A Diophantine equation related to the sum of powers of two consecutive generalized Fibonacci numbers

Abstract

Let (Fn)n≥0 be the Fibonacci sequence given by Fm+2=Fm+1+Fm, for m≥0, where F0=0 and F1=1. In 2011, Luca and Oyono proved that if Fms+Fm+1s is a Fibonacci number, with m≥2, then s=1 or 2. A well-known generalization of the Fibonacci sequence, is the k-generalized Fibonacci sequence (Fn(k))n which is defined by the initial values 0,0,…,0,1 (k terms) and such that each term afterwards is the sum of the k preceding terms. In this paper, we generalize Luca and Oyono's method by proving that the Diophantine equation(Fm(k))s+(Fm+1(k))s=Fn(k) has no solution in positive integers n,m,k and s, if 3≤k≤min⁡{m,log⁡s}.