Albanian J. Math. 17 (2), 3-17, (2023) DOI: 10.51286/albjm/1678110273
KEYWORDS: Wall-Sun-Sun prime, Monogenic, power-compositional, 11R04, 11B39, 11R09, 12F05

For positive integers $a$ and $b$, we let $\left[{U}_{n}\right]$ be the Lucas sequence of the first kind defined by

$${U}_{0}=0,{U}_{1}=1\mathrm{and}{U}_{n}=a{U}_{n-1}+b{U}_{n-2}\mathrm{for}n\ge 2,$$

and let $\mathrm{\pi}\left(m\right):={\mathrm{\pi}}_{(a,b)}\left(m\right)$ be the period length of $\left[{U}_{n}\right]$ modulo the integer $m\ge 2$, where gcd$(b,m)=1$. We define an $\left(a,b\right)$*-Wall-Sun-Sun prime* to be a prime $p$ such that $\mathrm{\pi}\left({p}^{2}\right)=\mathrm{\pi}\left(p\right)$. When $(a,b)=(1,1)$, such a prime $p$ is referred to simply as a *Wall-Sun-Sun prime*.

We say that a monic polynomial $f\left(x\right)\in \mathrm{\mathbb{Z}}\left[x\right]$ of degree $N$ is *monogenic* if $f\left(x\right)$ is irreducible over $\mathrm{\mathbb{Q}}$ and

$$\{1,\theta ,{\theta}^{2},\dots ,{\theta}^{N-1}\}$$

is a basis for the ring of integers of $\mathrm{\mathbb{Q}}\left(\theta \right)$, where $f\left(\theta \right)=0$.

Let $f\left(x\right)={x}^{2}-ax-b$, and let $s$ be a positive integer. Then, with certain restrictions on $a$, $b$ and $s$, we prove that the monogenicity of

$$f\left({x}^{{s}^{n}}\right)={x}^{2{s}^{n}}-a{x}^{{s}^{n}}-b$$

is independent of the positive integer $n$ and is determined solely by whether $s$ has a prime divisor that is an $\left(a,b\right)$-Wall-Sun-Sun prime. This result improves and extends previous work of the author in the special case $b=1$.