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CHAP. Ill]                 H'EHMAT'S AND WlLSON's THEOREMS.                               79
m—pN(l+np), when h = l. For the symmetric group on p letters, m=pl and N=p — 1, so that (p — 1)!= — 1 (mod p). There is exhibited a special group for which m = pap, N = a, whence ap=a (mod p).
G. Levi114' failed in his attempt to prove Wilson's theorem. Let b and a = (p — 1)6 have the least positive residues rx and r when divided by p. Then r+rj-p. Multiply b/p-q+r^p by p — 1. Thus r^p — 1) has the same residue as a, so that
He concluded that ri(p — l)=r, falsely, as the example p = 5, & = 7, shows. He added the last equation to r+r1=p and concluded that rx = l, r = p — 1, so that (a+l)/p is an integer. The fact that this argument is independent of Levi's initial choice that b — (p— 2) ! and his assumption that p is a prime shows that the proof is fallacious.
Axel Thue115 obtained Format's theorem by adding
[Paoli46]. Then the differences &lF(j) of the first order of F(x)=xp~1 are divisible by p f or j = 1 , . . . , p - 2 ; likewise A2F(1) , . . . , AP~2F(1) . By adding
A''+1F(0) =Atff(l) -Ay^(0)    (j = l, . . ., p-2), we get -Ap-1^l(0)*l+A1F(l)-A2F(l) + . . .+Ap-2F(l),    (p-l)!+l=0(modp).
N. M. Ferrers116 repeated Sylvester's78 proof of Wilson's theorem.
M. d'Ocagne117 proved the identity in r:
(r + l)W + <M^£p('I^^
ql   ğĞi
where ^ = [(/cH-l)/2] and P(^} is the product of n consecutive integers of which m is the largest, while P2 = l- Hence if k+1 is a prime, it divides (r+l)fc+1— r^4"1 — !, and Fermat's theorem follows. The case k = p — I shows that if p is a prime, q = (p — 1)/2, and r is any integer,
2i(-ryz=Q (mod g!).
T. del Beccaro118 used products of linear functions to obtain a very complicated proof of the generalized Wilson theorem.
A. Schmidt119 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
114Atti del R. Istituto Vencto di Sc., (7), 4, 1802-3, pp. 1816-42.
ll6Archiv Math, og Natur., Kriatiaiiia, 16, 1893, 255-265.
116Messenger Math., 23, 1893-4, 56.
117Jour. de 1'gcole.polyt., 64, 1894, 200-1.
118Atti R. Ac. Lincei (Fis. Mat.), 1, 1894, 344-371.