THE MULTIPLICATIVE ORDERS OF CERTAIN GAUSS FACTORIALS

A theorem of Gauss extending Wilson's theorem states the congruence (n - 1)(n)! equivalent to -1 (mod n) whenever n has a primitive root, and equivalent to 1 (mod n) otherwise, where N-n! denotes the product of all integers up to N that are relatively prime to n. In the spirit of this theorem, we study the multiplicative orders of (n-1/M)(n)! (mod n) for odd prime powers p(alpha). We prove a general result about the connection between the order for pa and for p(alpha+1) and study exceptions to the general rule. Particular emphasis is given to the cases M = 3, M = 4 and M = 6, while the case M = 2 is already known.