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. 
