On the $x$--coordinates of Pell equations that are products of two Padovan numbers

Abstract

Let $\{P_{n}\}_{n\geq 0}$ be the sequence of Padovan numbers defined by $P_0=0$, $P_1 = P_2=1$, and $P_{n+3}= P_{n+1} +P_n$ for all $n\geq 0$. In this paper, we find all positive square-free integers $d \ge 2$ such that the Pell equations $x^2-dy^2 = \ell$, where $\ell\in\{\pm 1, \pm 4\}$, have at least two positive integer solutions $(x,y)$ and $(x^{\prime}, y^{\prime})$ such that each of $x$ and $x^{\prime}$ is a product of two Padovan numbers