The multiplicative orders of certain Gauss factorials, II

Abstract

We study the multiplicative orders of $(\frac{n-1}{M})_n! \pmod{n}$ for odd prime powers $n=p^\alpha$, $p\equiv 1\pmod{M}$, where the Gauss factorial $N_n!$ denotes the product of all integers up to $N$ that are relatively prime to $n$. Departing from previously obtained results on the connection between the order for $p^\alpha$ and for $p^{\alpha+1}$, we obtain new criteria for exceptions to a general pattern, with particular emphasis on the cases $M=3$, $M=4$ and $M=6$. In the process we also obtain some results of independent interest. Most results are based on generalizations of binomial coefficient congruences of Gauss, Jacobi, and Hudson and Williams.