## 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.

##### Subject

##### Collections

##### License

Creative Commons Attribution - Non-Commercial - No Derivatives (CC BY-NC-ND)