GENERALIZED WALL-SUN-SUN PRIMES AND MONOGENIC POWER-COMPOSITIONAL TRINOMIALS

Abstract

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

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

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 $b=1$.