On the Diophantine equation $F_{n-2}^m+F_n^m+F_{n+2}^m=y^p$

Abstract

Let $p, m$ be positive integers and $p\geq 2, F_n$ is the Fibonacci number. In this paper, we consider the Diophantineequation $F_{n-d}^m+F_n^m+F_{n+d}^m=y^p$, where $q$ is a prime andwe prove that there are no integer solutions when $3\nmid d, 2\mid m$. Moreover, we study this equation in a particular case $d=2$, $m=1$ or $m=q$, where $q$ is a prime of the type $4k+3$.