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  On members of Lucas sequences which are products of Catalan numbers

Laishram, S., Luca, F., & Sias, M. (2021). On members of Lucas sequences which are products of Catalan numbers. International Journal of Number Theory, 17(6), 1487-1515. doi:10.1142/S1793042121500457.

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 Creators:
Laishram, Shanta, Author
Luca, Florian1, Author           
Sias, Mark, Author
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1Max Planck Institute for Mathematics, Max Planck Society, ou_3029201              

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Free keywords: Mathematics, Number Theory
 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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Language(s): eng - English
 Dates: 2021
 Publication Status: Issued
 Pages: -
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 Rev. Type: Peer
 Identifiers: arXiv: 2006.01756
DOI: 10.1142/S1793042121500457
 Degree: -

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Title: International Journal of Number Theory
Source Genre: Journal
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Publ. Info: World Scientific
Pages: - Volume / Issue: 17 (6) Sequence Number: - Start / End Page: 1487 - 1515 Identifier: -