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

TL;DR

This paper explores the properties of generalized Wall-Sun-Sun primes and their connection to the monogenicity of certain power compositional trinomials, extending previous results for specific cases.

Contribution

It establishes that the monogenicity of specific polynomial families depends solely on the existence of an $(a,b)$-Wall-Sun-Sun prime dividing the parameter $s$, generalizing earlier findings.

Findings

Monogenicity is independent of $n$ under certain conditions.

Presence of an $(a,b)$-Wall-Sun-Sun prime dividing $s$ determines monogenicity.

Extends previous work from the case $b=1$ to more general settings.

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 $b=1$.