Gauss Factorials, Jacobi Primes, and Generalized Fermat Numbers

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.