Generalized Wall-Sun-Sun primes and monogenic power compositional trinomials

Abstract

For positive integers $a$ and $b$, we let $[U_n]$ be the Lucas sequence of the first kind defined by \[U_0=0,\quad U_1=1\quad \mbox{and} \quad U_n=aU_{n-1}+bU_{n-2} \quad \mbox{ for $n\ge 2$},\] and let $\pi(m):=\pi_{(a,b)}(m)$ be the period length of $[U_n]$ modulo the integer $m\ge 2$, where $\gcd(b,m)=1$. We define an \emph{$(a,b)$-Wall-Sun-Sun prime} to be a prime $p$ such that $\pi(p^2)=\pi(p)$. When $(a,b)=(1,1)$, such a prime $p$ is referred to simply as a \emph{Wall-Sun-Sun prime}. We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is \emph{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 $f(x)=x^2-ax-b$, and let $s$ be a positive integer. Then, with certain restrictions on $a$, $b$ and $s$, we prove that the monogenicity of \[f(x^{s^n})=x^{2s^n}-ax^{s^n}-b\] is independent of the positive integer $n$ and is determined solely by whether $s$ has a prime divisor that is an $(a,b)$-Wall-Sun-Sun prime. This result improves and extends previous work of the author in the special case $b=1$.