On members of Lucas sequences which are products of Catalan numbers (2006.01756v1)

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 $n=1$.