A role for generalized Fermat numbers

Abstract:

We define a Gauss factorial $N_n!$ to be the product of all positive integers up to $N$ that are relatively prime to $n\in \mathbb N$. In this paper we study particular aspects of the Gauss factorials $\lfloor \frac {n-1}{M}\rfloor _n!$ for $M=3$ and 6, where the case of $n$ having exactly one prime factor of the form $p\equiv 1\pmod {6}$ is of particular interest. A fundamental role is played by those primes $p\equiv 1\pmod {3}$ with the property that the order of $\frac {p-1}{3}!$ modulo $p$ is a power of 2 or 3 times a power of 2; we call them Jacobi primes. Our main results are characterizations of those $n\equiv \pm 1\pmod {M}$ of the above form that satisfy $\lfloor \frac {n-1}{M}\rfloor _n!\equiv 1\pmod {n}$, $M=3$ or 6, in terms of Jacobi primes and certain prime factors of generalized Fermat numbers. We also describe the substantial and varied computations used for this paper.