A new condition for $k$-Wall-Sun-Sun primes

Abstract

Let $k\ge 1$ be an integer, and let $(U_n)$ be the Lucas sequence of the first kind defined by \begin{equation*}\label{Eq:Lucas} U_0=0,\quad U_1=1\quad \mbox{and} \quad U_n=kU_{n-1}+U_{n-2} \quad \mbox{ for $n\ge 2$}. \end{equation*} 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. A prime $p$ is called a $k$-Wall-Sun-Sun prime if $\pi(p^2)=\pi(p)$. 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 monogenic if $\Theta=\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of integers ${\mathbb Z}_K$ of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. If $\Theta$ is not a basis for ${\mathbb Z}_K$, we say that $f(x)$ is non-monogenic. Define ${\mathcal D}:=k^2+4$ if $k\equiv 1 \pmod{2}$, and ${\mathcal D}:=(k/2)^2+1$ if $k\equiv 0 \pmod{2}$. Suppose that $k\not \equiv 0 \pmod{4}$ and that ${\mathcal D}$ is squarefree. In this article, we prove that $p$ is a $k$-Wall-Sun-Sun prime if and only if ${\mathcal F}_p(x)=x^{2p}-kx^p-1$ is non-monogenic. This result, combined with previous work, shows that ${\mathcal F}_p(x)$ is monogenic if $p$ is a prime divisor of $k^2+4$.