Chap. Ill] FeKMAT's AND WiLSON's THEOREMS. 69 
a single term of a row is =1 (mod p). If this term be TkO^rk, replace it 
by (p—rk)a\^-l. Next, if r^^a^^ =f1, r„aVi=±l, then rk-^ri=p and 
one of the r„ is replaced by p—r^. Hence we may separate riO, . . ., r^a 
into q/2 pairs such that the product of the two of a pair is = ± 1 (mod p) . 
Taking a = 1, we get ri . . .rg= ± 1 (mod p). The sign was determined only 
for the case p a prime (by Gauss' method). 
A. Cauchy^*^ derived Wilson's theorem from (1), page 62 above. 
*Caraffa^^ gave a proof of Fermat's theorem. 
E. Midy^^ gave Ivory's^^ proof of Fermat's theorem, 
W. G. Horner^^ gave Euler's^^ proof of his theorem. 
G. Libri^^ reproduced Euler's proof^^ without a reference. 
Sylvester^^ gave the generalized Wilson theorem in the incomplete form 
that the residue is ± 1. 
Th. Schonemann^^ proved by use of symmetric functions of the roots 
that if s"+6i2;""^+ ... =0 is the equation for the pth powers of the roots 
of x^+aiX^~^-{- ... =0, where the a's are integers and p is a prime, then 
hi=af (mod p). If the latter equation is (x — 1)'' = 0, the former is 
2'*-(nP+pQ)2"-^+. ..=0, and yet is evidently (2:-l)'* = 0. Hence 
71^=71 (mod p). 
W. Brennecke^^ elaborated one of Gauss'^^ suggestions for a proof of the 
generalized Wilson theorem. For a>2, x^=l (mod 2°) has exactly four 
incongruent roots, =•= 1, ='= (l+2"~^), since one of the factors x=^l, of differ- 
ence 2, must be divisible by 2 and the other by 2""^. For p an odd prime, 
let ri, . . ., r^ be the positive integers <p" and prime to p", taking ri = l, 
r^ = p"— 1. For 2^s^)u — 1, the root x of r^x^l (mod p") is distinct from 
Ti, r^, r^. Thus 7-2, ... , r^_i may be paired so that the product of the two 
of a pair is =1 (mod p"). Hence ri . . .r^= — 1 (mod p"). This holds also 
for modulus 2p". For a > 2, 
(2-i-i)(2»-i+l)=-l, ri. . .r^=-\-l (mod 2"). 
Finally, let N=p''M, where M is divisible by an odd prime, but not by p. 
Then m=(f>{M) is even. The integers <N and prime to p are 
rj>rj+p'^, rj+2p%.. ., r,.+(M-l)p» (i = l,. . ., m). 
For a fixed j, we obtain m integers <iV and prime to N. Hence if \N\ 
denotes the product of all the integers < N and prime to iV, 
\N\^{n. . .r^)'^=l (mod p"). 
ForiV = pV.-., \N\=1 (mod O,--, whence jiVt^l (modiV). 
"R6suin6 analyt., Turin, 1, 1833, 35. 
"Elem. di mat. commentati da Volpicelli, Rome, 1836, I, 89. 
"De quelques propri^t^s des nombres, Nantes, 1836. 
"London and Edinb. Phil. Mag., 11, 1837, 456. 
"M6m. divers savants ac. sc. Institut de France (math.), 5, 1838, 19. 
"Phil. Mag., 13, 1838, 454 (14, 1839, 47); Coll. Math. Papers, 1, 1904, 39. 
"Jour, fiir Math., 19, 1839, 290; 31, 1846, 288. Cf. J. J. Sylvester, Phil. Mag., (4), 18, 1859, 281. 
"Jour, fiir Math., 19, 1839, 319. 
