Research PaperNo Access

On members of Lucas sequences which are products of Catalan numbers

Shanta Laishram

Stat-Math Unit, Indian Statistical Institute, 7, S. J. S. Sansanwal Marg, New Delhi, 110016, India

E-mail Address: [email protected]

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Florian Luca

https://orcid.org/0000-0003-1321-4422

School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa

Research Group in Algebraic Structures and Applications, King Abdulaziz University, Jeddah, Saudi Arabia

Centro de Ciencias Matemáticas UNAM, Morelia, Mexico

E-mail Address: [email protected]

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, and

Mark Sias

Department of Pure and Applied Mathematics, University of Johannesburg, PO Box 524, Auckland Park 2006, South Africa

E-mail Address: [email protected]

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https://doi.org/10.1142/S1793042121500457Cited by:1 (Source: Crossref)

Abstract

We show that if ${U_{n}}_{n \geq 0}$ is a Lucas sequence, then the largest $n$ suc 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 $n = 1$ .

Keywords:

AMSC: 11B39, 11B65, 11D72, 11D45

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