On the Diophantine Equation $F_n=F_l^k(F_l^m-1)$

Abstract

In this paper, we examine the Diophantine problem given by the equation $F_n=F_l^k(F_l^m-1)$, where $n,l,m\ge 1$ and $k\ge 3$. Here, $\{F_t{\}}_{t=0}^{\infty }$ denotes the Fibonacci numbers, defined by the recurrence relation $F_0=0$, $F_1=1$, and $F_t=F_{t-1}+F_{t-2}$ for $t\ge 2$. By applying Matveev's theorem, which provides lower bounds for linear forms in logarithms of algebraic numbers, along with a modified Baker-Davenport reduction method and a divisibility property of Fibonacci numbers, we show that $(n,l,k,m)=(6,3,3,1)$ is the only positive integer quadruple that satisfies this equation.