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. 
