PERFECT POWERS THAT ARE SUMS OF TWO POWERS OF FIBONACCI NUMBERS

Abstract

In this paper, we consider the Diophantine equations $$\begin{eqnarray}\displaystyle F_{n}^{q}\pm F_{m}^{q}=y^{p} & & \displaystyle \nonumber\end{eqnarray}$$ with positive integers $q,p\geq 2$ and $\gcd (F_{n},F_{m})=1$, where $F_{k}$ is a Fibonacci number. We obtain results for $q=2$ or $q$ an odd prime with $q\equiv 3\;(\text{mod}\;4),3<q<1087$, and complete solutions for $q=3$.