68 HISTORY OF THE THEOKY OF NUMBERS. [CHAP, in Change m to mp, . . . , mpn~2. Thus Hence o^w — 1 is divisible by JV for N=pn and so for any JV. For x odd, x2-! is divisible by 8, and x4m-l by 2(x2m— 1). As above, he found that 3*- 1 is divisible by 2' for <=m-2*"2, i>2. Thus, if Ar=2*n, n odd, zfc-l is divisible by N for & = 2*~2<Kn). A. L. Crelle47 employed a fixed quadratic non-residue v of the prime p, and set j^r,-, vj2=v;- (mod p). By multiplication of (P - j)2ssfy t^stj, (mod p) (j = 1, • • . J2-^ and use of t?(p"1)/ss -1, we get -{(p-l)!}8snWs(p-l)l (modp). F. Minding48 proved the generalized Wilson theorem. Let P be the product of the TT integers a, ft. . ., <A and relatively prime to A. Let J. = 2"pmqnrk . . . , where p, #, r, . . . are distinct odd primes, and ra> 0. Take a quadratic non-residue t of p and determine a so that a=2 (mod p), a=l (mod 2<jr. . .). Then a is an odd quadratic non-residue of A. Let ax^a (mod A). For ft 7*x} a, let /fy=a (mod A). Then 2/5^a, 3, ft In this way the T numbers a, ft . . . can be paired so that the product of the two in any pair is =a (mod A), whence P=oT/2 (mod A). First, let A = 2Mpm. Then a'^-1 (mod pm), s = pm~1(p-l)/2, whence P=-l (mod A) if/i = 0or 1. But, if /*>!, a5= (-i)*"1-! (mod p1*), £=af~*=l (mod 2"), P= +1 (mod A). Next, let m>l, n>l, in A. Raising the above a8=— 1 to the power 2"~V~1 (?-].)• . ., we get a*/2=+l (mod pn). A like congruence holds moduli qn, rfc, . . . , and 2M, whence P= + l (mod A). Finally, let A =2^ /z>l. Then o=— 1 is a quadratic non-residue of 2" and, as above, PS ( - 1) l (mod A) , Z = 2A""2. The proof of Fermat's theorem due to Ivory33 is given by Minding on p. 32. J. A. Grunert49 gave Horner's37 proof of Euler's theorem, attributing the case of a prime to Dirichlet instead of Ivory.33 A part of the generalized Wilson theorem was proved as follows : Let rlt. . ., rq denote the positive integers <p and prime to p. Let a be prime to p. In the table i, r2a2r1,...,raa2r1 r1a2rq)r2a2rg)...Jrqa2rq <7Abh. Ak. Wiss. BerUn (Math.), 1832, 66. Reprinted.65 48Anfangsgrunde der Hoheren Arith., 1832, 75-78. "Math. Worterbuch, 1831, pp. 1072-3; Jour, flir Math. 8, 1832, 187.