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 \[\{1,θ,θ^2,\ldots, θ^{N-1}\}\] 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 \[U_{π_k(p)}\equiv 0 \pmod{p^2},\] 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 $k\equiv 1(\mathrm{mod}2)$, and $\mathcal{D}=(k/2{)}^2+1$ if $k\equiv 0(\mathrm{mod}2)$. Suppose that $k\not\equiv 0(\mathrm{mod}4)$ 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 $x^{2s^n}-k{x}^{s^n}-1$ is monogenic for all integers $n\ge 1$ if and only if no prime divisor of $s$ is a $k$-Wall-Sun-Sun prime.