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


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.


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John B. Cosgrave. Karl Dilcher. "The multiplicative orders of certain Gauss factorials, II." Funct. Approx. Comment. Math. 54 (1) 73 - 93, March 2016.


Published: March 2016
First available in Project Euclid: 22 March 2016

zbMATH: 1218.11003
MathSciNet: MR3477736
Digital Object Identifier: 10.7169/facm/2016.54.1.7

Primary: 11A07
Secondary: 11B65

Keywords: binomial coefficient congruences , factorials , Gauss factorials , Gauss-Wilson theorem

Rights: Copyright © 2016 Adam Mickiewicz University


Vol.54 • No. 1 • March 2016
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