Abstract
Given positive integers N and n, we define the Gauss factorial
Nn! as the product of all positive integers from 1 to N and coprime to
n. In this expository paper we begin with the classical theorem of Wilson, extending it in various different but related directions, mostly modulo
composite integers. Most of the results presented in this paper involve the
multiplicative orders, and in particular order 1, of certain Gauss factorials.
In the process we define two types of special primes, the Gauss and Jacobi
primes, and some of the results involve large-scale computations, including factoring certain generalized Fermat numbers. The main tools in most
of the results are the well-known binomial coefficient theorems of Gauss
and Jacobi, along with other related congruences and their generalizations.
John B. Cosgrave, Karl Dilcher. (2018) Gauss Factorials, Jacobi Primes, and Generalized Fermat Numbers, Punjab University Journal of Mathematics, Volume 50, Issue 4.
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