# Lucas sequences and repdigits

Mathematica Bohemica (2022)

• Volume: 147, Issue: 3, page 301-318
• ISSN: 0862-7959

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## Abstract

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Let ${\left({G}_{n}\right)}_{n\ge 1}$ be a binary linear recurrence sequence that is represented by the Lucas sequences of the first and second kind, which are $\left\{{U}_{n}\right\}$ and $\left\{{V}_{n}\right\}$, respectively. We show that the Diophantine equation ${G}_{n}=B·\left({g}^{lm}-1\right)/\left({g}^{l}-1\right)$ has only finitely many solutions in $n,m\in {ℤ}^{+}$, where $g\ge 2$, $l$ is even and $1\le B\le {g}^{l}-1$. Furthermore, these solutions can be effectively determined by reducing such equation to biquadratic elliptic curves. Then, by a result of Baker (and its best improvement due to Hajdu and Herendi) related to the bounds of the integral points on such curves, we conclude the finiteness result. In fact, we show this result in detail in the case of ${G}_{n}={U}_{n}$, and the remaining case can be handled in a similar way. We apply our result to the sequences of Fibonacci numbers $\left\{{F}_{n}\right\}$ and Pell numbers $\left\{{P}_{n}\right\}$. Furthermore, with the first application we determine all the solutions $\left(n,m,g,B,l\right)$ of the equation ${F}_{n}=B·\left({g}^{lm}-1\right)/\left({g}^{l}-1\right)$, where $2\le g\le 9$ and $l=1$.

## How to cite

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Hashim, Hayder Raheem, and Tengely, Szabolcs. "Lucas sequences and repdigits." Mathematica Bohemica 147.3 (2022): 301-318. <http://eudml.org/doc/298498>.

@article{Hashim2022,
abstract = {Let $(G_\{n\})_\{n \ge 1\}$ be a binary linear recurrence sequence that is represented by the Lucas sequences of the first and second kind, which are $\lbrace U_n\rbrace$ and $\lbrace V_n\rbrace$, respectively. We show that the Diophantine equation $G_n=B \cdot (g^\{lm\}-1)/(g^\{l\}-1)$ has only finitely many solutions in $n, m \in \mathbb \{Z\}^+$, where $g \ge 2$, $l$ is even and $1 \le B \le g^\{l\}-1$. Furthermore, these solutions can be effectively determined by reducing such equation to biquadratic elliptic curves. Then, by a result of Baker (and its best improvement due to Hajdu and Herendi) related to the bounds of the integral points on such curves, we conclude the finiteness result. In fact, we show this result in detail in the case of $G_n=U_n$, and the remaining case can be handled in a similar way. We apply our result to the sequences of Fibonacci numbers $\lbrace F_n\rbrace$ and Pell numbers $\lbrace P_n\rbrace$. Furthermore, with the first application we determine all the solutions $(n,m,g,B,l)$ of the equation $F_n=B \cdot (g^\{lm\}-1)/(g^l-1)$, where $2 \le g \le 9$ and $l=1$.},
author = {Hashim, Hayder Raheem, Tengely, Szabolcs},
journal = {Mathematica Bohemica},
keywords = {Diophantine equation; Lucas sequence; repdigit; elliptic curve},
language = {eng},
number = {3},
pages = {301-318},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Lucas sequences and repdigits},
url = {http://eudml.org/doc/298498},
volume = {147},
year = {2022},
}

TY - JOUR
AU - Hashim, Hayder Raheem
AU - Tengely, Szabolcs
TI - Lucas sequences and repdigits
JO - Mathematica Bohemica
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 147
IS - 3
SP - 301
EP - 318
AB - Let $(G_{n})_{n \ge 1}$ be a binary linear recurrence sequence that is represented by the Lucas sequences of the first and second kind, which are $\lbrace U_n\rbrace$ and $\lbrace V_n\rbrace$, respectively. We show that the Diophantine equation $G_n=B \cdot (g^{lm}-1)/(g^{l}-1)$ has only finitely many solutions in $n, m \in \mathbb {Z}^+$, where $g \ge 2$, $l$ is even and $1 \le B \le g^{l}-1$. Furthermore, these solutions can be effectively determined by reducing such equation to biquadratic elliptic curves. Then, by a result of Baker (and its best improvement due to Hajdu and Herendi) related to the bounds of the integral points on such curves, we conclude the finiteness result. In fact, we show this result in detail in the case of $G_n=U_n$, and the remaining case can be handled in a similar way. We apply our result to the sequences of Fibonacci numbers $\lbrace F_n\rbrace$ and Pell numbers $\lbrace P_n\rbrace$. Furthermore, with the first application we determine all the solutions $(n,m,g,B,l)$ of the equation $F_n=B \cdot (g^{lm}-1)/(g^l-1)$, where $2 \le g \le 9$ and $l=1$.
LA - eng
KW - Diophantine equation; Lucas sequence; repdigit; elliptic curve
UR - http://eudml.org/doc/298498
ER -

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