A Connection Between the Monogenicity of Certain Power-Compositional Trinomials and $k$-Wall-Sun-Sun Primes

Abstract

We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and \[\{1,\theta,\theta^2,\ldots, \theta^{N-1}\}\] is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. Let $k$ be a positive integer, and let $U_n:=U_n(k,-1)$ be the Lucas sequence $\{U_n\}_{n\ge 0}$ of the first kind defined by \[U_0=0,\quad U_1=1\quad \mbox{and} \quad U_n=kU_{n-1}+U_{n-2} \quad \mbox{ for $n\ge 2$}.\] A $k$-Wall-Sun-Sun prime is a prime $p$ such that \[U_{\pi_k(p)}\equiv 0 \pmod{p^2},\] where $\pi_k(p)$ is the length of the period of $\{U_n\}_{n\ge 0}$ modulo $p$. Let ${\mathcal D}=k^2+4$ if $k\equiv 1 \pmod{2}$, and ${\mathcal D}=(k/2)^2+1$ if $k\equiv 0 \pmod{2}$. Suppose that $k\not \equiv 0 \pmod{4}$ and ${\mathcal D}$ is squarefree, and let $h$ denote the class number of ${\mathbb Q}(\sqrt{{\mathcal D}})$. Let $s\ge 1$ be an integer such that, for every odd prime divisor $p$ of $s$, ${\mathcal D}$ is not a square modulo $p$ and $\gcd(p,h{\mathcal D})=1$. In this article, we prove that $x^{2s^n}-kx^{s^n}-1$ is monogenic for all integers $n\ge 1$ if and only if no prime divisor of $s$ is a $k$-Wall-Sun-Sun prime.