On the $x-$coordinates of Pell equations which are sums of two Padovan numbers

Abstract

Let $\{P_n{\}}_{n\ge 0}$ 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 $n\ge 0$. In this paper, we find all positive square-free integers $d$ such that the Pell equations $x^2-d{y}^2=\pm 1$, $X^2-d{Y}^2=\pm 4$ have at least two positive integer solutions $(x,y)$ and $(x^{\prime },y^{\prime })$, $(X,Y)$ and $(X^{\prime },Y^{\prime })$, respectively, such that each of $x,x^{\prime },X,X^{\prime }$ is a sum of two Padovan numbers.

arXiv comment30 pages