80 HlSTOKY OF THE THEORY OF NUMBERS. [CHAP. HI
unity to each element and replacing p+l 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—l cases in which m = l. Hence (p _ 1) 1 _ (p _ 1) is divisible by p. Cf . Petersen.94
Many proofs of (3), p. 63, have been given.120
D. von Sterneck121 gave Legendre's proof of Wilson's theorem.
L. E. Dickson122 noted that, if p is a prime, p(p— 1) of the pi substitutions on p letters have a linear representation x'=ax+b, a^O (mod p), while the remaining ones are represented analytically by functions of degree >1 which fall into sets of p2(p — 1) each, viz., af(x-i-b)+c, where a is prime top. Hencep!— p(p — 1) is a multiple of p2(p — l),and therefore (p— 1)1+1 is a multiple of p.
C. Moreau123 gave without references Schering's102 extension to any modulus of Dirichlet's40 proof of the theorems of Fermat and Wilson.
H. Weber124 deduced Euler's theorem from the fact that the integers <m and prime to m form a group under multiplication, whence every integer belongs to an exponent dividing the order 4>(m) of the group.
E. Cahen125 proved that the elementary symmetric functions of 1,. . ., p-1 of order <p — 1 are divisible by the prune p. Hence
(s-l)(s-2). . .(»-p+l)sa*-i+(p-l)! (mod p),
identically in x. The case x-1 gives Wilson's theorem, so that also Fer-mat's theorem follows.
J. Perott126 gave Petersen's94 proof of Fermat's theorem, using qp "configurations'7 obtained by placing the numbers 1, 2,..., q into p cases, arranged in a line. It is noted that the proof is not valid for p composite; for example, if p = 4, # = 2, the set of configurations derived from 1212 by cyclic permutations contains but one additional configuration 2121.
L. Kronecker127 proved the generalized Wilson theorem essentially as had Brennecke.67
G. Candido128 made use of the identity
1 ' 2i . . .7*
Take pa prime and 6=-l. Thus ap-a=(a~l)p-(a-l) (mod p).
110L'interm6diaire des math., 3, 1896, 26-28, 229-231; 7, 1900, 22-30; 8, 1901, 164. A. Capelli
Giornale di Mat., 31, 1893, 310. S. Pincherle, ibid., 40, 1902, 180-3. mMonatshefte Math. Phys., 7, 1896, 145. mAnnals of Math., (1), 11, 1896-7, 120. 1MNouv. Ann. Math., (3), 17, 1898, 296-302. 1MLehrbuch der Algebra, II, 1896, 55; ed. 2, 1899, 61. mEtements de la th€orie des nombres, 1900, 111-2 ""Bull, dea Sc. Math., 24, 1, 1900, 175. U7Vorlesungen iiber Zahlentheorie, 1901, I, 127-130. mGiornale di Mat., 40, 1902, 223.