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.
Citation
John B. Cosgrave. Karl Dilcher. "The multiplicative orders of certain Gauss factorials, II." Funct. Approx. Comment. Math. 54 (1) 73 - 93, March 2016. https://doi.org/10.7169/facm/2016.54.1.7
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