Generalized Wieferich primes and monogenic polynomials

Abstract

Let $b,p\in \mathbb{Z}$ with $b\ge 2$ and $p\ge 3$ a prime. If $b^{p-1}\equiv 1(\mathrm{mod}{p}^2)$, then $p$ is called a {\em generalized Wieferich prime base $b$}, or more succinctly, a {\em base-$b$ Wieferich prime}. When $b=2$, $p$ is also known simply as a Wieferich prime. We say that a monic polynomial $f(x)\in \mathbb{Z}[x]$ is {\em monogenic} if $f(x)$ is irreducible over $\mathbb{Q}$ and $\{1,\theta ,{\theta }^2,\dots ,{\theta }^{°(f)-1}\}$ is a basis for the ring of integers of $\mathbb{Q}(\theta )$, where $f(\theta )=0$. Recently, necessary and sufficient conditions for the monogenicity of the trinomials $x^{2n}+b{x}^n+b$ were given that included base-$b$ Wieferich prime congruences, and also congruences involving a certain Lucas sequence. In this article, we prove a similar result for a different class of trinomials that does not rely on any congruence condition involving a Lucas sequence. Furthermore, we extend this result to a related class of $N$-nomials, for any $N\ge 4$.