72 History of the Theory of Numbers. [Chap. hi 
similarly a new A^-gon, etc., until the initial polygon is reached.®^ The 
number )U of distinct polygons thus obtained is seen to be a divisor of <t>{N), 
the number of polygons corresponding to the various a:'s. If in the initial 
polygon we take the x^th vertex following any one, etc., we obtain the 
initial polygon. Hence of and thus also x"^^^ has the remainder unity when 
divided by N. [When completed this proof differs only shghtly from that 
by Euler."] 
E. Prouhet^^ modified Poinsot's method and obtained a correct proof 
of the generalized Wilson theorem. Let r be the number of roots of x^=l 
(mod N), and w the number of ways of expressing iV as a product of two 
relatively prime factors. If AT = 2'"pi" . . . p/", where the p's are distinct 
odd primes, evidently w; = 2*' if m>0, 1^ = 2""^ if m = 0. By considering 
divisors of a: =*= 1, it is proved that r = 2u' if ttz = or 2, r = w; if w = 1, r = 4iy if 
m>2. Hence r = 2" if m = or 1, 2"+^ if m = 2, 2"+^ if m> 2. By Crelle,^^ 
the product P of the integers <A'' and prime to N is =( — 1)''''^ (mod N). 
Thus for jLt>0, P= + l unless m = or l,/i = l, viz., N = p^ or 2p'; while, for 
^ = 0, N = 2"', m>2, we have r = 4, P=-\-l. 
Friderico Arndt'^° elaborated Gauss'^^ second suggestion for a proof of 
the generalized Wilson theorem. Let gf be a primitive root of the modulus 
p" or 2p", where p is an odd prime. Set y=0(p"). Then g, g^,. . ., g" are 
congruent to the numbers less than the modulus and prime to it. If P is 
the product of the latter, P^g''-'^^^'^ But g"^=-l. Hence P=-l. 
Next, if n>2, the product of the incongruent numbers belonging to an 
exponent 2"""* is =1 (mod 2"). 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 0:^ are <A and prime 
to A, the yi are <B and prime to B. But, if a=0(A), 
a 
7ri = 'n.{Ayi+BXj)=B''xi. . .Xa=Xi. . .x^ (mod A), 
3 = 1 
P=riTr2. ..^{x^.. .xJ'^^^Hniod A). 
By resolving M into a product of powers of primes and applying the above 
results, we determine the sign in P=±l (mod M). 
J. A. Grunert^^ proved that if a prime n+l>2 divides no one of the 
integers ai, . . ., a„, nor any of their differences, it divides aia2. . .a„+l, and 
stated that this result is much more general than Wilson's theorem (the 
case aj=j). But the generalization is only superficial since ai,. . ., a„ are 
congruent modulo n+1 to 1,..., n in 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. Heather^^ gave without reference the first results of Grunert.^^ 
osCf. P. Bachmann, Die Elemente der Zahlentheorie, 1892, 19-23. 
«»Nouv. Ann. Math., 4, 1845, 273-8. 
"Jour, fvir Math., 31, 1846, 329-332. 
"Archiv Math. Phys., 10, 1847, 312. 
"The Mathematician, London, 2, 1847, 296. 
