On the x-coordinates of Pell equations that are sums of two Padovan numbers.

Abstract

Let ${\left(P_n\right)}_{n\ge 0}$ 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 $x^2-d{y}^2=N$ with $N\in \{\pm 1,\pm 4\}$ , have at least two positive integer solutions (x, y) and $(x^{\text{'}},y^{\text{'}})$ such that both x and $x^{\text{'}}$ are sums of two Padovan numbers.