On members of Lucas sequences which are products of Catalan numbers

Shanta Laishram; Florian Luca; Mark Sias

paper

On members of Lucas sequences which are products of Catalan numbers

arXiv:2006.01756

Abstract

We show that if ${U_{n}}_{n \geq 0}$ is a Lucas sequence, then the largest $n$ such that $∣ U_{n} ∣ = C_{m_{1}} C_{m_{2}} \dots C_{m_{k}}$ with $1 \leq m_{1} \leq m_{2} \leq \dots \leq m_{k}$ , where $C_{m}$ is the $m$ th Catalan number satisfies $n < 6500$ . In case the roots of the Lucas sequence are real, we have $n \in {1, 2, 3, 4, 6, 8, 12}$ . As a consequence, we show that if ${X_{n}}_{n \geq 1}$ is the sequence of the $X$ coordinates of a Pell equation $X^{2} - d Y^{2} = \pm 1$ with a nonsquare integer $d > 1$ , then $X_{n} = C_{m}$ implies $n = 1$ .