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. 
