On the monogenicity of power-compositional Shanks polynomials

Let $f(x)\in \mathbb{Z}[x]$ be a monic polynomial of degree $N$ that is irreducible over $\mathbb{Q}$. We say $f(x)$ is \emph{monogenic} if $\mathrm{\Theta }=\{1,\theta ,{\theta }^2,\dots ,{\theta }^{N-1}\}$ is a basis for the ring of integers ${\mathbb{Z}}_K$ of $K=\mathbb{Q}(\theta )$, where $f(\theta )=0$. If $\mathrm{\Theta }$ is not a basis for ${\mathbb{Z}}_K$, we say that $f(x)$ is \emph{non-monogenic}. Let $k\ge 1$ be an integer, and let $(U_n)$ be the sequence defined by U_0=U_1=0,\quad U_2=1 \quad and\quad U_n=kU_{n-1}+(k+3)U_{n-2}+U_{n-3} \quad for $n\ge 3$. It is well known that $(U_n)$ is periodic modulo any integer $m\ge 2$, and we let $\pi (m)$ denote the length of this period. We define a \emph{$k$-Shanks prime} to be a prime $p$ such that $\pi (p^2)=\pi (p)$. Let ${\mathcal{S}}_k(x)=x^3-k{x}^2-(k+3)x-1$. Let $\mathcal{D}=(k\mathrm{/}3{)}^2+k\mathrm{/}3+1$ if $k\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}3)$, and $\mathcal{D}=k^2+3k+9$ otherwise. Suppose that $k\equiv ̸3(\mathrm{m}\mathrm{o}\mathrm{d}9)$ and that $\mathcal{D}$ is squarefree. In this article, we prove that $p$ is a $k$-Shanks prime if and only if ${\mathcal{S}}_k(x^p)$ is non-monogenic, for any prime $p$ such that ${\mathcal{S}}_k(x)$ is irreducible in ${\mathbb{F}}_p[x]$. Furthermore, we show that ${\mathcal{S}}_k(x^p)$ is monogenic for any prime divisor $p$ of $k^2+3k+9$. These results extend previous work of the author on $k$-Wall-Sun-Sun primes.