THE MULTIPLICATIVE ORDERS OF CERTAIN GAUSS FACTORIALS

Abstract

A theorem of Gauss extending Wilson's theorem states the congruence (n - 1)n! ≡ -1 (mod n) whenever n has a primitive root, and ≡ 1 (mod n) otherwise, where Nn! 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 [Formula: see text] (mod n) for odd prime powers pα. We prove a general result about the connection between the order for pα and for pα+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.