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arXiv · 2006.01756

On members of Lucas sequences which are products of Catalan numbers

Shanta Laishram·Florian Luca·Mark Sias
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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 $n=1$.

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Shanta Laishram, Florian Luca, Mark Sias. 2020-06-02. On members of Lucas sequences which are products of Catalan numbers. https://arxiv.org/abs/2006.01756

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