On a Diophantine equation involving powers of Fibonacci numbers

Abstract

This paper deals with the diophantine equation $F_{1}^{p}+2F_{2}^{p}+\cdots +kF_{k}^{p}=F_{n}^{q}$, an equation on the weighted power terms of Fibonacci sequence. For the exponents $p,q\in\{1,2\}$ the problem has already been solved in ad hoc ways using the properties of the summatory identities appear on the left-hand side of the equation. Here we suggest a uniform treatment for arbitrary positive integers $p$ and $q$ which works, in practice, for small values. We obtained all the solutions for $p,q\le 10$ by testing the new approach.