72 HISTORY OF THE THEORY OP NUMBERS. [CHAP, in similarly a new N-gon, etc., until the initial polygon is reached.68 The number M of distinct polygons thus obtained is seen to be a divisor of 4>(N)f the number of polygons corresponding to the various x's. If in the initial polygon we take the afth vertex following any one, etc., we obtain the initial polygon. Hence 3? and thus also x*(N) has the remainder unity when divided by N. [When completed this proof differs only slightly from that by Euler.14] E. Prouhet69 modified Poinsot's method and obtained a correct proof of the generalized Wilson theorem. Let r be the number of roots of x2=l (mod N), and w the number of ways of expressing AT as a product of two relatively prime factors. If N=2mpil. . .pfa where the p's are distinct odd primes, evidently w = 2" if w>0, w = 2M~1 if ra = 0. By considering divisors of &=*=!, it is proved that r = 2w if ra = 0 or 2, r = w if m = l, r~kw if m> 2. Hence r = 2* if m = 0 or 1, 2*+1 if m = 2, 2"+2 if m> 2. By Crelle,58 the product P of the integers <N and prime to N is =(— l)r/2 (mod N). Thus for ju>0,P=+l unless m = 0 or l,ju = l, viz^N^p* or 2p*; while, for M=S0, N = 2m, w>2, we have r = 4, P^-fl. Friderico Arndt70 elaborated Gauss'31 second suggestion for a proof of the generalized Wilson theorem. Let g be a primitive root of the modulus pn or 2pn, where p is an odd prime. Set v=<£(pn). Then g, g2, . . ., 0" are congruent to the numbers less than the modulus and prime to it. If P is the product of the latter, p=^^)/2. But ^/2=-l. Hence P--1. Next, if n>2, the product of the incongruent numbers belonging to an exponent 2n~m is =1 (mod 2n). Next, consider the modulus M = AB, where A and B are relatively prime. The positive integers < M and prime to M are congruent modulo M to Ayi+Bxj, where the xt are < A and prime to A, the 2/i are <£ and prime to B. But, if a=< ?y)s5.Bfl:ci. . .^s^. . .za (mod By resolving M into a product of powers of primes and applying the above results, we determine the sign in P==t=l (mod M). J. A. Grunert71 proved that if a prime n-\-l>2 divides no one of the integers a1}. . ., an, nor any of their differences, it divides a^ . . .an+l, and stated that this result is much more general than Wilson's theorem (the case ay=j). But the generalization is only superficial since a1; . . ., an are congruent modulo n+ltol,..., nin some order. His proof employed Fermat's theorem and certain complex equations involving products of differences of n numbers and sums of products of n numbers taken m at a time. J. F. Heather72 gave without reference the first results of Grunert.36 68Cf. P. Bach-mann, Die Elemente der Zahlentheorie, 1892, 19-23. 69Nouv. Ann. Math., 4, 1845, 273-8. 70Jour. fiir Math., 31, 1846, 329-332. 7lArchiv Math. Phya., 10, 1847, 312. "The Mathematician, London, 2, 1847, 296.