Sums of Fibonacci numbers that are perfect powers

Abstract

Let us denote by Fn the n-th Fibonacci number. In this paper we show that for a fixed integer y there exists at most one integer exponent  a > 0 such that the Diophantine equation Fn + Fm = y^a has a solution (n, m, a) in positive integers satisfying n > m > 0, unless y = 2, 3, 4, 6 or 10.