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Journal Article

#### On members of Lucas sequences which are products of Catalan numbers

##### External Resource

https://dx.doi.org/10.1142/S1793042121500457

(Publisher version)

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##### Fulltext (public)

2006.01756.pdf

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##### Citation

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.

Cite as: https://hdl.handle.net/21.11116/0000-0009-1FDC-E

##### 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 $n=1$.

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 $n=1$.