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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 $n=1$.