On the exponential Diophantine equation $F_{n}^{x}+F_{n+1}^{x}+\cdots+F_{n+k-1}^{x}=F_{m}$

Abstract

In this paper, we find all the solutions of the title Diophantine equation in positive integer variables $(m, n, k, x)$, where $F_i$ is the $i^{th}$ term of the Fibonacci sequence.