68 History of the Theory of Numbers. [Chap, hi 
Change m to mp, . . . , T/zp""^. Thus 
2/-^ - 1 = p(^-^ - 1) F^, = p^X^y^F^,, 
Hence x^^^ — 1 is divisible by N for iV" = p" and so for any N. 
For X odd, x^ — 1 is divisible by 8, and x*'" — 1 by 2(x^*"— 1). As above, 
he found that x' - 1 is divisible by 2* for t = m-2'-^, i> 2. Thus, if iV = 2'n, 
n odd, X*— 1 is divisible by N for A: = 2*"^0(n). 
A. L. Crelle^^ employed a fixed quadratic non-residue v of the prime p, 
and set j^=ry, vf=Vj (mod p). By multipUcation of 
ip-jf=rj, vf=v^ (mod p) (i = 1, • • • »^^) 
and use of v^^~'^^'^= — \, we get 
-](p-l)!f2=nr,v,= (p-l)! (modp). 
F. Minding^* proved the generaUzed Wilson theorem. Let P be the 
product of the tt integers a, /3, . . ., <A and relatively prime to A. Let 
A = 2''p"'g"r* . , . , where p, q,r,. . . are distinct odd primes, and m> 0. Take 
a quadratic non-residue t of p and determine a so that a=t (mod p), o=l 
(mod 2qr. . .). Then a is an odd quadratic non-residue of A. Let ax=a 
(mod A). For ^9^x, a, let i3?/=a (mod A). Then y^^a, x, jS. In this way 
the TT numbers a, j8, . . . can be paired so that the product of the two in any 
pair is =a (mod A), whence P=a''^^ (mod A). 
First, let A = 2''p^ Then a'=-l (mod p"^), s = p"*-^(p-l)/2, whence 
P=-l (mod^) if M = Oor L But, if m>1, 
a^={-iy =l(modp'"), a^=/ =l(mod2''), P= + l(modA). 
Next, let m>l, n>l, in A. Raising the above a*=— 1 to the power 
2''"V~^(? — !)• • •> we get a'^^=-\-l (mod p"). A like congruence holds 
moduli g", r^ . . ., and 2", whence P=4-l (mod A). 
Finally, let A = 2", /x>L Then a=— 1 is a quadratic non-residue of 
2" and, as above, P= ( — 1)^ (mod A),l = 2""^. The proof of Fermat's theorem 
due to Ivory^^ is given by Minding on p. 32. 
J. A. Grunert*^ gave Horner's^^ proof of Euler's theorem, attributing 
the case of a prime to Dirichlet instead of Ivory .^^ A part of the generalized 
Wilson theorem was proved as follows: Let ri,. . ., r, denote the positive 
integers <p and prime to p. Let a be prime to p. In the table 
riflVg, r2a^rg, . . . , rqa\ 
«'Abh. Ak. Wiss. Berlin (Math.), 1832, 66. Reprinted." 
"Anfangsgriinde der Hoheren Arith., 1832, 75-78. 
"Math. Worterbuch, 1831, pp. 1072-3; Jour, fiir Math. 8, 1832, 187. 
