Lucas sequences and repdigits

Abstract

Let $(G_n)_{n \geq1}$ be a binary linear recurrence sequence that is represented by the Lucas sequences of the first and second kind, which are $\{U_n\}$ and $\{V_n\}$, 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 $g \geq2$, $l$ is even and $1 \leq B \leq g^l-1$. 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 $\{F_n\}$ and Pell numbers $\{P_n\}$. 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 $2 \leq g \leq9$ and $l=1$.

Keywords

• diophantine equation
• lucas sequence
• repdigit
• elliptic curve