On members of Lucas sequences which are products of Catalan numbers

TL;DR

This paper investigates when members of Lucas sequences can be expressed as products of Catalan numbers, establishing bounds on such indices and characterizing specific cases for real roots and Pell equation solutions.

Contribution

It provides explicit bounds on the index n for Lucas sequence members that are products of Catalan numbers and characterizes cases with real roots and Pell solutions.

Findings

Largest n with product of Catalan numbers in Lucas sequences is less than 6500.

For real roots, n belongs to {1,2,3,4,6,8,12}.

X-coordinate solutions of Pell equations equal to Catalan numbers only at n=1.

Abstract

We show that if $\{U_n\}_{n\geq 0}$ is a Lucas sequence, then the largest $n$ such that $|U_n|=C_{m_1}C_{m_2}\cdots C_{m_k}$ with $1\leq m_1\leq m_2\leq \cdots\leq m_k$, 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 $X^2-dY^2=\pm 1$ with a nonsquare integer $d>1$, then $X_n=C_m$ implies $n=1$.