divisible by 7. R. W. D. Christie (p. 247) extended the test to the divisors 11, 13, 17, 37, the respective multiphers being 1, 9, 5, 11, provided always the number tested ends with 1, 3, 7 or 9. R. Perrin^^ would find the minimum residue of N modulo p as follows. Decompose N, written to base x, into any series of digits, each with any number of digits, say A, Bi, Cj,. . ., where Bi has i digits. Let p be any integer prime to x and find qi so that qiX^^ 1 (mod p). Let a be any one of the integers prime to p and numerically <p/2. Let j8 be the ith integer following a in that one of the series containing a which are defined thus: as the first series take the residues modulo p of 1, g, g^, . . . ; as the second series take the products of the preceding residues by any new integer prime to p; etc. Let y be the jth integer following /S in the same series, etc. Then N' = Aa-\-BS-{-Cjy-\-... is or is not divisible by p according as A'' is or not. By repetitions of the process, we get the minimum residue of N modulo p. The special case A-{-Biqi, with p a prime, is due to Loir.^^ Dietrichkeit^^ would test Z = \Ok-\-a for the divisor n by testing k — xa, where 10a:+l is some multiple of n. To test Z (pp. 316-7) for the divisor 7, test the sum of the products of the units digit, tens digit, ... by 1, 3, 2, 6, 4, 5, taken in cyclic order beginning with any term (the remainders on con- verting 1/7 into a decimal fraction) . Similarly for 1/n, when n is prime to 10. J. Pontes^ ^ would test N for a divisor M by using a number <iV and = N (mod M), found as follows. For the base B, let q be the absolutely least residue of B"" modulo M. Commencing at the right, decompose N into sets of m digits, as X,„, . . . , a^, and set f{x)=a^x"'+^jn^"'~^-\- . . . +X;;,, whence N=f{B'^). By expanding N=f{q+M^), we see that f{q) is the desired number < N and = N (mod M) . S. Levanen^^ gave a table showing the exponent to which 10 belongs for siLemberg Museum (Polish), 1886. "Comptes Rendus Paris, 106, 1888, 1070-1; errata, 1194. ^'Nature, 40, 1889, 115-6. 34 Assoc, franp. avanc. sc, 18, 1889, II, 24-38. '^Zeitschr. Math. Phys., 36, 1891, 64. 3«Comptes Rendus Paris, 115, 1892, 1259-61. "Ofversigt af finska vetenskaps-soc. forhandUngar, 34, 1892, 109-162. Cf . Jahrbuch Fortschr. Math., 24, 1892, 164-5. 342 History of the Theory of Numbers. [Chap, xii primes 6<200 and certain larger primes, from which are easily deduced tests for the divisor 6. Several"" noted that if 10 belongs to the exponent n modulo d, and if Si, Si, ■ ■ .denote the sums of every nth digit of N beginning with the first, second, ... at the right, the remainder on the division of A^ by d is that of S1 + IOS2+IO-S3+ . . . J. Fontes^^ would find the least residue of A^ modulo M. If 10" has the residue q modulo M, we do not change the least residue of N if we multiply a set of n digits of A^ by the same power of q as of 10". Thus for M = 19, iV=10433 = 10'+4-10H33, 10" has the residue 5 modulo 19 and we may replace N by 5- +4 ',5+ 33. The method is applied to each prime M^ 149. Fontes^^ gave a history of the tests for divisibility, and an "extension of the method of Pascal," similar to that in his preceding paper. P. Valerio*° would test the divisibility of N by 39, for example, by sub- tracting from N a multiple of 39 with the same ending as N. F. Belohldvek^^ noted that 10A-\-B is divisible by 10p±l if A=FpB is. C. Borgen'^^ ^oted that Z = a„-10"+ . . . +ai-10+ao is divisible by A^ if "T' (a_„+rlO''-'+ . . . +a,)(10"-A^)''/'' ..=0 is divisible by N. For A'' = 7, take a = 1 ; then 10"— AT = 3 and Z is divisible by 7 if ao+3ai+2a2 — ^3 — 3a4 — 2a5+ ... is divisible by 7. J. J. Sylvester^^" noted that, if the r digits of A'^, read from left to right, be multiplied by the first r terms of the recurring series 1, 4, 3, — 1,-4, — 3; 1, 4, . . . [the residues, in reverse order, of 10, 10^, . . ., modulo 13], the sum of the products is divisible by 13 if and only if N is divisible by 13. C. L. Dodgson^^** discussed the quotient and remainder on division by 9 or 11. L. T. Riess^' noted that, if p is not divisible by 2 or 5, 106+a(a<10) is divisible by p if b — xa is divisible by p, where mp = 10x-\-a (a< 10) and m = l, 7, 3, 9 according as p=l, 3, 7, 9 (mod 10), respectively. A. Loir^^ gave tests for prime divisors < 100 by uniting them by twos or threes so that the product P ends in 1 , as 7 -43 = 30 1 . To test N, multiply the number formed of the last two digits of A'^ by the number preceding 01 in P, subtract the product from A^, and proceed in the same manner with the difference. Then P is a factor if we finally get a difference which is zero. If a difference is a multiple of a prime factor p of P, then N is divisible by p. Plakhowo"*^ gave the test by Bougaief, but without using congruences. '"'Math. Quest. Educ. Times, 57, 1892, 111. "Assoc, frang. avanc. sc, 22, 1893, II, 240-254. »M4m. ac. sc. Toulouse, (9), 5, 1893, 459-475. "La Revue Scientifique de France, (3), 52, 1893, 765 "Casopis, Prag, 23, 1894, 59. "Mature, 57, 1897-8, 54. «MEducat. Times, March, 1897. Proofs. Math. Quest. Educ. Times, 66, 1897, 108. Cf. W. E. Heal, Amer. Math. Monthly, 4, 1897, 171-2. "'^Nature, 56, 1897, 565-6. «Russ. Nat., 1898, 329. Cf. Jahrb. Fortschritte Math., 29, 1898, 137. "Assoc, frang. avanc. sc, 27, 1898, II, 144-6. «»Bull. des sc. math, et phys. 61(§mentaires, 4, 1898-9, 241-3, 1 Chap. XII] CRITERIA FOR DIVISIBILITY. 343 To testiV = ao+Oi-S+ • • • +On5"for the divisor D prime to B, determine d and X so that Bd = Dx+l. Multiply this equation by Oq and subtract from N. Thus N=-BN'-DaoX, N' = aod-\-{ai+a2B-\- . . . +a^B''-^)B. Hence N is divisible by D if and only if N' is divisible by D. Now, N' is derived from N by supressing the units digit do and adding to the result the product aod. Next operate with N' as we did with A^. J. Malengreau^^ would test N for a factor q prime to 10 by seeking a multiple 11 ... 1 (to m digits) of q, then an exponent t such that the number of digits of lO'-A^ is a multiple of m. From each set of m digits of lO'-A^ subtract the nearest multiple of 1 ... 1 (to m digits) . The sum of the resi- dues is divisible by q if and only if A'' is divisible by q. G. Loria^^ proved that N = aQ-\-gai-\- . . .-\-g''ak is divisible by a if and only if a divides the sum ao+ • • . +a^ of the digits of N written to a base g of the form /ca+ 1 ; or if a divides Oq — ai +02 — • • • when the base g is of the form ka — 1. Taking g = IC", we have the test, in Gelin's Arithm^tique, in terms of groups of m digits. We may select m to be |0(a) or a number such that lO'^il has the factor a. Inplace of 00+^1+ • • . wheng' = 10'", we may employ pao+Xai + 10Xa2+ . . . +10"-'Xa^_i + s\o^™-^(a,^+10a,^+i+ . . . +10"'-'a,^+m-i), k = l where X = l, 2 or 5, and p is determined by 10p/X=l (mod o). Taking a = 7, 13, 17, 19, 23, special tests for divisors are obtained. G. Loria^^ proved that, if ao, ai,. . . are successive sets of t digits of N, counted from the right, and o- = ao='=cti+02=^«3+ • • •, then N-(T = a,{10'=Fl)-\-a2{10^'-l)+as{10^'=pl) + . . ., so that a factor of 10'=f1 divides A^ if and only if it divides a. A. Tagiuri^^ extended the last result to any base g. We have N = ao+ga,+ . . . =Nom+g"'Nr^+g''^N2^-\- . . . if N,m = o,pm+apm+i9-\- • • • +«pm+m-i^"'"^ Heuce, if 9"^= ± 1 (mod a), N=Nom^Nim+N2m=^... (mod a). L. Ripert^" noted that lOD-\-uis divisible by lOS+i if Di—bu is divisible, and gave many tests for small divisors. G. Biase^^ derived tests that \Od-\-u has the factor 7 or 19 from 2{l{)d+u)^2u-d (mod 7), 2{lQd+u)=2u+d (mod 19). O. Meissner^^ reported on certain tests cited above. "Mathesis, (3), 1, 1901, 197-8. *^Rendiconti Accad. Lincei (Math.), (5), 10, 1901, sem. 2, 150-8. Mathesis, (3), 2, 1902, 33-39. "II Boll. Matematica Gior. Sc.-Didat., Bologna, 1, 1902. Cf. A. Bindoni, ibid., 4, 1905, 87. "Periodico di Mat., 18, 1903, 43-45. ^oL'enseignement math., 6, 1904, 40-46. "II Boll. Matematica Gior. Sc.-Didat., Bologna, 4, 1905, 92-6. ^''Math. Naturw. Blatter, 3, 1906, 97-99. 344 History of the Theory of Numbers. [Chap, xii E. NanneF employed ri=Oi — aoX, r2 = a2—riX,. . . (a-<10). Then, if r„ = 0, A'' = 10"a„4- . . . +10ai+Oo is divisible by lOx-fl and the quotient has the digits r„_i, r„_2, . . ■ , 7*i, Qo- The cases x = 1, 2 are discussed and several tests for 7 deduced. For a:= 1/3, we conclude that, if r„ = 0, N is divisible by 13 and the digits of the quotient are r„_i/3, . . . , r^/S, ao/3. A. Chiari^ employed D'Alembert's^ method for 10+6, 6 = 3, 7, 9. G. Bruzzone^^ noted that, to find the remainder R when N is divided by an integer x of r digits, we may choose y such that x-\-y = 10'', form the groups of r digits counting from the right of N, and multiply the successive groups (from the right) hy l,y,y^,. . . or by their residues modulo x; then R equals the remainder on dividing the sum of the products by x. If we choose x — y = lO^, we must change alternate signs before adding. For practical use, take y = l. Fr. Schuh^^ gave three methods to determine the residue of large numbers for a given modulus. Stuyvaert^^ let a, 6, ... be the successive sets of n digits of A'' to the baseB, so that iV = a+6jB'*+c52''+ rj.^^^ ^ -^ (^^.^isibie ^^y ^ factor D of B''=pR'' if and only if a=^bR''+cR~''^ ... is divisible by D. For R = l, B = 10, n = 1, 2, . . ., we obtain tests for divisors of 9, 99, 11, 101, etc. A divisor, prime to B, of niB+l divides N = a+bB if and only if it divides h—ma. Further Papers Giving Tests for a Given Divisor d. J. R. Young and Mason for d = l, 13 [Pascal^], Ladies' Diary, 1831, 34-5, Quest. 1512. P. Gorini [Pascal^], Annali di Fis., Chim. Mat., (ed., Majocchi), 1,1841, 237. A. Pinaud for d = l, 13, Mem. Acad. Sc. Toulouse, 1, 1844, 341, 347. *Dietz and Vincenot, Mem. Acad. Metz, 33, 1851-2, 37. Anonymous writer for d = 9, 11, Jour, fiir Math., 50, 1855, 187-8. *H. Wronski, Principes de la phil. des math. Cf. de Montferrier, Encyclop^die math., 2, 1856, p. 95. O. Terquem for d^l9, 23, 37, 101, Nouv. Ann. Math., 14, 1855, 118-120. A. P. Reyer for d = l, Archiv Math. Phys., 25, 1855, 176-196. C. F. Lindman for d = l, 13, ibid., 26, 1856, 467-470. P. Buttel for d = 7, 9, 11, 17, 19, ibid., 241-266. De Lapparent [Herter^^], Mem. soc. imp. sc. nat. Cherbourg, 4, 1856, 235-258. Karwowski [Pascal^], Ueber die Theilbarkeit . . ., II, Progr., Lissa, 1856. *D. van Langeraad, Kenmerken van deelbarheid der geheele getallen, Schoonho- ven, 1857. Flohr, Ueber Theilbarkeit und Reste der Zahlen, Progr., Berlin, 1858. V. Bouniakowsky for d = 37, 989, Nouv. Ann. Math., 18, 1859, 168. Elefanti for d = l-n, Proc. Roy. Soc. London, 10, 1859-60, 208. A. Niegemann for d = 10'"-n+a, Archiv Math. Phys., 38, 1862, 384-8. J. A. Grunert for d = 7, 11, 13, ibid., 42, 1864, 478-482. V. A. Lebesgue, Tables diverses pour la decomposition des nombres, Paris, 1864, p. 13. "II Pitagora, Palermo, 13, 1906-7, 54-9. »/6r(f., 14, 1907-8, 35-7. "/6td., 15, 1908-9, 119-123. "Supplem. De Vriend der Wiskunde, 24, 1912, 89-103. "Les Nombres Positifs, Gand, 1912, 59-62. Chap. XII] CrITEEIA FOR DIVISIBILITY. 345 C. M. Ingleby for d = 9, 11, British Assoc. Report, 35, 1865, 7 (trans.). M. Jenkins for any prime d, Math. Quest. Educ. Times, 8, 1868, 69, 111. F. Unferdinger [Gergonnei^], Sitzungsber. Ak. Wiss. Wien (Math.), 59, 1869, II, 465-6. H. Anton for d = 9, 11, 13, 101, Archiv Math. Phys., 49, 1869, 241-308. W. H. Walenn, British Assoc. Report, 40, 1870, 16-17 (trans.); Phil. Mag., (4), 36, 1868, 346-8; (4), 46, 1873, 36-41; (4), 49, 1875, 346-351; (5), 2, 1876, 345; 4, 1877, 378; 9, 1880, 56, 121, 271. M. A. X. Stouff for d< 100, Nouv. Ann. Math., (2), 10, 1871, 104. J. Lubin, ibid., (2), 12, 1874, 528-30 (trivial). Szenic for d = 7, 9, 37, Von der Kongruenz der Z., Progr. Schrimm, 1873. E. Brooks for d = 7, Des Moines Analyst, 2, 1875, 129. W. J. Greenfield and M. Collins for d = 47, 73, Math. Quest. Educ. Times, 22, 1875, 87. F. da Ponte Horta for d = 7, 9, 11, 13, Jornal de Sciencias Mat. Ast., 1, 1877, 57-62. Mennesson for d = 7, Nouv. Corresp. Math., 4, 1878, 151; generahzation by Cesaro, p. 156. C. Lange, for d = 7, 13, 17, 19, Ueber die Teilbarkeit der Zahlen, Progr., Berlin, 1879. F. Jorcke for d = 7, 9, 11, Ueber Zahlenkongruenzen . . ., Progr, Fraustadt, 1878. K. Broda for any base, Archiv Math. Phys., 63, 1879, 413-428. A. Badoureau for d = 19, Nouv. Ann. Math., (2), 18, 1879, 35-6. S. M. Drach for d = 7, Math. Quest. Educ. Times, 35, 1881, 71-2. W. A. Pick for d = 7, ibid., 38, 1883, 64. A. Evans for d = 7, Des Moines Analyst, 10, 1883, 134. K. Haas, Theilbarkeitsregeln . . ., Progr., Wien, 1883. G. Wertheim, Elemente der Zahlentheorie, 1887, 31-33. B. Adam for d<100, Ueber die Teilbarkeit. . ., Progr. Gym. Clausthal, 1889. A. Loir for d<138. Jour, de math, elem., 1889, 66, 107-10, 121-3. A. G. Fazio [SchlegeP^], Sui caratteri. . ., Palermo, 1889. E. Gelin, Mathesis, (2), 2, 1892, 65, 93; (2), 12, 1902, 65-74, 93-99 (extract in Mathesis, (3), 10, 1910, Suppl. I); Ann. Soc. Sc. Bruxelles, 34, 1909-10, 66; Recueil de problemes d'arith., 1896. Extracts by M. Nasso, Revue de Math. (ed., Peano), 7, 1900-1, 42-52. Speckmann, Dorsten, Haas, Dorr, Zeitschrift Math. Phys., 37, 1892, 58, 63, 128, 192, 383. Lalbaletrier, Jour, de Math, (ed., de Longchamps), 1894, 54. H. T. Burgess [Pascal*], Nature, 57, 1897-8, 8-9, 30, 55. A. Conti [Pascal*], Periodico di Mat., 13, 1898, 180-6, 207-9. F. Mariantoni, ibid., 149-151, 191-2, 217-8. T. Lange for d<30, Archiv. Math. Phys., (2), 16, 1898, 220-3. W. J. Greenstreet, Math. Gazette, 1, 1900, 186-7. Christie for d = 2^p,5'^p (p prime). Math. Quest. Educ. Times, 73, 1900, 119. A. Cunningham and D. Biddle for d = rp=i=l, ibid., 75, 1901, 49-50. M. Zuccagni for d = 7, Suppl. al Periodico di Mat., 6, fasc. V. Calvitti for d = 7, ibid., 8, fasc. IV. S. Dickstein, Wiad. Mat., Warsaw, 6, 1902, 253-7 (Pohsh). B. Niewenglowski, ibid., 252-3. Pietzker ford = 7, 11, 13, 27, 37, Unterrichtsblatter Math. Naturwiss., 9, 1903, 85-110. A. Church for d = 7, 13, 17, Amer. Math. Monthly, 12, 1905, 102-3. E. A. Cazes, Assoc, frang., 36, 1907, 55-63. A. Gerardin for d = 7, 13, 17, 37, 43, Sphinx-Oedipe, 1907-8, 2. M. Morale for d = 7, Suppl. al Periodico di Mat., 11, 1908, 103. *T. Ghezzi, ibid., 12, 1908-9, 129-130. Lenzi, II Boll. Matematica Gior. Sc.-Didat., 7, 1908. 346 History of the Theory of Numbers. [Chap, xiii R. Polpi, ibid., 8, 1909, 281-5. M. Morale for d = 7, 13, Suppl. al Periodico di Mat., 13, 1909-10, 38-9. A. L. Csada, ibid., 56-8. *A. La Paglia, ibid., 14, 1910-11, 136-7, extension of Morale to any d. A. V. Filippov, 8 methods for d = 9, Kagans Bote, 1910, 88-92, No. 520. P. Cattaneo for rf= 11, II Boll. Matematica Gior. Sc.-Didat., 9, 1910, 305-6. *L. Miceli, Condizioni di divisibility di un numero N per un numero a . . . , Matera, 1911, 8 pp. R. Ayza for d = a-10''±l, Revista sociedad mat. espanola, Madrid, 1, 1911, 162-6. *Paoletti, II Pitagora, Palermo, 18, 1911-12, 128-132. *R. La Marca, Criteri di congruenza e criteri di divisibilita, Torre del Greco, 1912, 30 pp. K. W. Lichtenecker, Zeitschr. fur Realschulwesen, 37, 1912, 338-49. R. E. Cicero, Sociedad Cientifica Antonio Alzate, 32, 1912-3, 317-331. J. G. Gal6 for d = 7, Revista sociedad mat. espanola, 3, 1913-4, 46-7. .. C. F. lodi for d = 7, 13, 17, 19, Suppl. al Periodico di Mat., 18, 1914, 20-3. > E. Kylla for d=ll, Unterrichtsblatter Math. Naturwiss., 20, 1914, 156. R. Krahl for d = 7, Zeitschrift Math. Naturw. Unterricht, 45, 1914, 562. P. A. Fontebasso, II Boll. Matematica, 13, 1914-5. G. M. Persico, Periodico di Mat., 32, 1917, 105-124. Sammlung der Aufgaben in Zeitschrift Math. Naturw. Unterricht, 1898: ford=7, II, 337; IV, 404, 407; for d = 9, 11, XXIV, 606; XXV, 587-8; for d = 37, etc., XXVI, 18, 25-27. Criteria for di\dsibility in connection with tables were given by Barlow ,^^ Tarry«6 ^nd Lebon" of Ch. XIII, and Harmuth^^ of Ch. XIV. Papers on Divisibility not Available for report. Joubin, Jour. Acad. Soc. Sc. France et de I'Etranger, Paris, 2, 1834, 230. J. Lenth^ric, Th^orie de la divisibility des nombres, Paris, 1838. R. Volterrani, Saggio sulla divisione ragionata dei n. interi, Pisa, 1871. F. Tirelli, Teoria della divisibilita de' numeri, Napoli, 1875. E. Tiberi, Teoria generale sulle condizioni di divisibility . . . , Arezzo, 1890. J. Kroupa, Casopis, Prag, 43, 1914, 117-120. G. Schroder, Unterrichtsblatter fiir Math. Naturwiss., 21, 1915, 152-5. CHAPTER XIII. FACTOR TABLES. LISTS OF PRIMES. Eratosthenes (third century B.C.) gave a method, called the sieve or crib of Eratosthenes, of determining all the primes under a given limit I, which serves also to construct the prime factors of numbers <l. From the series of odd numbers 3, 5, 7, ... , strike out the square of 3 and every third number after 9, then the square of 5 and every fifth number after 25, etc. Proceed until the first remaining number, directly following that one whose multiples were last cancelled, has its square >l. The remaining numbers are primes. Nicomachus and Boethius^ began with 5 instead of with 5^, 7 instead of with 7^, etc., and so obtained the prime factors of the numbers <l. A table containing all the divisors of each odd number ^113 was printed at the end of an edition of Aratus, Oxford, 1672, and ascribed to Eratos- thenes by the editor, who incorrectly considered the table to be the sieve of Eratosthenes. Samuel Horsley^ believed that the table was copied by some monk in a barbarous age either from a Greek commentary on the Arithmetic of Nicomachus or else from a Latin translation of a Greek manuscript, published by Camerarius, in which occurs such a table to 109. Leonardo Pisano^ gave a table of the 21 primes from 11 to 97 and a table giving the factors of composite numbers from 12 to 100; to determine whether n is prime or not, one can restrict attention to divisors ^ ^/n. Ibn Albanna in his Talkhys^ (end of 13th century) noted that in using the crib of Eratosthenes we may restrict ourselves to numbers ^ -y/l. Cataldi^ gave a table of all the factors of all numbers up to 750, with a separate list of primes to 750, and a supplement extending the factor table from 751 to 800. Frans van Schooten® gave a table of primes to 9979. J. H. Rahn^ (Rhonius) gave a table of the least factors of numbers, not divisible by 2 or 5, up to 24000. T. Brancker^ constructed a table of the least divisors of numbers, not divisible by 2 or 5, up to 100 000. [Reprinted by Hinkley.^^] *Introd. in Arith. Nicomachi; Arith. Boethii, lib. 1, cap. 17 (full titles in the chapter on perfect numbers). Extracts of the parts on the crib, with numerous annotations, were given by Horsley.2 Cf. G. Bernhardy, Eratosthenica, Berlin, 1822, 173-4. 2Phil. Trans. London, 62, 1772, 327-347. 311 Liber Abbaci di L. Pisano (1202, revised 1228), Roma, 1852, ch. 5; Scritti, 1, 1857, 38. *Transl. by A. Marre, Atti Accad. Pont. Nuovi Lincei, 17, 1863-4, 307. "Trattato de' numeri perfetti, Bologna, 1603. Libri, Histoire des Sciences Math, en Italic, ed. 2, vol. 4, 1865, 91, stated erroneously that the table extended to 1000. «Exercitat. Math., libri 5, cap. 5, p. 394, Leiden, 1657. ^Algebra, Zurich, 1659. WaUis,!"* p. 214, attributed this book to John Pell. *An Introduction to Algebra, translated out of the High-Dutch [of Rahn's' Algebra] into EngHsh by Thomas Brancker, augmented by D. P. [=Dr. Pell], London, 1668. It is cited in Phil. Trans. London, 3, 1668, 688. The Algebra and the translation were de- scribed by G. Wertheim, BibUotheca Math., (3), 3, 1902, 113-126. 347 348 History of the Theory of Numbers. (Chap, xiii D. Schwenter^ gave all the factors of the odd numbers < 1000. John Wallis^° gave a list of errata in Brancker's^ table. John Harris," D. D., F. R. S., reprinted Brancker's^ table. De Traytorens^^ emphasized the utility of a factor table. To form a table showing all prime factors of numbers to 1000, begin by multiplying 2, 3, .. . by all other primes < 1000, then multiply 2X3 by all the primes, then 2X3X5, etc. Joh. Mich. Poetius^^ gave a table (anatomiae numerorum) of all the prime factors of numbers, not divisible by 2, 3, 5, up to 10200. It was reprinted by Christian Wolf," Willigs,^^ and Lambert. ^- Johann Gottlob Krliger^* gave a table of primes to 100 999 (not to 1 million, as in the title), stating that the table was computed by Peter Jager of Niirnberg. James Dodson^® gave the least di\'isors of numbers to 10000 not divisible by 2 or 5 and the primes from 10000 to 15000. Etienne FranQois du Tour^^ described the construction of a table of all composite odd numbers to 10000 by multiplying 3, 5, ... , 3333 by 3, ... , 99. Giuseppe Pigri^^ gave all prime factors of numbers to 10000. Michel Lorenz Willigs^^ (Willich) gave all di\dsors of numbers to 10000. Henri Anjema-° gave all divisors of numbers to 10000. Rallier des Ourmes-^ gave as if new the sieve of Eratosthenes, placing 3 above 9 and every third odd number after it, a 7 above 49, etc. He expressed each number up to 500 as a product of powers of primes. J. H. Lambert^^ described a method of making a factor table and gave Poetius'^^ table and expressed a desire for a table to 102 000. Lagrange called his attention to Brancker's^ table. Lambert-^ gave [Ivriiger's^^] table showing the least factor of numbers not di\'isible by 2, 3, 5 up to 102000, and a table of primes to 102 000, errata in which were noted by KliigeP^. •Geometria Practica, Numb., 1667, I, 312. loTreatise of Algebra, additional treatise, Ch. Ill, §22, London, 1685. "Lexicon Technicum, or an Universal English Dictionary of Arts and Sciences, London, vol. 2, 1710 (under Incomposite Numbers). In ed. 5, London, 2, 1736, the table was omitted, but the text describing it kept. WaUis, Opera, 2, ,511, listed 30 errors. "Histoire de I'Acad. Roy. Science, ann6e 1717, Paris, 1741, Hist., 42-47. "Anleitung zu der Arith. Wissenschaft vermittelst einer parallel Algebra, Frkf . u. Leipzig, 1728. "VoUst. Math. Lexicon, 2, Leipzig, 1742, 530. '*Gedancken von der Algebra, nebst den Primzahlen von 1 bis 1 000 000, Halle im Magd., 1746, Cf. Lambert. ^a "The Calculator. . .Tables for Computation, London, 1747. •"Histoire de I'Acad. Ro>. Sc, Paris, ann6e 1754, Hist., 8&-90. "Nuove tavole degli elementi dei numeri dall' 1 al 10 000, Pisa, 1758. "Griindhche Vorstellung der Reesischen allgemeinen Regel . . . Rechnungsarten, Bremen u. Gottingen, 2, 1760, 831-976. ^Table des diviseurs de tous les nombres naturels, depuis 1 jusqu'4 10 000, Leyden, 1767, 302 pp. "M^m. de math, et de physique, Paris, 5, 1768, 485-499. "Bej-trage znm Gebrauche der Math. u. deren Anwendung, Berlin, 1770, II, 42. "Zusatze zu den logarithm ischen imd trig. Tabellen, BerUn, 1770. "Math. Worterbuch, 3, 1808, 892-900. Chap. XIII] FACTOR TABLES, LiSTS OF PrIMES. 349 J. Ozanam^^ gave a table of primes to 10000. A. F. Marci^^ gave in 1772 a list of primes to 400 000. Jean Bernoulli^^" tabulated the primes 16n+l up to 21601. L. Euler" discussed the construction of a factor table to one miUion. Given a prime p = 30a±i (^ = 1, 7, 11, 13), he determined for each r = l, 7, 11, 13, 17, 19, 23, 29, the least q for which SOq+r is divisible by p, and arranged the results in a single table with p ranging over the primes from 7 to 1000. He showed how to use this auxiliary table to construct a factor table between given limits. C. F. Hindenburg^^ employed in the construction of factor tables a "patrone" or strip of thick paper with holes at proper intervals to show the multiples of p, for the successive primes p. A. FelkeP^ gave in 1776 a table of all the prime factors (designated by letters or pairs of letters) of numbers, not divisible by 2, 3, or 5, up to 408 000, requiring for entry two auxiliary tables. In manuscript^", the table extended to 2 million; but as there were no purchasers of the part printed, the entire edition, except for a few copies, was used for cartridges in the Turkish war. The imperial treasury at Vienna, at the cost of which the table was printed, retained the further manuscript. [See Felkel.^^] L. Bertrand^^ discussed the construction of factor tables. The Encyclopedie of d'Alembert, ed. 1780, end of vol. 2, contains a factor table to 100 000. Franz Schaffgotsch^^ gave a method, equivalent to that of a stencil for each prime p, for entering the factor p in a factor table with eight headings SOm+k, /c = 1, 7, 11, 13, 17, 19, 23, 29, and hence of numbers not divisible by 2, 3, or 5. Proofs were given by Beguelin and Tessanek, ibid., 362, 379. The strong appeals by Lambert^^ that some one should construct a fac- tor table to one million led L. Oberreit, von Stamford, Rosenthal, Felkel, and Hindenburg to consider methods of constructing factor tables and to prepare such tables to one million, with plans for extension to 5 or 10 ^^Recreations Math., new ed., Paris, 1723, 1724, 1735, etc., I, p. 47. ^^Primes "in quater centenis millibus," Amstelodami, 1772. 26aNouv. M6m. Ac. Berlin, ann^e 1771, 1773, 323. "Novi Comm. Acad. Petrop., 19, 1774, 132; Comm. Arith., 2, 64. ''^Beschreibung einer ganz neuen Art nach einem bekannten Gesetze fortgehende Zahlen durch Abzahlen oder Abmessen bequem u. sicher zu finden. Nebst Anwendung der Methode auf verschiedene Zahlen, besonders auf eine damach zu fertigende Factorentafel . . . , Leipzig, 1776, 120 pp. 2*Tabula omnium factorum simphcium, numerorum per 2, 3, 5 non divisibilium ab 1 usque 10 000 000 [!]. Elaborata ab Antonio Felkel. Pars I. Exhibens factores ab 1 usque 144 000, Vindobonae, 1776. Then there is a table to 408 000, given in three sections. There is a copy of this complete table in the Graves Library, University College, London. Tafel aller einfachen Factoren der durch 2, 3, 5 nicht theilbaren Zahlen von 1 bis 10 000 000. Entworfen von Anton Felkel. I. Theil. Enthaltend die Factoren von 1 bis 144 000, Wien, 1776. There is a copy of this incomplete table in the hbraries of the Royal Society of London and Gottingen University. "Cf. Zach's Monatliche Correspondenz, 2, 1800, 223; Allgemeine deutsche BibUothek, 33, II, 495. '^Develop, nouveau de la partie ^1. math., Geneve, 1774. '''Gesetz, welches zur Fortsetzung der bekannten Pellischen Tafehi dient, Abhand. Privatgesell- schaft in Bohmen, Prag, 5, 1782, 354-382. 350 History of the Theory of Numbers. [Chap, xiii | '^ ' million. Their extended correspondence with Lambert^^ was published. Of the tables constructed by these computers, the only one published is that by Felkel.-^ The history of their connection with factor tables has been treated by J. W. L. Glaisher.^ Johann Neumann^^ gave all the prime factors of numbers to 100 100. Desfaviaae gave a like table in the same year. F. Maseres^^ reprinted the table of Brancker.^ G. Vega^^ gave all the prime factors of numbers not divisible by 2, 3, or 5 to 102 000 and a list of primes from 102 000 to 400 031. Chernac hsted errors in both tables. In Hiilsse's edition, 1840, of Vega, the Ust of primes extends to 400 313. A. Felkel,^^ in his Latin translation of Lambert's'^^ Zusatze, gave all the prime factors except the greatest of numbers not divisible by 2, 3, 5 up to 102 000, large primes being denoted by letters. In the preface he stated that, being unable to obtain his extensive manuscript^" in 1785, he calculated again a factor table from 408 000 to 2 856 000. J. P. Griison^^ gave all prime factors of numbers not divisible by 2, 3, 5 to 10500. He^^'' gave a table of primes to 10000. F. W. D. Snell^° gave the prime factors of numbers to 30000. A. G. Kastner^^ gave a report on factor tables. K. C. F. Krause'*- gave a table of 22 pages showing all products < 100 000 of two primes, a table of primes < 100 000 with letters for 01, 03, ... , 99, and (pp. 25-28) a factor table to 10000 by use of letters for numbers < 100. N. J. Lidonne^^ gave all prime factors of numbers to 102 000. Jacob Struve"*^" made a factor table to 100 by de Traytorens'^^ method. L. Chernac^ gave all the prime factors of numbers, not divisible by 2, 3 or 5, up to 1 020 000. J. C. Burckhardt*^ gave the least factor of numbers to 3 million. He did not compute the first million, but compared Chernac's table with a manu- script (mentioned in Briefwechsel,^^ p. 140) by Schenmarck which extended to 1 008 000. Cf. iVIeissel.''^ ^'Joh. Heinrich Lamberts deutscher gelehrter Briefwechsel, herausgegeben von Joh. Bernoulli, Berlin, 1785, Leipzig, 1787, vol. 5. "Proc. Cambridge Phil. Soc, 3, 1878, 99-138. '^Tabellen der Primzahlcn und der Faktoren der Zahlen, welche unter 100 100, und durch 2, 3 Oder 5 nicht theilbar sind, Dessau, 1785, 200 pp. '*The Doctrine of Permutations and Combinations. . ., London, 1795. '^Tabulae logarithmico-trigonometricae, 1797, vol. 2. **J. H. Lambert, Supplementa tab. log. trig., Lisbon, 1798. "Pinaeoth6que, ou collection de Tables. . ., Berlin, 1798. ''"Enthiillte Zaubereyen u. Geheimnisse d. Arith., Berlin, 1796, I, 82-4. *°Ueber eine neue und bequeme Art, die Factorentafeln einzurichten, nebst einer Kupfertafel der einfachen Factoren von 1 bis 30000, Gicssen and Darmstadt, 1800. "Fortsetzung der Rechenkunst, ed. 2, Gottingen, 1801, 566-582. ^'Factoren- und Primzahlentafel von 1 bis 100 000 neu berechnet, Jena u. Leipzig, 1804. "Tables de tous les diviseurs des nombres < 102 000, Paris, 1808. ^'''Handbuch der Math., Altona, II, 1809, 108. **Cribrum Arithmeticum . . . Daventriae, Isil, 1020 pp. Reviewed by Gauss, Gottingische gelehrte Anzeigen, 1812; Werke 2, 181-2. Errata, Cunningham.*' "Tables des diviseurs. . . 1 ^ 3 036 000, Paris, 1817, 1814, 1816 (for the respective three milliona), and 1817 (in one volume). •'I;i Chap. XIII] FACTOR TABLES, LiSTS OF PrIMES. 351 P. Barlow^® gave the prime and power of prime factors of numbers to 10000 and a list of primes to 100 103. C. Hutton^^ gave the least factor of numbers to 10000. Rees' Cyclopaedia, 1819, vol. 28, Hsts the primes to 217 219. Peter Barlow^^ gave a two-page table for finding factors of a number iV< 100 000. The primes p = 7 to p = 313 are at the head of the columns, while the 18 numbers 1000, . . . , 9000, 10000, 20000, . . . , 90000 are in the left- hand column. In the body of the table is the remainder of each of the latter when divided by the primes p. To test if p is a factor of N, add its last two digits to the remainders in the line of hundreds and thousands in the column headed p and test whether the sum is divisible by p. J. P. Kulik'^^ gave a factor table to 1 million. J. HantschP° gave a factor table to 18277; J. M. Salomon,^^ to 102 Oil. A. L. Crelle^^ gave the number of primes 4n+ 1 and the number of primes 4n+3 in each thousand up to the fiftieth. A. Guyot^^ hsted the primes to 100 000. A. F. Mobius,^^" using square ruled paper, inserted from right to left 0, 1, 2, ... in the top row of cells, and inserted n in each cell of the nth row below the top row whenever the corresponding number in the top row is divisible by n. We thus have a factor table. Certain numbers of the table Ue in straight lines, others in parabolas, etc. P. A. G. Colombier^^^ discussed the determination of the primes <V, given those < I. H. G. Kohler'^ gave a factor table to 21524. E. Hinkley^^ gave a factor table to 100 000, listing all factors of odd numbers to 20000 and of even numbers to 12500. F. Schallen^^"gave the prime and prime-power factors of numbers < 10000. F. Landry^*^ gave factor and prime tables to 10000. A. L. Crelle^^ discussed the expeditious construction of a factor table, and in particular a method of extending Chernac's^ table to 7 million. J. HoiieP^ gave a factor table to 10841. Jacob Philip Kulik (1773-1863) spent 20 years constructing a factor *^New Mathematical Tables, London, 1814. Errata, Cunningham.^ <Thil. and Math. Dictionary, 1815, vol. 2, 236-8. ^8New Series of Math. Repository (ed., Th. Leybourn), London, 4, 1819, II, 30-39. ^'Tafeln der einfachen Faktoren aUer Zahlen unter 1 million, Graz, 1825. 60Log.-trig. Handbuch, Wien, 1827. "Log. Tafeln, Wien, 1827. ^2Jour. fur Math., 10, 1833, 208. '^Thdorie g^nerale de la divisibihte des nombres, suivie d'applications varices et d'une table de nombres premiers compris entre et 100 000, Paris, 1835. 63a Jour, fvir Math., 22, 1841, 276-284. "^Nouv. Ann. Math., 2, 1843, 408-410. "Log.-trig. Handbuch, Leipzig, 1848. Errata, Cunningham.** ^^Tables of the prime numbers and prime factors of the composite numbers from 1 to 100 000, Baltimore, 1853. Reproduction of Brancker's* table. ssaprirjizahlen-Tafel von 1 bis 10000. . ., Weimar, 1855. For 99 errata, see Cunningham.*^ 6*Tables des nombres entiers non divisibles par 2, 3, 5, et 7, jusqu' k 10201, avec leurs diviseurs simples en regard, et des carres des 1000 premiers nombres, Paris, 1855. Tables des nombres premiers, de 1 £l 10000, Paris, 1855. "Jour, ftir Math., 51, 1856, 61-99. "Tables de log., Paris, 1858. 352 History of the Theory of Numbers. [Ch.u'. xiii table to 100 million; the manuscripts^ has been in the library of the Vienna Royal Academy since 1867. Lehmer^- gave an account of the first of the eight volumes of the manuscript, listed 226 errors in the tenth million, and concluded that Kulik's manuscript is certainly not accurate enough to warrant publication, though of inestimable value in checking a newly constructed table. Lehmer^^ gave a further account of this manuscript which he examined in Vienna. Volume 2, running from 12 642 600 to 22 852 800 is missing. The eight volumes contained 4,212 pages. B. Goldberg*^" gave all factors of numbers prime to 2, 3, 5, to 251 647. Zacharias Dase,^^ in the introduction to the table for the seventh million, printed a letter from Gauss, dated 1850, giving a brief history- of previous tables and referring to the manuscript factor table for the fourth, fifth and sixth milUons presented to the Berlin Academy by A. L. Crelle. Although Gauss was confident this manuscript would be pubhshed, and hence urged Dase to undertake the seventh million, etc., the Academy found the manu- script to be so inaccurate that its publication was not ad\'isable. Dase died in 1861 lea\'ing the seventh million complete and remarkably accurate, the eighth nearly complete, and a large part of the factors for the ninth and tenth millions. The work was completed by Rosenberg, but ^vith numerous errors. The table for the tenth million has not been printed ; the manuscript was presented to the Berlin Academy in 1878, but no trace of it was found when Lehmer^- desired to compare it with his table of 1909. C. F. Gauss^- gave a table showing the number of primes in each thousand up to one million and in each ten thousand from one to three million, with a comparison with the approximate formula jdx/log x. V. A. Lebesgue^^ discussed the formation of factor tables and gave that to 115500 constructed by Hoiiel. W. H. Oakes^ used a complicated apparatus consisting of three tables on six sheets of various sizes and nine perforated cards (cf. Committee, ^^ p. 39). W. B. Da\as^s considered numbers in the vicinity of 10^, and of 10^^ E. MeisseP^ computed the number of primes in the successive sets of 100 000 numbers to one million and concluded that Burckhardt's*^ table gives correctly the primes to one million. ••Cited by Kulik. Abh. Bohm. Gesell. Wiss., Prag, (5), 11, 1860, 24, footnote. A report on the manuscript was made by J. Petzval, Sitzungsberichte Ak. Wiss. Wien (Math.), 53, 1866, II, 460. Cited by J. Perott, I'interm^iaire des math., 2, 1895, 40; 11, 1904, 103. •"Primzahlen- u. Faktortafeln von 1 bis 251 647, Leipzig, 1862. Errata, Cunningham." •'Factoren-Tafeln fur alle Zahlcn der siebenten MilUon . . . , Hamburg, 1862; . . .der achten Mil- Uon, 1863;. . .der neuhten MilUon (erganzt von H. Rosenberg), 1865. •^Posthumous manuscript, Werke, 2, 1863, 435-447. ••Tables diverses pour la decomposition des nombres en leurs facteurs premiers, M6m. soc. sc. phys. et nat. de Bordeaux, 3, cah. 1, 1864, 1-37. •♦Machine table for determining primes and the least factors of composite numbers up to 100 000, London, 1865. ••Jour.de Math., (2), 11, 1866, 188-190; Proc. London Math. Soc, 4, 1873, 416-7. Math. Quest. Educ. Times, 7, 1867, 77; 8, 1868, 30-1. ••Math. Annalen, 2, 1870, 63&-642. Cf. 3, p. 523; 21, 1883, p. 304; 25, 1885, p. 251. I Chap. XIII] FACTOR TABLES, LiSTS OF PRIMES. 353 J. W. L. Glaisher" gave for the second and ninth millions the number of primes in each interval of 50000 and a comparison with lix' — lix, where lix = jdx/log X [more precise definition at the end of Ch. XVIII]. A committee^^ consisting of Cayley, Stokes, Thompson, Smith, and Glaisher prepared the Report on Mathematical Tables, which includes (pp. 34-9) a list of factor and prime tables. J. W. L. Glaisher^^ described in detail the method used by his father^" and gave an account of the history of factor tables. Glaisher^^" enumerated the primes in the tables of Burckhardt and Dase. Glaisher^^^ tabulated long sets of consecutive composite numbers. He^^" enumerated the prime pairs (as 11, 13) in each successive thousand to 3 million and in the seventh, eighth, and ninth millions. E. Lucas^^'' wrote P(q) for the product of all the primes ^ q, where q is the largest prime < n. If xP(g)±l are both composite, xP{q)—n,. . ., xP{q),. . ., xP{q)-\-n give 2n+l composite numbers. Glaisher^^' enumerated the primes 4n4-l and the primes 4n+3 for inter- vals of 10000 in the kth milUon for k = l,2, 3, 7, 8, 9. James Glaisher'^° filled the gap between the tables by Burckhardt^^ and Dase". The introduction to the table for the fourth million gives a history of factor tables and their construction. Lehmer^^ praised the accuracy of Glaisher's table, finding in the sixth million a single error besides two mis- prints. Tuxen'^^ gave a process to construct tables of primes. Groscurth and Gudila-Godlewksi, Moscow, 1881, gave factor tables. *V. Bouniakowsky'^^" gave an extension of the sieve of Eratosthenes. W. W. Johnson'^^'' repeated Glaisher's'^° remarks on the history of tables. P. Seelhoff^^ gave large primes /c-2"+l {k< 100) and composite cases. Simony'^^ gave the digits to base 2 of primes to 2^^ = 16384. L. Saint-Loup^^ gave a graphical exposition of Eratosthenes' sieve. H. Vollprecht'^^ discussed the construction of factor tables. "Report British Association for 1872, 1873, trans., 19-21. Cf. W. W. Johnson, Des Moines Analyst, 2, 1875, 9-11. 68Report British Association for 1873, 1874, pp. 1-175. Continued in 1875, 305-336; French transl., Sphinx-Oedipe, 8, 1913, 50-60, 72-79; 9, 1914, 8-14. "Proc. Cambridge Phil. Soc, 3, 1878, 99-138, 228-9. 69'»/6id., 17-23, 47-56; Report British Assoc, 1877, 20 (sect.). Extracts by W. W. Johnson, Des Moines Analyst, 5, 1878, 7. s'^-Messenger Math., 7, 1877-8, 102-6, 171-6; French transl., Sphinx-Oedipe, 7, 1912, 161-8. ^^'^Ibid., 8, 1879, 28-33. «9«*/6id., p. 81. C. Gill, Ladies' Diary, 1825, 36-7, had noted that xP(q)+j is composite for j = 2,...,q-l. BseReport British Assoc, 1878, 470-1; Proc. Roy. Soc. London, 29, 1879, 192-7. ^"Factor tables for the fourth, fifth and sixth millions, London, 1879, 1880, 1883. "Tidsskrift for Mat., (4), 5, 1881, 16-25. 'i«Memoirs Imperial Acad. Science, St. Petersburg, 41, 1882, Suppl, No. 3, 32 pp. "^Annals of Math., 1, 1884-5, 15-23. "Zeitschrift Math. Phys., 31, 1886, 380. Reprinted, Sphinx-Oedipe, 4, 1909, 95-6. "Sitzungsber. Ak. Wiss. Wien (Math.), 96, II, 1887, 191-286. ^^Comptes Rendus Paris, 107, 1888, 24; Ann. de I'^cole norm., (3), 7, 1890, 89-100. "Ueber die Herstellung von Faktorentafehi, Diss. Leipzig, 1891. 354 History of the Theory of Numbers. [Chap, xiii C. A. Laisant'^^" would exhibit a factor table by use of shaded and un- shaded squares on square-ruled paper without using numbers for entries. G. Speckmann'^'' made tri\aal remarks on the construction of a list of primes. P. Valerio'® arranged the odd numbers prime to 5 in four columns according to the endings 1, 3, 7, 9. From the first column cross out the first multiple 21 of 3, then the third following number 51, etc. Similarly for the other columns. Then use the primes 7, 11, etc., instead of 3. J. P. Gram" pubhshed the computation by N. P. Bertelsen of the number of primes to ten million in intervals of 50000 or less, which led to the detection of numerous errors in the tables of Burckhardt^^ and Dase." G. L. Bourgerel"^ gave a table with 0, 1, . . . , 9 in the first row, 10, . . . , 19 in the second row (with 10 under 0), etc. Then all multiples of a chosen number lie in straight lines forming a paralellogram lattice, with one branch through 0. For example, the multiples of 3 appear in the Une through 0, 12, 24, 36, ... , the parallel through 3, 15, 27, ... , the parallel 21, 33, 45, ... ; also in a second set of parallels 3, 12, 21, 30; 6, 15, 24, 33, 42, 51, 60; etc. E. Suchanek" continued to 100 000 Simony's^^ table of primes to base 2. D. von Sterneck^" counted the number of primes 100 n-|- 1 in each tenth of a million up to 9 million and noted the relatively small variation from one- fortieth of the total number of primes in the interval. H. Vollprecht^^ discussed the determination of the number of primes <N by use of the primes < v^- A. Cunningham and H. J. WoodalP^ discussed the problem to find all the primes in a given range and gave many successive primes >9 million. They^^a ^^^^^^ jj^y primes between 224±1020. H. Schapira^^'' discussed algebraic operations equivalent to the sieve of Eratosthenes. *V. Di Girio, Alba, 1901, applied indeterminate analysis of the first degree to define a new sieve of Eratosthenes and to factoring. John Tennant^ wrote numbers to the base 900 and used auxiliary tables. A. Cunningham^" gave long lists of primes between 9-10^ and 10^^ Ph. Jolivald^ noted that a table of all factors of the first 2n numbers serves to tell readily whether a number <4n+2 is prime or not. ^'"Assoc. frang., 1891, II, 165-8. 7**Archiv Math. Phys., (2), 11, 1892, 439-441. '«La revue scientifique de France, (3), 52, 1893, 764-5. "Acta Math., 17, 1893, 301-314. List of errors reproduced in Sphinx-Oedipe, 5, 1910, 49-51. ^*La revue scientifique de France, (4), 1, 1894, 411-2. "Sitzunpsber. Ak. Wiss. Wien (Math.), 103, II a, 1894, 443-610. '"Anzeiger K. Akad. Wiss. Wien (Math.), 31, 1894, 2-4. Cf. Kronecker, p. 416 below. "Zeitschrift Math. Phys., 40,. 1895. 118-123. "Report British Assoc, 1901, 553; 1903, 561; Messenger Math., 31, 1901-2, 165; 34, 1904-5, 72, 184; 37, 1907-8, 6.5-83; 41, 1911, 1-16. s^^Report British Assoc, 1900, 646. »«'Jahresber. d. Deutschen Math. Verein., 5, 1901, I, 69-72. "Quar. Jour. Math., 32, 1901, 322-342. »*^Ibid., 35, 1903, 10-21; Mess. Math., 36, 1907, 145-174; 38, 1908, 81-104; 38, 1909, 145-175; 39, 1909, 33-63, 97-128; 40, 1910, 1-36; 45, 1915, 49-75; Proc London Math. Soc, 27, 1896, 327; 28, 1897, 377-9; 29, 1898, 381-438, 518; 34, 1902, 49. •♦L'intermfidiaire des math., 11, 1904, 97-98. Chap. XIII] FaCTOK TaBLES, LiSTS OF PRIMES. 355 A. Cunningham^^ noted errata in various factor tables. *J. R. Akerlund^^" discussed the determination of primes by a machine. Gaston Tarry^^ would use an auxiliary table (as did Barlow in 1819) to tell by the addition of two entries (< |p) if a given number < iV is divisible by a chosen prime p. For N = 10000, he used the base 6 = 100, and gave a table showing the numerically least residues of the numbers r<h and the multiples of b for each prime p<b. Then nh-\-r is divisible by p if the residues of nh and r are equal and of opposite sign. For A^ = 100 000, he used 6 = 60060 = 2-91-330 and wrote numbers in the form m6+330g+r, q<90, r<330; or, again, 6 = 20580. Ernest Lebon" used such tables with the base 30030 = 2-3-5-7-lM3, or its product by 17. Ernest Lebon,^^ J. Deschamps,^^ and C. A. Laisant^'''' discussed the con- struction of factor tables. J. C. Morehead^° extended the sieve of Eratosthenes to numbers ma^+6 (m = l, 2, 3,. . .) in any arithmetical progression. The case a = 2, 6= ±1, is discussed in detail, with remarks on the construction of a table to serve as a factor table for numbers m-2''=t 1. L. L. Dines^^ treated the case a = 6, 6 = =fcl, and the factorization of numbers m-Q'^^l. D. N. Lehmer^^ gave a factor table to 10 million and listed the errata in the tables by Burckhardt, Glaisher, Dase, Dase and Rosenberg, and Kulik's tenth million, and gave references to other (shorter) lists of errata. E. B. Escott^^" listed 94 pairs of consecutive large numbers all of whose prime factors are small. L. Aubry^^° proved that a group of 30 consecutive odd numbers does not contain more than 15 primes or numbers all of whose prime factors exceed 7. Cunningham^^" listed the numbers of 5 digits with prime factors ^ 11 . 85Messenger Math., 34, 1904-5, 24-31; 35, 1905-6, 24. ss^Nyt Tidsskrift for Mat., Kjobenhavn, 16A, 1905, 97-103. 8«Bull. Soc. Philomathique de Paris, (9), 8, 1906, 174-6, 194-6; 9, 1907, 56-9. Sphinx-Oedipe, Nancy, 1906-7, 39-41. Tablettes des Cotes, Gauthier-Villars, Paris, 1906. Assoc, frang. avanc. sc, 36, 1907, II, 32-42; 41, 1912, 38-43. "Comptes Rendus Paris, 151, 1905, 78. Bull. Amer. Math. Soc, 13, 1906-7, 74. L'enseignement math., 9, 1907, 185. Bull. Soc. PhUomathique de Paris, (9), 8, 1906, 168, 270; (9), 10, 1908, 4-9, 66-83; (10), 2, 1910, 171-7. Assoc, frang. avanc. sc, 36, 1907, II, 11-20, 49-55; 37, 1909, 33-6; 41, 1912, 44-53; 43, 1914, 29-35. Rend. Accad. Lincei, Rome, (5), 15, 1906, I, 439; 26, 1917, I, 401-5. Sphinx-Oedipe, 1908-9, 81, 97. BuU. Sc. Math. El6m., 12, 1907, 292-3. II Pitagora, Palermo, 13, 1906-7, 81-91 (table serving to factor numbers from 30030 to 510 510). Table de caract6ristiques relatives a le base 2310 des facteurs premiers d'un nombre inf^rieur k 30030, Paris, 1906, 32 pp. Comptes Rendus Paris, 159, 1914, 597-9; 160, 1915, 758-760; 162, 1916, 346-8; 163, 1916,259-261; 164, 1917, 482-4. *8Jomal de sciencias math., phys. e nat., acad. sc. Lisbona, (2), 7, 1906, 209-218. 89Bull. Soc. Philomathique de Paris, (9), 9, 1907, 112-128; 10, 1908, 10-41. s'^Assoc. frang., 41, 1912, 32-7. soAnnals of Math., (2), 10, 1908-9, 88-104. ^Ubid., pp. 105-115. s^Factor table for the first ten milhons, Carnegie Inst. Wash. Pub. No. 105, 1909. '^''Quar. Jour. Math., 41, 1910, 160-7; I'interm^diaire des math., 11, 1904, 65; Math. Quest, Educ. Times, (2), 7, 1905, 81-5. "bSphinx-Oedipe, 6, 1911, 187-8; Problem of Lionnet, Nouv. Ann. Math., (3), 2, 1883, 310. '^'^Math. Quest. Educ. Times, (2), 21, 1912, 82-3. 356 History of the Theory of Numbers. [Chap, xni E. Lebon'' stated that he constructed in 1911 a table of residues p, p' permitting the rapid factorization of numbers to 100 million, the manuscript being in the Bibliotheque de I'lnstitut. H. W. Stager^ gave theorems on numbers which contain no factors of the form p{kp-\-l), where k>0 and p is a prime, and listed all such numbers < 12230. Lehmer'^ listed the primes to ten million. A. G^rardin^^ discussed the finding of all primes between assigned limits by use of stencils for 3, 5, 7, 11,. . .. He^^ described his manuscript of an auxiliary table permitting the factoring of numbers to 200 million. He^^" gave a five-page table serving to factor numbers of the second million. Cor- responding to each prime M^ 14867 is an entr\' P such that A^ = 1 000 000+P is diWsible by M. If a value of P is not in the table, A^ is prime (the P's range up to 28719 and are not in their natural order). By a simple division one obtains the least odd number in any million which is divisible by the given prime M^ 14867. C. Boulogne^^ made use of lists of residues modulis 30 and 300. H. E. Hansen^^ gave an impracticable method of forming a table of primes based on the fact that all composite numbers prime to 6 are products of two numbers 6x± 1, while such a product is QN=^ 1, where N = 6xy='X+y or Qxy—x — y. A table of values of these A^'s up to k serves to find the com- posite numbers up to 6A-. To apply this method to factor 6N=^ 1, seek an expression for A^ in one of the above three forms. N. AUiston^°° described a sieve (a modification of that by Eratosthenes) to determine the primes 4n+l and the primes An — l. H. W. Stager^"^ expressed each number < 12000 as a product of powers of primes, and for each odd prime factor gave the values >0 of A: for all divisors of the form p{kp-{-l). The table thus gives a list of numbers which include the numbers of Sylow subgroups of a group of order ^ 12000. In Ch. XVI are cited the tables of factors of a^+1 by Euler,^' Escott,^^ Cunningham^^ and WoodalP; those of a--\-k- (^* = 1, . . . , 9) of Gauss"; those of ?/" + l, 2/^±2, y'=t. 1, x^zti/^, 2«=tg, etc., of Cunningham.^^- ^"^ Concern- ing the sieve of Eratosthenes, see No\'iomagus-^ of Ch..I, Poretzkj^^ of Ch. V, :MerUn"^ and de Polignac^*^"-^ of Ch. XVIII. Saint-Loup" of Ch. XI, Re>Tnond^^^ and Kempner^^^ of Ch. XIV, represented graphically the divi- sors of numbers, while Kulik^^ gave a graphical determination of primes. »»L'interin6diaire des math., 19, 1912, 237. •♦University of California Public, in Math., 1, 1912, No. 1, 1-26. •*LiHt of prime numbers from 1 to 10,006,721. Carnegie Inst. Wash. Pub. No. 165, 1914. The introduction gives data on the distribution of primes. "Math. Gazette, 7, 1913-4, 192-3. •'Assoc, frang. avanc. sc, 42, 1913, 2-8; 43, 1914, 26-8. »«/bw/., 43, 1914, 17-26. ••"^Sphinx-Oedipe, s^rie sp4ciale. No. 1, Dec, 1913. ••L'enseignement math., 17, 1915, 93-9. Cf. pp. 244-5 for remarks by G^rardin. """Math. Quest. Educat. Times, 28, 1915, 53. '"A Sylow factor table of the first twelve thousand numbers. Carnegie Inst. Wash. Pub. No. 151, 1916. CHAPTER XIV. METHODS OF FACTORING. Factoring by Method of Difference of Two Squares. Fermat^ described his method as follows: "An odd number not a square can be expressed as the difference of two squares in as many ways as it is the product of two factors, and if the squares are relatively prime the factors are. But if the squares have a common divisor d, the given number is divisible by d and the factors by Vrf- Given a number n, for example 2027651281, to find if it be prime or composite and the factors in the latter case. Extract the square root of n. I get r = 45029, with the remainder 40440. Subtracting the latter from 2r+l, I get 49619, which is not a square in view of the ending 19. Hence I add 90061 = 2+2r+l to it. Since the sum 139680 is not a square, as seen by the final digits, I again add to it the same number increased by 2, i. e., 90063, and I continue until the sum becomes a square. This does not happen until we reach 1040400, the square of 1020. For by an inspection of the sums mentioned it is easy to see that the final one is the only square (by their endings except for 499944). To find the factors of n, I subtract the first number added, 90061, from the last, 90081. To half the difference add 2. There results 12. The sum of 12 and the root r is 45041. Adding and subtracting the root 1020 of the final sum 1040400, we get 46061 and 44021, which are the two numbers nearest to r whose product is n. They are the only factors since they are primes. Instead of 11 additions, the ordinary method of factoring would require the division by all the numbers from 7 to 44021." Under Fermat,^^^ Ch. I, was cited Fermat's factorization of the number 100895598169 proposed to him by Mersenne in 1643. C. F. Kausler^ would add 1^, 2^, . . . to iV to make the sum a square. C. F. Kausler^ proceeded as follows to express 4m+l in the form p^ — q^. Then q is even, q = 2Q. Set p-q = 2^-\-l. Then w = Q(2i3+l)+/3(|8+l). Subtract from m in turn the pronic numbers i8(/3+l), a table of which he gave on pp. 232-267, until we reach a difference divisible by 2/3+1. Ed. Collins,^ in factoring N by expressing it as a difference of two squares, let g^ be the least odd or even square > A^, according as N= 1 or 3 (mod 4), and set N = g^ — r. If r is not a square, set r = h^ — c, where h^ is the even or odd square just >r, according as r is even or odd, whence c = 4d, N = g^ — h^-\-4:d. By trial find integers x, y such that both g^-\-x and h^-\-y are squares, while x — y = 4id. Then N will be a difference of two squares. ^Fragment of a letter of about 1643, Bull. Bibl. Storia Sc. Mat., 12, 1879, 715; Oeuvres de Fer- mat, 2, 1894, 256. At the time of his letter to Mersenne, Dec. 26, 1638, Oeuvres, 2, p. 177, he had no such method. "Euler's Algebra, Frankfort, 1796, III, 2. Anhang, 269-283. Cf. Kausler, De Cribro Eratos- thenis. 1812. »Nova Acta Acad. Petrop., 14, ad annos 1797-8 (1805), 268-289. <BuU. Ac. Sc. St. Pdtersbourg, 6, 1840, 84-88. 357 358 History of the Theory of Numbers. Chap, xivi F. Landr>'* used the method of Ferinat, eliminating certain squares by their endings and others by the use of moduU. C. Henry^ stated that Landry's method is merely a perfection of the method given in the article "nombre premier" in the Dictionnaire des Math^matiques of de Montferrier. It is improbable that the latter in- vented the method (based on the fact that an odd prime is a difference of two squares in a single way), since it was given by Fermat. F. Thaarup^ gave methods to limit the trials for x in x'^ — y^ = n. We may multiply n by f = a^ — b^ and investigate nf = X" — Y^, X = ax — by, Y = bx — ay. We may test small values of y, or apply a mechanical test based on the last digit of n. C. J. Busk^ gave a method essentially that by Fermat. It was put into general algebraic form by W. H. H. Hudson.^ Let N be the given number, n" the next higher square. Then N=^n'-ro={n+iy-r,= ..., where ri, r^,... are formed from ro by successive additions of 2?i + l, 2nH-3, 2n+5, . . .. Thus r^ = ro+27?27i+??r. If r^ is a square, iV is a difference of two squares. A. Cunningham {ibid., p. 559) discussed the conditions under w^hich the method is practical, noting that the labor is prohibitive except in favorable cases such as the examples chosen by Busk. J. D. Warner^*" would make N = A~—B'^ by use of the final two digits. A. Cunningham^^ gave the 22 sets of last two digits of perfect squares, as an aid to expressing a number as a difference of two squares, and described the method of Busk, which is facilitated by a table of squares. F. W. Lawrence^ ^ extended the method of Busk (practical only when the given odd number iV is a product of two nearly equal factors) to the case in which the ratio of the factors is approximately l/m, where I and m are small integers. If I and ??? are both odd, subtract from bnN in turn the squares of a, a+1, . . . , where o^ just exceeds ImN, and see if any remainder is a perfect square (6") . If so, ImN = (a+ T)'^ — 6^. G. Wertheim^^ expressed in general form Fermat's method to factor an odd number ?n. Let a^ be the largest square <m and set m = a~+r. If p=2a+l— r is a square (n^), we eliminate r and get m = {a-{-l-\-n) X (a+1 — n). If p is not a square, add to p enough terms of the arithmetic progression 2a +3, 2a+5, . . . to give a square: p+(2a+3)-}-...H-(2a+2n-l)=s". 'Aux math^maticiens de toutes les parties du monde: communication sur la decomposition des nombres en leurs facteurs simples, Paris, 1867. Letter from Landry to C. Henry, Bull. Bibl. Storia Sc. Mat.. 13, 1880, 469-70. •Assoc, frang. av. sc, 1880, 201; Oeuvres de Fermat, 4, 1912, 208; Sphinx-Oedipe, 4, 1909, 3« Trimestre, 17-22. ^Tidsskrift for Mat., (4), 5, 1881, 77-85. •Nature, 39, 1889, 413-5. 'Nature, 39, 1889, p. 510. »'Proc. Amer. Assoc. Adv. Sc, 39, 1890, 54r-7. "Mess. Math., 20, 1890-1, 37-45. Cf. Meissner.i" 137-8. "/Wd., 24, 1894-5, 100. »»Zeit8chrift Math. Naturw. Unterricht, 27, 1896, 256-7. Chap. XIV] METHODS OF FACTORING. 359 Then 2an+n^-r = s^ and m = {a-\-ny-s^. The method is the more rapid the smaller the difference of the two factors. M. Neumann^^ proved that this process of adding terms leads finally to a square and hence to factors, one of which may be 1. F. W. Lawrence^^ denoted the sum of the two factors of n by 2a and the difference by 2b, whence n = a^ — 6^. Let q be the remainder obtained by dividing n by a chosen prime p, and write down the pairs of numbers < p such that the product of two of a pair is congruent to q modulo p. If p = 7, q = S, the pairs are 1 and 3, 2 and 5, 4 and 6, whence 2a=4, or 3 (mod 7). Using various primes p and their powers, we get limitations on a which together determine a. The work may be done with stencils. The method was used by Lawrence^^ to show that five large numbers are primes, including 10, 11 and 12 place factors of 3^^ — 1, 10^^ — 1, 10^^ — 1, respectively. The same examples were treated by other methods by D. Biddle.^^ A. Cunningham^'^ remarked that in computing by Busk's method a k for which {s+ky—N is a, square, we may use the method of Lawrence, just described, to limit greatly the number of possible forms of k. F. J. Vaes^^ expressed N in the form a^ — b"^ by use of the square a^ just >A^ and then increasing a by 1, 2, ... , and gave (pp. 501-8) an abbreviation of the method. He strongly recommended the method of remainders (p . 425) : If p is a factor oiG = h^ — g^, and iig={G — l)/2 has the remainder r when divided by p, then h={G+l)/2 must have the remainder r+1, so that p is a factor of 2r-\-l=G. For example, let G = 80047, whence ^ = 200H23 = 20M99+24 = 202-198+27,.... For r = 24, 27, 32, . . . we see that 2r+l is not a multiple of 201, 202, . . . until we reach gr = 209-191 +p, p=104, 2p+ 1 = 209. Thus 209 divides G. P. F. Teilhet^^ wrote N = a^-b in the form (a+kY-P, where P = k^ -{-2ak-{-b. Give to k successive values 1, 2, . . . (by additions to P), until P becomes a square v^. To abbreviate consider the residues of P for small prime moduli. E. Lebon^° proceeded as had Teilhet^^ and then set /=a+A; — y. Then 2kf={a-fy-b, and we examine primes /< a to see if k is an integer. M. Kraitchik^^ would express a given odd number A in the form if—x^ by use of various moduli p. Let A = r (mod p) and let Oi, . . . , a„ be the "Zeitschrift Math. Naturw. Unfcerricht, 27, 1896, 493-5; 28, 1897, 248-251. "Quar. Jour. Math., 28, 1896, 285-311. French transl., Sphinx-Oedipe, 5, 1910, 98-121, with an addition by Lawrence on g^ +1. isProc. Lond. Math. Soc, 28, 1897, 465-475. French transl., Sphinx-Oedipe, 5, 1910, 130-6. i^Math. Quest. Educat. Times, 71, 1899, 113-4; cf. 93-99. "/bid., 69, 1898, 111. i«Proc. Sect. Sciences Akad. Wetenschappen Amsterdam, 4, 1902, 326-336, 425-436, 501-8 (EngUsh); Verslagen Ak. Wet., 10, 1901-2, 374-384, 474-486, 623-631 (Dutch). i^L'intermediaire des math., 12, 1905, 201-2. Cf. Sphinx-Oedipe, 1906-7, 49-50, 55. "Assoc, franc, av. sc, 40, 1911, 8-9. ^^Sphinx-Oedipe, Nancy, Mai, 1911, num^ro special, pp. 10-16. 360 History of the Theory of Numbers. [Chap, xiv quadratic residues of p. Then r-\-x'^=ai (mod p). Thus a, — r must be a quadratic residue. Reject from Oi, . . . , a^ the terms for which a^ — r is not in the set. We get the possible residues of x modulo p. His method to fac- tor 0"=*= 1 is the same as Dickson's^^^ and is applied to show that the factor (273_|_237 4-i)/(5.239-9929) of 2"^+l is a prime in case it has no factor between 10500 and 108000. Kraitchik" extended the method of Lawrence. F. J. Vaes^^ applied his^^ method to factor Mersenne's^ number. The same was factored by various methods in L'lnterm^diaire des Math6mat- iciens, 19, 1912, 32-5. J. Petersen, ibid., 5, 1898, 214, noted that its product by 8 equals k^+k, where A: = 898423. Method of Factoring by Sum of Two Squares. Frenicle de Bessy^^ proposed to Fermat that he factor h given that h = a^+b''-=c'+(f, as 221 = 100+121 = 196+25. In 1647, Mersenne^^ (of Ch. I) noted that a number is composite if it be a sum of two squares in two ways. L. Euler^^ noted that iV is a prime if it is expressible as a sum of two squares in a single way, while if iV = a^+5^ = c^+d^, N is composite : {{a-cY+{h-d)'} {{a+cy+ih-d)'] 4(6 -d)2 Euler^' proved, that, if a number A'" = 4n+1 is expressible as the sum of two relatively prime squares in a single way, it is a prime. For, if iV were composite, then N={d^-{-b^){c^+d^) is the sum of the squares of ac^bd and ad=f^bc, contrary to hypothesis. If iV = a^+6^ = c^+d^, N is composite; for if w^e set a = c-\-x, d = b-\-y, and assume* that the common value of 2cx-\-x^ and 2by-\-'if' is of the form xyz, we get 2c = yz-x, 2b = xz-y, N = b'^+c^+xyz = \{x'^-]-y^){l+z^), whence x^-\-y^ or {x^+y^)/4: is a factor of N. To express iV as a sum of two squares in all possible ways, use is made of the final digit of N to limit the squares x^ to be subtracted in seeking differences N — x^ which are squares. Several numerical examples of factoring are treated in full. Euler-^ gave abbreviations of the work of applying the preceding test. For example, if 4n+l=5m+2 = x^+!/^, then x and y are of the form "Sphinx-Oedipe, 1912, 61-4. i^L'enseigncment math., 15, 1913, 333-4. "Oeuvres de Fermat, 2. 1894, 232, Aug. 2, 1641. "Letters to Goldbach, Feb. 16, 1745, May 6, 1747; Corresp. Math. Phys. (ed., Fuss), I, 1843, 313, 416-9. "Novi Comm. Ac. Petrop., 4, 1752-3, p. 3; Comm. Arith., 1, 1849, 165-173. *Euler gave a faultless proof in the margin of his posthimious paper, Tractatus, §570, Comm. Arith., 2, 573; Opera postuma, I, 1862, 73. We have {a+c)ia-c) = {b+d){d-b) =pqrs, a+c = pq,a—c = rs, b+d = pr, d—b = qs [since, if pbe theg. c. d. of a+c, b+d, then g(a—c) is divisible by r, whence a—c = rs]. Hence a2+6^ = (p*+a^)(g'+r')/4. »»Novi Comm. Ac. Petrop., 13, 1768, 67; Comm. Arith., 1, 379. Chap. XIV] METHODS OF FaCTOKING. 361 5p± 1. To express a number as x^+y^, subtract squares in turn and seek a remainder which is a square. N. Beguehn^^ proposed to find x such that 4pV+ 1 is a prime by exclud- ing the values x making the sum composite. The latter is the case if 4pV+l=462+(2c+l)2, a;2 = ^!±^. Set X = q+b/p. Then b is expressed rationally in terms of c and the known p. Taking p = l, he derived a tentative process for finding a prime, of the form 4x^+1, which exceeds a given number a. L. Euler^° proved that 1000^+3^ is prime since not expressible as a sum of two squares another way. A. M. Legendre^"" factored numbers represented as a sum of two squares in two ways. J. P. Kulik's^"^ tables VIII and IX, relating to the ending of squares, serve to test if 4n+l is a sum of two squares and hence to test if it be prime or composite. Th. Harmuth^^ suggested testing a^-j-b^ for factors, where a and b are relatively prime, by noting that it is divisible by 5 if a= =t 1, 6= ± 2 (mod 5) , and similar facts for p = 13, 17, 29, 37, there being p — 1 sets of values of a, b for each prime p = 4n+l. G. Wertheim^^ explained in full Euler's^'' method of factoring. R. W. D. Christie and A. Cunningham^^ granted N = A^+B^ = C^+D'^ and showed how to find a,...,dso that N={a^+b^){c^-\-cP). Similarly, if N = x^-\-Py'^ in two ways. Factoring by Use of Binary Quadratic Forms. L. Euler^^ noted that a number is composite if it be expressible in two v^ /* .fi'*^ ways in the form/ = ax^+i(32/^. The product of two numbers of the form/ is of the form g = aPx^+y^; the product of a number of the form / by one of the form g is of the form /. If for m>2 a composite number mp is express- ible in a single way in the form /, there exist an infinitude of composite numbers mq expressible in a single way by /. He called (§34) a number A'' idoneal (numerus idoneus) if, for a^ = N, every number representable hy f = ax^+^y^ (with ax relatively prime to ^y) is a prime, the square of a prime, the double of a prime or a power of 2, so that a number representable by / in a single way is a prime. It suffices to test N+y^<4N, y prime to N. He gave (§39, p. 208) the 65 idoneal numbers 1, 2, . . ., 1848 less than 10000. "Nouv. M^m. Acad. Sc. BerUn, 1777, ann6e 1775, 300. soNova Acta Petrop., 10, 1792 (1778), 63; Comm. Arith., 2, 243-8. ^'^Theorie des nombres, ed. 3, i830, I, 310. Simplification by Vuibert, Jour, de math. 6\em., 10, p. 42. Cf. I'interm^diaire des math., 1, 1894, 167, 245; 18, 1911, 256. 3obTafebi der Quadrat und Kubik Zahlen ... bis hundert Tausend, Leipzig, 1848. "Archiv Math. Phys., 67, 1882, 215-9. 32Elemente der Zahlentheorie, 1887, 295-9. "Math. Quest. Educat. Times, (2), 11, 1907, 52-3, 65-7, 89-90. "Nova Acta Petrop., 13, 1795-6 (1778), 14; Comm. Arith., 2, 198-214. 362 History of the Theory of Numbers. [Chap, xiv Euler^* used the idoneal number 232 to find all values of a < 300 for which 232a- + 1 is a prime, by excluding the values of a for which 232a^-hl = 232x~-\-y-,y>l. Euler^^ noted that N = a~ -\-\b- = x^ +\y^ imply N = l(\nr+n'^){\p^-\-q^), a=^x = \mp, nq; y=^h = mq, np, so that Xp^ + g-, or its half or quarter, is a factor of A''. He gave (p. 227) his^' former table of 65 idoneal numbers. Given one representation by cur 4-/3?/", where a/3 is idoneal, he sought a second representation. If A^ = 4n + 2 is idoneal, 4iV is idoneal. Euler^*^ called mx--\-ny- a congruent form if everj' number representable by it in a single way (with x, y relatively prime) is a prime, the square of a prime, the double of a prime, a power of 2, or the product of a prime by a factor of vin. Then also vinx~-\-y~ is a congruent form and conversely. The product mn is called an idoneal or congruent number. His table of 65 idoneal numbers is reproduced (§18, p. 253). He stated rules for deducing idoneal numbers from given idoneal numbers. He factored numbers expressed in two ways by ax^+/3i/-, where a/S is idoneal, and noted that a composite number may be expressible in a single way in that form if a^ is not idoneal. Euler^^ proved that the first five squares are the only square idoneal numbers. C. F. Kausler^- proved Euler's theorem that a prime can be expressed in a single way in the form mx'^-\-ny'^ ii m, n are relatively prime. To find a prime v exceeding a given number, see whether 38a:- +5?/^ = v has a single set of positive solutions x, y; or use 1848x"+?/^. As the labor is smaller the larger the idoneal number 38-5 or 1848, it is an interesting question if there be idoneal numbers not in Euler's list of 65. Cf. Cunningham. ^^ Euler"*^ gave the 65 idoneal numbers n (with 44 a misprint for 45) such that a number representable in a single way by nx~-\-y'^ {x, y relatively prime) is a prime. By using n = 1848, he found primes exceeding 10 million. N. Fuss^ stated the principles due to Euler.^^ E. Waring^^ stated that a number is a prime if it be expressible in a single way in the form ar-\-mh^ and conversely. A. M. Legendre"*^ would express the number A to be factored, or one of its multiples kA , in the form t~+air, where a is as small as possible and within the limits of his Tables III-YII of the linear forms of divisors of f^au^. »»Nova Acta Petrop., 14, 1797-8 (1778). 3; Ck)mm. Arith., 2, 215-9. »»/Wd., p. 11; Comm. Arith., 2, 220-242. For X = 2, Opera postuma, I, 1862, 159. "/Wd.. 12, 1794 (1778), 22; Comm. Arith., 2, 249-260. "/6wf., 15, ad annos 1799-1802 (1778), 29; Comm. Arith., 2, 261-2. "Ibid., 156-180. "Nouv. M^m. Berlin, ann6e 1776, 1779, 337; letter to Beguelin, May, 1778; Comm. Arith., 2, 270-1. **Ibid., 340-6 «Medit. Algebr., ed. 3, 1782, 352. «^h6orie des nombres, 1798, pp. 313-320; ed. 2, 1808, pp. 287-292. German transl. by Maser, 1, 329-336. Cf. Sphinx-Oedipe, 1906-7, 51. Chap. XIV] METHODS OF FACTORING. 363 Then the divisors of A are included among these linear forms. When VS is converted into a continued fraction, let (VM +/)/!) be a complete quotient, and p/q the corresponding convergent. Then ^D = p^ — kAq^, so that the divisors of A are divisors oi p^=f=D. C. F. Gauss'*^ stated that the 65 idoneal numbers n of Euler and no other numbers have the two properties that all classes of quadratic forms of determinant —n are ambiguous and that any two forms in the same genus (Geschlecht) are both properly and improperly equivalent. Gauss^^ gave a method of factoring a number M based on the deter- mination of various small quadratic residues of M. Gauss^^ gave a second method of factoring M based on the finding of representations of M by forms x^+D, where D is idoneal. F. Minding^° gave an exposition of the method of Legendre.^^ P. L. Tchebychef^^ gave a rapid process to find many forms x^^ay^ which represent a given number A or a multiple of A. Then a table of the linear forms of the divisors of x^^ay^ serves to limit the possible factors of A. Tchebychef^^ gave theorems on the limits between which lie at least one set of integral solutions of x^ — Dy^ = ± iV. If there are two sets of solu- tions within the limits, N is composite. There are given various tests for primality by use of quadratic forms. C. F. Gauss^^ left posthumous tables to facilitate factoring by use of his*^ second method. F. Grube^^ criticized and completed certain of Euler's proofs relating to idoneal numbers, here called Euler numbers. While Gauss^^ said it is easy to prove Euler's*^ criterion for idoneal numbers, Grube could prove only the following modification: Let Q, be the set of numbers D+n^^4Z) in which n is prime to D. According as all or not all numbers of 12 are of the form q, 2q, q"^, 2^ {q a prime), D is or is not an idoneal number. E. Lucas^^ proved that if p is a prime and A; is a positive integer, and p = x^+ky^, then pT^Xi^+ky^^ for values Xi, yi distinct from ^x, ^y. P. Seelhoff^^ made use of 170 determinants (including the 65 idoneal numbers of Euler and certain others of Legendre), such that every reduced form in the principal genus is of the type ax^-\-by^. To factor A^ seek among the numbers m of which iV is a quadratic residue several values <^Disq. Arith., 1801, Art. 303. "7btd., Arts. 329-332. *mid., Arts. 333-4. s^Anfangsgriinde der Hoheren Arith., 1832, 185-7. "Theorie der Congruenzen (in Russian), 1849; German transl. by H. Schapira, 1889, Ch. 8, pp. 281-292. "Jour, de Math., 16, 1851, 257-282; Oeuvres, 1, 73. «Werke, 2, 1863, 508-9. "Zeitschrift Math. Phys., 19, 1874, 492-519. "Nouv. Corresp. Math., 4, 1878, 36. [Euler."] "Archiv Math Phys., (2), 2, 1885, 329; (2), 3, 1886, 325; Zeitschrift Math. Phys., 31, 1886, 166, 174, 306; Amer. Jour. Math., 7, 1885, 264; 8, 1886, 26-44. 364 History of the Theory of Numbers. [Chap, xiv for which N is represent able by x^-\-my-. For example, if iV = 31-2^*-f 1, Eliminating 19-83 between the first two, we get nN = w'^ — 7f. This with the third leads to factors of N. In general, when elimination of common factors of the 77i's has led to representations of two multiples of N by the same form x~+ny^, we may factor N unless it be prime. H. Weber^^ computed the class invariants for the 65 determinants of Euler and remarked that there is no known proof of the fact found by induction by Euler and Gauss that there are only 65 determinants such that all classes belonging to the determinant are ambiguous and hence each genus has only one class. T. Pepin^^ developed the theory of Gauss'^^ posthumous tables and the means of deducing complete tables from the given abridged tables. Pepin^* showed how to abridge the calculations in using the auxiliary tables of Gauss in factoring a" — 1, where a and n are primes. D. F. Seliwanoff^° noted that the factoring of numbers of the form t' — Du- reduces to the solution of {D/x) = \, all solutions of which are easily found by use of six relations by Euler on these Jacobi symbols {D/x) . E. Lucas" gave a clear proof of Euler's remark that a prime can not be expressed in two ways in the form Ax~-\-By-, if A, B are positive integers. S. Levanen^^ showed and illustrated by examples and tables how binary quadratic forms may be applied to factoring. G. B. ^Mathews*"^ gave an exposition of the subject. T. Pepin^ applied determinants — 8n — 3 for which each genus has three classes of quadratic forms. The paper is devoted mainly to the solution of X-+ (871+3)?/"" = 4A, where A is the number to be factored. T. Pepin^^ assumed that the given number N had been tested and found to have no prime factor ^p. Let Xx+1, Xy+1 be the two factors of N, each between p and N/p. The sum of the factors lies between 2VW and p+N/p. Let x—y = u, x+y=pz. Then {N — l)/\ = Xxy+x+y gives in which special values are assigned to p. This equation yields a quadratic congruence for w" with respect to an arbitrary prime modulus, used as an excludant. The method applies mainly to numbers a^=^l. E. Cahen^^ used the linear divisors of x^ + Dy'^. '"Math. Annalen, 33, 1889. 390-410. "Atti Accad. Pont. Nuovi Lincei, 48, 1889, 135-156. "Ibid., 49, 1890. 163-191. "Moscow Math. Soc, 15, 1891, 789; St. Petersburg Math. See, 12, 1899. •'Th6orie des nombres, 1891, 356-7. "^vereigt af finska Vetenskaps-Soc. forhandlingar, 34, 1892, 334-376. "Theory of Numbers, 1892, 261-271. French transl., Sphinx-Oedipe, 1907-8, 155-8, 161-70. "Memorie Accad. Pontif. Nuovi Lincei, 9, I, 1893, 46-76. Cf. Pepin," 332. «;6iV/., 17, 1900-1, 321-344; Atti, 54, 1901, 89-93. Cf. Meissner"*, 121-2. "filaments de la th^orie des nombres, 1900, 324-7. Sphinx-Oedipe, 1907-8, 149-155. Chap. XIV] METHODS OF FACTORING. 365 A. Cunningham" and J. Cullen listed the 188 prime numbers x^+18482/^ between 10"^ and 101-10^, with x prime to 1848?/. A. Cunningham^ ^ noted that two representations of N by ixx^-\-vy^ lead to factors of N under certain conditions. A. Cunningham^^ recalled that an idoneal number / has the property that, if an odd number A'' is expressible in only one way in the form N = mx^-\-ny^, where mn = I, and mx^ is prime to ny^, then A" is a prime or the square of a prime. Euler's largest I is 1848. There is no larger I under 50000, a computation checked by J. Cullen. Cunningham noted on the proof-sheets of this history that this limit has been extended to 100 000. A. Cunningham^" noted conditions that an odd prime be expressible by f^qu" when q or —q'l^ idoneal. F. N. Cole'^^ discussed Seelhoff's^^ method of factoring. Al. Laparewicz^^ described and applied Gauss' ^^'^^ two methods. P. Meyer^^ discussed Euler's theorem that, if n is idoneal, a number representable only once by x^-\-ny'^ is a prime. R. BurgwedeF^ gave an exposition and completion of the method of Euler^^'^^ and an exposition of the methods of Gauss.^^'*^ L. Valroff stated and A. Cunningham'^^" proved that (Dx'^ — a^) (Dy"^ — a^) = Dz^ — a^ implies that one factor is composite unless x^ = y^ = 4: when o = 1, D = 2, and in the remaining cases if the two factors are distinct and > 1. A. Gerardin'^^ gave a method illustrated for N = a^ — 5-29^, where a = 6326. We shall have a second such representation N = (a-{-5xy — 5y^ if E=5x^+2ax+841=y^. Use is made of various moduli ?« = 4, 3, 7, 25, ... . On square-ruled paper, mark x = 0, 1, 2, . . . at the head of the columns. On the line for modulus m, shade the square under the heading x when x makes E a quadratic non- residue of m. Then examine the column in which occurs no shaded square. Up to x^l5, these are x = (excluded), and x = 4, which gives A = 6346^ — 5-227^ and the factor 99^ — 5-2^. The same diagram serves for all num- bers 1050 ff +671, our N being given by i7 = 38108. To apply the method to A = (2a;)*+1 = (4x^+1)^ — 2(2x)^, seek a second representation N=(4x^ +2p + lY-2{2uy. The condition is {2p-\-l)x^+Mp-j-l)=u^, solutions of which are found for p = 1, 8, 9, . . . , 6^, 35^, ... Or we may choose x, say X = 48, and find p = S, u = 198. s^Brit. Assoc. Reports, 1901, 552. The entry 10098201 is erroneous. "Proc. London Math. Soc, 33, 1900-1, 361. «976id., 34, 1901-2, 54. ■"^Ibid., (2), 1, 1903, 134. 7iBuU. Amer. Math. Soc, 10, 1903-4, 134-7. "Prace mat. fiz., Warsaw, 16, 1905, 45-70 (PoUsh). ^'Beweis eines von Euler entdeckten Satzes, betreffend die Bestimmung von Primzahlen, Diss., Strassburg, 1906. '^Ueber die Eulerschen und Gausschen Methoden der Primzahlbestimnaung, Diss., Strassburg, 1910, 101 pp. '*»Sphinx-Oedipe, 7, 1912, 60, 77-9. "Wiskundig Tijdschrift, 10, 1913, 52-62. 366 History of the Theory of Numbers. [Chap, xiv G^rardin^® gave a note on his machine to factor large numbers, espe- cially those of the form 2x^ — 1. Factoring by Method of Final Digits. Johann Tessanek^° gave a tedious method of factoring N, not divisible by 2, 3, or 5, when A^/10 is within the limits of a factor table. For example, let N=10a+l; its factors end in 1,1 or 3,7 or 9,9. To treat one of the four cases, consider a factor lOx+3, the quotient being 102+7. Then z is the quotient of a — 2 — 7x by lOx+3. Give to x the values 1, 2, . . ., and test a — 9 for the factor 13, a — 16 for 23, etc., by the factor table. He gave a lengthy extension^^ to di\'isors 100j+10/+gr. Again, to factor iV = 2a + l, given a table extending to N/2, note that if 2xH-l is a divisor of N, it di\'ides a—x, which falls in the table. F. J. Studnicka^^ quoted the last result. N. Beguelin^ would factor iV = 4p+3 by considering the final digit of TT = (A'' — 1 1)/4 and hence find the proper line in an auxihary table (pp. 291-2) , each line containing four fractional expressions. Proceed with each until we reach a fraction whose numerator is zero. Then its denominator is a factor of A^. Georg Simon KliigeP noted that a number, not divisible by 2, 3 or 5, is of the form 30x+m (m = l, 7, 11, 13, 17, 19, 23, 29). Suppose 10007 = (30x+w)(30!/+n). Then {m, n) = (l, 17), (7, 11), (13, 29) or (19, 23). For m = 1, n = 17, we get 333-?/ ^=3o^Ti7' ^<^' y<^' But X is not integral for y = 0, 1, 2, 3. Johann Andreas von Segner {ibid., 217-225) took two pages to prove that any number not divisible by 2 or 3 is of the form 6n='= 1 and noted that, given a table of the least prime factor of each 6n=tl, he could factor any number within the limits of the table! Sebastiano Canterzani^^ would factor 10^ + 1, by noting the last digits 1, 1 or 3, 7 or 9, 9 of its factors. If one factor ends in 7, there are 10 possibilities for the digit preceding 7; if one ends in 1 or 9, there are five cases; hence 20 cases in all. A. Niegemann^^" used the same method. Anton Niegemann^® gave a method of computing a table of squares arranged according to the last two digits. Thus, if A76 = {l0x — Qy, then ™\s30c. fran?. avanc. sc, 43, 1914, 26-8. Proc. Fifth Internat. Congress, II, 1913, 572-3; Brit. Assoc. Reports, 1912-3, 405. ••Abhandl. einer Privatgesellschaft in Bohmen, zur Aufnahme der Math., Geschichte, . . . , Prag, I, 1775, 1-64. "M. Cantor, Geschichte Math., 4, 1908, 179. "Casopis, 14, 1885, 120 (Fortschr. der Math., 17, 1885, 125). "Nouv. Mdm. Ac. Berhn, ann^e 1777 (1779), 265-310. •*Leipziger Magazin fiir reine u. angewandte Math, (eds., J. Bernoulli und Hindenburg), 1, 1787, 199-216. "Memorie dell' Istituto Xazionale Italiano, Classe di Fis. e Mat., Bologna, 2, 1810, II, 445-476. ""Entwickelung . . . Theilbarkeit, Jahresber. Kath. Gymn. Kobi., 1847-8, 23. "Archiv Math. Phys., 45, 1866, 203-216 . ^'11 Chap. XIV] METHODS OF FACTORING. 367 A0 = 10x^ — 12x — 4:, whence 12a:+4 is divisible by 10, so that a: = 5(i — 2. Then A=25d'^ — 2Qd+Q. Thus if we delete the last two digits 7 , 6 of squares A7Q, we obtain numbers A whose values for d = l, 2, . . . can be derived from the initial one 5 by successive additions of 49, 49+50, 49+2-50, . . . . He gave such results for every pair of possible endings of squares. A similar method is applied to any composite number. One case is when the last two digits are m, 1 and Aml = (10a; — 1)(10?/ — 1). Then A0 = 10xy — y — x — m, y+x+m = 10a, A = 10ax—x'^—mx — a. The discriminant of the last equation must be a square. A table of values of A for each a may be formed by successive additions. G. Speckmann^^ noted that the two factors of AT" = 2047 end in 1 and 7 or 3 and 9. Treating the first case, we see that, if a and b are the digits in tens place, 6+7a=4 (mod 10), so that the factors end in 01 and 47, or 11 and 77, etc. G. Speckmann^^ wrote the given number prime to 3 in the form 9a +6 (6<9), so that the sum of its digits is =6 (mod 9). By use of a small auxiliary table we have the residues modulo 9 of the sums of the digits of every possible pair of factors. R. W. D. Christie^^ and D. Biddle^° made an extensive use of terminal digits. E. Barbette^ ^ noted that lOd+u has a divisor lOw — 1 if and only if d+?nw has that divisor. Set d-\-'mu = n(10m — l), d = 10d'+u'. Then mn = d'-\-x, lOx = 'mu-\-n-\-u'. Eliminating n, we get a quadratic for m. Its discriminant is a quadratic function of x which is to be made a square. Similarly for lOm + 1, 10m ±3. A. Gerardin^^" developed Barbette's^^ method. R. Rawson^^ found Fermat's^ factors of a number proposed by Mersenne by writing it to the base 100 and expressing it as (a- 10^ +23) (6 -10^ +3). J. Deschamps^^ would use the final digits and auxiliary tables. A. Gerardin®^ would factor N (prime to 2, 3, 5) by use of iV=120n+i^=(120x+a)(120i/+6), and a table showing, for each of the 32 values of K< 120, the 16 pairs o, b (each< 120) such that ab=K (mod 120). He factored Mersenne's number.^ Factoring by Continued Fractions or Fell Equations. Franz von Schaffgotsch^"" would factor a by solving az^-^l = x^ (having "Archiv Math. Phys., (2), 12, 1894, 435. ssArchiv. Math. Phys., 14, 1896, 441-3. "Math. Quest. Educat. Times, 69, 1898, 99-104. Cf. Meissner,"^ 138-9. '"/feid., 87-88, 112-4; 71, 1899, 93-9; Mess. Math., 28, 1898-9, 120-149, 192 (correction). Cf. Meissner,"8 137-8. siMathesis, (2), 9, 1899, 241. "«Sphinx-Oedipe, 1906-7, [1-2, 17, 33], 49-50, 54, 65-7, 77-8, 81-4; 1907-8, 33-5; 5, 1910, 145-7; 6, 1911, 157-8. s^Math. Quest. Educat. Times, 71, 1899, 123-4. "^BuU. See. Philomathique de Paris, (9), 10, 1908, 10-26. s'Assoc. frang., 38, 1909, 145-156; Sphinx-Oedipe, Nancy, 1908-9, 129-134, 145-9; 4, 1909, 3« Trimestre, 17-25. i»»Abh. Bohmischen Gesell. Wiss., Prag, 2, 1786, 140-7. 368 History of the Theory of Numbers. [Chap, xiv solutions if a is not a square) and testing x" — 1 for a factor in common with a. Further, U ay-\-l=x'^ does not hold for l<x<a — 1, then a is a power of a prime and conversely [false if a = 10]. Marcker^*'^ noted that if there are 2n terms in the period of and Q = 0, Q' = a, Q" = a'P' -Q',. . .,^ p_i p/_ ^~Q'' ptf _ ^~Q"^ r — 1, r — p ' pf ) • ■ • i then the nth P or its half is a factor of A. If A is a prime, then the nth Pis 2. J. G. Birch^°- derived a factor of N from a solution x of x'^ = Ny+l. The continued fraction for x/{N —x) is of the form 1 J_ J_ ill Go- 1 +01+02+ ■ • • +a2+ai+ao' and N is the continuant defined as the determinant with Oq, Oi,. . ., a„_i, On) On-i>- •■> Oi, Oo in the main diagonal, elements +1 just above this diagonal, elements —1 just below, and zeros elsewhere. Then the continu- ant with the diagonal ao> • • •> o„_i is a factor of N. W. W. R. BalP°^ appHed this method to a number of Mersenne.^ A. Cunningham^*^ noted that a set of solutions of if—Dx^ = — 1 gives at sight factors of 7/" + l. M. V. Thielmann^o^ illustrated his method by factoring /c = 36343817. The partial denominators in the continued fraction for \/^ are 1, 1, 2, 1, 1, 12056. Drop the last term and pass to the ordinary fraction 7/12. Hence set (12x+7)^ = 12^i/+l. The least solution is x = 4, y = 2\. Using the part of the period preceding the middle term it» = 2, we get Y^ = -^, P = l, M = 2, Q = iyM+2P = 6, u = MQ = \2. Hence ^" — 21m^ = 1 has the solution < = 55. For a suitably chosen n, ifc = wV+2^n+21= (2g2n+^^ Uu^n+^-^X where q is the largest integer ^ Q/2. Here n = 502 and the factors of k are 2-3"n + 7 and 2-22n+3. D. N. Lehmer^''^ noted that if jR = pg' is a product of two odd factors whose difference is <2y/R, so that Kp— 9)^<V7^, then x'-Rf = \{V-q? has the integral solutions x = (p+g')/2, y = \. Hence i(p — g')^ is a denomi- nator of a complete quotient in the expansion of y/R, as a continued fraction, ""Jour, fur Math., 20, 1840, 355-9. Cf. I'interm6diaire dea math., 20, 1913, 27-8. J^Mcsa. Math., 22, 1892-3. 52-5. 'o'/6id., p. 82-3. French transl., with Birch"^, Sphinx-Oedipe, 1913, 86-9. >"/6id., 35, 1905-6, 166-185; abat. in Proc. London Math. Soc. 3, 1905, xxii. '""Math. Annalen, 62, 1906, 401. Bull. Amer. Math. Soc, 13, 1906-7, 501-2. French transl., Sphinx-Oedipe, 6, 1911, 138-9. IM Chap. XIV] Methods of Factoring. 369 in view of the theorem of Lagrange: If x^—Ry'^=^D has relatively prime integral solutions x, y, where D< \/r, then D is a denominator of a com- plete quotient in the expansion of \/r as a continued fraction. Factoring by use of Various Moduli. C. F. Gauss^^° gave a "method of exclusion," based on the use of various small moduli, to express a given number in a given form mx^+ny^. V. Bouniakowsky^^^ noted that information as to the prime factors of a number N may be obtained by comparing the solution x = (l){N) of 2''=1 (mod N) with the least positive solution x = a found by a direct process such as the following : Since 2" = NK+l, multiply the given N by the unknown K, each expressed in the binary scale (base 2), add 1 and equate the result to 10. . .0. The digits of K are found seriatim and very simply. H. J. WoodalP^^ expressed the number N to be factored in the form ^a_|_^6_^ . . . +r, where r< 1000, while a, /3, . . . are small, but not necessarily distinct. Hence the residues of N with respect to various moduli are readily found by tables of residues. F. Landry^^^ employed the method of exclusion by small moduli. D. Biddle^^^ investigated factors 2Ap + l by using moduli A^, 4A^. C. E. Bickmore, A. Cunningham and J. CuUen^^^ each treated the large factor of 10^^+1 by use of various moduli, and proved it is prime. J. Cullen"^" gave an effective graphical process to factor numbers by the use of various moduli; the numbers to be searched for in a diagram are all small. Alfred Johnsen^^^ used Rt(p) to denote the numerically least residue of p modulo t. Then, for every p, t, k, [Rmf+Ri(p-k')^Rt(p) (modO. If Hs a factor of the given number p, the left member will be divisible by t. In practice take k^ to be the nearest square to p, larger or smaller. For example, let p = 4699, k^ = 4624 = 68^ p-k'' = 75. Then t [^.(68)P Rt(75) Sum 7 13 37 4 9 36 -2 -3 "i 2 6 37 Thus 37 is the least factor of p. ""DJsq. Arith., 1801, Arts. 323-6. i"M6m. Ac. Sc. St. P^tersbourg, Math.-Phys., (6), 2, 1841, 447-69. Extract in Bull. Ac. Sc, 6, p. 97. Cf. Nordlund.i" "2Math. Quest. Educat. Times, 70, 1899, 68-71; 71, 1899, 124. "sproc^d^s nouveaux . . . , Paris, 1859. Cf. A. Aubry,"^ pp. 214-7. i"Mess. Math., 30, 1900-1, 98, 190. Math. Quest. Educat. Times, 74, 1901, 147-152. "'Math. Quest. Educat. Times, 72 1900, 99-103. "sa/bid., 73, 1900, 133-5; 75, 1901, 10^4. Proc. London Math. Soc, 34, 1901-2, 323-334; (2), 2, 1905, 138-141. "'Nyt Tidsskrift for Mat., 15 A, 1904, 109-110. 370 History of the Theory of Numbers. [Chap, xiv K. P. Nordlund^^^ would use the exponent e to which 2 belongs modulo iV [Bouniakowsky^^^]. For A^ = 91, e = 12 is not a divisor of iV — 1, so that A^ is composite, and we expect the factor 13. L. E. Dickson"^ found the factors of 56'±1, 26'Hl, 34^^-}-l, 52^Hl by an expeditious method. For example, each factor of is =l(mod 14). Let 6 = (l + 14A;)(l + 14A'i). Then A'+;i-i + 14A'A-i=4Ar, /:+A'i = 4+14/1. Thus h and A-j are the roots of a quadratic whose discriminant Q is of the second degree in In. By use of various moduli which are powers of small primes, the form of h is limited step by step, until finally at most a half dozen values of h remain to be tested directly. L. E. Dickson^ ^® gave further illustrations of the last method. J. Schatunovsky^^° reduced to a minimum the number of trials in Gauss'^^° method of exclusion, taking the simplest case m = \. He gave theorems on the linear forms of the factors of a"~H-D6^, which lead easily to all its odd factors when D is an odd prime. H. C. Pocklington^^^ would use Fermat's theorem to tell whether A" is prime or composite. Choose an integer x and find the least positive residue of x^"^ modulo A''; if p^l, A'' is composite. But if it be unity, let p be a prime factor (preferably the largest) of A^— 1 and contained a times in it. Find the remainder r when x"* is divided by A', where 7n = (A' — l)/p. If Tt^I, let 6 be the g. c. d. of r— 1 and A^. If 6>1, we have a factor of A^. If 6 = 1, all prime factors of A^ are of the form Ap^ + l. But if r = 1, replace m by mlq^ where q is any prime factor of m and proceed as before. D. Biddle^^^" made use of various small moduli. A. Gerardin^^^'' used various moduli to factor 77073877. See papers 14, 15, 21, 22, 48, 65. Factoring Into Two Numbers 6n±l. G. W. Kraft^" ^^^^^ ^hat 6a+l = (6w+l)(6n+l) implies _ a—m ^~6m + l' Find which 7?? = 1, 2, 3, . . . makes n an integer. Ed. BartP^ tested 6-186+5 for a prime factor less than 31, just less than its square root, by noting that 186, 185, 184, 183, 182 are not divisible by 5, 11, 17, 23, 29, respectively; while the last of 7, 13, 19 is a factor. '"Gotcborps Kunpcl. Vetenskaps. Hand!., (4), 1905, VII-VIII, pp. 21-4. »«Amer. Math. Monthly, 15, 1908, 217-222. "'Quar. Jour. Math., 40, 1909, 40-43. ""Der Grosste GemeinschaftUche Teller von Algebr. Zahlen zweiter Ordnung, Diss. Strassburg, Leipzig, 1912. >"Proc. Cambridge Phil. Sec, 18, 1914-5, 29-30. "'"Math. Quest. Educat. Times, (2), 25, 1914, 43-6. "'"L'enseignement math., 17, 1915, 244-5. '"Novi Comm. Ac. Petrop., 3, ad annos 1750-1, 117-8. ""Zur Theorie der Primzahlen, Progr. Mies, Pilsen, 1871. Chap. XIV] Methods of Factoring. 371 F. Landry ^^^ treated the possible pairs 6n± 1 and 6n'± 1 of factors of N. Taking for example the case of the upper signs, we have Set n+n' = 6/i+r. Qnn'+n-\-n' = — - — = Qq-\-r. 6 Then nn' = q — h, whence - q—n'(r—n') h = - 6n' + l Give to n' values such that 6n'+l is a prime < ViV. K. P. Nordlund^25 treated Qp-l = {Qm+l){Qn-l) solved for m. D. Biddle^^^ applied the method to 6n±l. Hansen,^^ of Ch. XIII, used this method. Miscellaneous Methods of Factoring. Matsunaga^^^ wrote the number to be factored in the form r^+R. For r odd, set r = Bi, Bi — 2 = B2, Bg — 2 = ^3, ... and perform the following calculations : R = Q,B,+Au Ai+K2' = Q2B2+A2, A2-\-Ks' = Q3Bs+As, etc., until we reach A„ = 0; then Bn is a factor, and replace i? by jR + 1 in what precedes. J. H. Lambert ^^° used periodic decimals [see Lambert,^ Ch. VI]. Jean Bernoulli^^^ gave a method based on that of Lambert (Mem. de Math. Allemands, vol. 2). Let A=a^+b have the factors a—x and a+x+y. Then x^ = ay— xy — b. Solve for a:. Thus 2/^+ 4a?/ — 46 must be a square. Take y = l, 2, . . . and use a table of squares. J. Gough^^^ gave a method to find the factors r, s of each number f^ — c between (/— 1)^ and/^. For example, let/=3 and make a double row for each r = l,. . ., /. In the upper row for r = l, insert 2/— 1,. .., 1, 0; in the lower, (/— 1)^, . . ., /^. In the upper row for r = 2, insert 1 (the remainder K2 = 2Q2-\-K2', K3 = 2Qs+Ks', K2'^^K,+4, K3' = 7^2+8, If r is even, set r — 1 = Bi r = l c = 5 s = 4 4 5 3 6 2 7 1 8 9 r = 2 c = 5 s = 2 3 3 1 4 r = 3 c = s = 3 i2'»Assoc. fran?. avanc. sc, 9, 1880, 185-9. i25Nyt Tidsskrift for Mat., Kjobenhavn, 15 A, 1904, 36-40. i^^Math. Quest. Educ. Times, 69, 1898, 87-8; (2), 22, 1912, 38-9, 84-6. i^'Japanese manuscript, first half eighteenth century, Abhandl. Geschichte Math. Wiss., 30, 1912, 236-7. isoNoya Acta Eruditorum, 1769, 107-128. "iNouv. Mem. Ac. BerHn, ann6e 1771, 1773, 323. "2Jour. Nat. Phil. Chem. Arts (ed., Nicholson), 1, 1809, 1-4. 372 History of the Theory of Numbers. [Chap, xiv on dmding/^ by 2), 1+2, 1+2+2 under 1, 3, 5 of the first row for r = l; in the lower row, insert 4 (the quotient), 4 — 1, 4 — 2. To factor/^ — c, locate the column headed by the given c; thus, for c = 3, the factors are s = 6, r = 1 and s = 3, r = 2. Since c = 2 occurs only in the first row, 9 — 2 is prime. Joubin,^^^ J. P. Kulik,^^^ 0. V. Kielsen,i36 ^^d G. K. Winter^^e published papers not accessible to the author. E. Lucas gave methods of factoring and tests for primes (Ch. XVII). D. Biddle^" wrote the proposed number N in the form S^-\-A, where S- is the largest square < A^. Write three rows of numbers, the first begin- ning with A, or A —S if A >S; the second beginning with S (or S+1) and increasing by 1 ; the third beginning with S and decreasing by 1. Let A^, B„, C„ be the nth elements in the respective rows. Then except that, when i4„>C„, we subtract C„ from An as often (say k times) as will leave a positive remainder, and then B„ = 5„_i + 1+A:. WTien we reach a value of n for which i4„ = 0, we have N = BJJn- For example, if iV = 589 = 24H 13, the rows are 13 14 17 1 9 24 25 26 28 29 31 (factors 31, 19). 24 23 22 21 20 19 It may prove best to start with 2N instead of with N. O. Meissner^^^ reviewed many methods of factoring. R. W. D. Christie^^^ gave an obscure method by use of "roots." Christie"" noted that, if N = AB, A = {4hN+d''-d)/i2b), B = (46Ar+dHd)/2, d=a-hc, whence d'' = {B-h Ay. D. Biddle"^ gave a method of finding the factors of N given those of AT+l. SetL = iV-l. Try to choose i^ and M so that i(:iV/ = iV+ 1 and so that 1 +Z is a factor of N. Since 2N={l-\-K) M-\-L — M,we will have L-M=il+K)m, whence 2N={l+K){M+m). For iV=1829, A^+1 = 2-3-5-61. Take K = 30, M = 61. Then m = 57, M+w = 2-59, iV = 31-59. He gave {ibid., p. 43) the theoretical test that N = S^+A is composite if the sum of r terms of is an integer for some value of r. '»*Sur lea facteura num6riques, Havre, 1831. >«Abh. K. Bohm. Gesell. Wiss., 1, 1841 (2, 1842-3, 47, graphical determination of primes). "HDra et heel tals upplosning i factorer, Kjobenhavn, 1841. "•Madras Jour. Lit. Sc, 1886-7, 13. "'Mesa. Math., 28, 1898-9, 116-20; Math. Quest. Educat. Timea, 70, 1899, 100, 122; 75, 1901, 48; extension, (2), 29, 1916, 43-6. "«Math. Naturw. Blatter, 3, 1906, 97, 117, 137. "'Math. Quest. Educat. Times, (2), 12, 1907, 90-1, 107-8. "o/bid., (2), 13, 1908, 42-3, 62-3. "i/btd., (2), 14, 1908, 34. The process is well adapted to factoring 2P-1, (2), 23, 1913, 27-8. I Chap. XIV] METHODS OF FACTORING. 373 E. Lebon^^^" would first test AT for prime factors P just < VN. Let Q be the quotient and R the remainder on dividing N by P. If Q and R have a common factor, it divides N) if not, N is not divisible by any factor of Q or of R. D. Biddle^^^ considered N = S'^+A = {S+u){S-v), wrote w = iVi and obtained hke equations in letters with subscripts unity. Then treat UiVi = N2 similarly, etc. A. Cunningham^^^ noted that the number of steps in Biddle's^^^ process is approximately the value of A; in 2''' = N, and developed the process. E. Lebon^^^ treated the decomposition of forms a:"±a:^±a;T± . . .±1 (a>|8>7...) of degrees ^ 9 into two such forms, using a table of those forms of degrees ^ 4 with all coefficients positive which are not factorable. The base most used in the examples is x = 10. But bases 2 and 3 are considered. E. Barbette^^^ quoted from his^^^ text the theorem that any integer N can be expressed in each of the four forms where ^j: = x{x-\-l)/2. The resulting new methods of factoring are now simplified by use of triangular and quadratic residues. The first formula implies N={x—y){x-\ry+\)/2. In his text, he considered the sum iv=(2/+i)+(y+2)+ . ..+{x-\)+x=^,-^y of consecutive integers. Treating four types of numbers N, he proved that this equation has 1, 2 or more than 2 sets of integral solutions x, y, according as iV is a power of 2, an odd prime, or a composite number not a power of 2. He proved independently, but again by use of sums of consecutive integers, that every composite number not a power of 2 can be given the form* N = u (2y — w+l)/2, where u and v are integers and v^u'^S. Solving for u, and setting x = 2v-\-l, we get 2u = x+(x'^ — 8Ny^^. Hence x'^—SN = y^ is solva- ble in integers [evidently by a: = 2A'+l, y = 2N—l]. Finally, Nz=Ax is equivalent to (2a;+l)^ = 8A^2;+l. For four types of numbers N, the solu- tions of y^ = SNz-\-l are found and seen to involve at least two arbitrary constants. A. Aubry^^^ reviewed various methods of factoring. "wil Pitagora, Palermo, 14, 1907-8, 96-7. i«Math. Quest. Educat. Times, (2), 19, 1911, 99-100; 22, 1912, 38-9; Educat. Times, 63, 1910, 500; Math. Quest, and Solutions, 2, 1916, 36-42. "»/6id., (2), 20, 1911, 59-64; Educat. Times, 64, 1911, 135. i^BuU. soc. philomathique de Paris, (10), 2, 1910, 45-53; Sphinx-Oedipe, 1908-9, 81-3, 97-101 "^L'enseignement math., 13, 1911, 261-277. "^Les sommes de p-i&mes puissances distinctes ^gales k une p-ieme puissance, Paris, 1910, 20-76. *This follows from the former result N = {x —y)(x+y +1) /2 by setting x=v, y = v—u. To give a direct proof, take u to be the least odd factor > 1 of the composite number N not a power of 2; then q = N /u can be given the form y — (u— 1)/2 by choice of y. If t;<M, then g<(«+l)/2<u, so that q has no odd factor and 2 = 2''. But N='2^U is of the desired form if we take v^u/2'^N. "'Sphinx-Oedipe, num^ro sp6c., June, 1911, 1-27. Errata and addenda, num6ro 3p6c., Jan., 1912, 7-9, 14. L'enseignement math., 15, 1913, 202-231. 374 History of the Theory of Numbers. [Chap, xiv S. Bisman^"** noted that N is composite if and only if there exist two integers A, B such that A +2 B and A-\-2BN divide 2{N-1) and (A''-1)A, respectively. But there is no convenient maximum for the smaller integer B. To find the factor 641 of 2'^^ + l there are 16 cases. A. G^rardin^^^ gave a report on methods of factoring. J. A. Gmeiner,^^° to factor a, prime to 6, determined b and e so that 9a = 166+e, 0^€<16. Let cu^ be the largest square <6 and set b = oi^-\-p, cr = p— CO. Hence 9a = 16(co — a:)(w+a;H-l)+r(a:), where t(x) = 16(7+€+16x(x+1). Since t{x)=t{x — 1)+S2x, we may rapidly tabulate the values of r(x) for x = 0, 1, 2, . . . . If we reach the value zero, we have two factors of a. To prove that a is a prime, we need extend the table until co+x + l is the largest square <a. To modify the process, use 4a = 76 + €. A. Reymond^^^ used the graphs of y = x/n (n = l, 2, 3, 5, . . .)> marking on each the points with integral coordinates. He o