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

Let (F-n) n >= 0 be the Fibonacci sequence given by Fm+2 = Fm+1 + F-m, for m >= 0, where F-0 = 0 and F-1 = 1. In 2011, Luca and Oyono proved that if F-m(s) + F-m+1(s) 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 (F-n((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 (F-m((k)))(s) + (F-m+1((k)))(s) = F-n((k)) has no solution in positive integers n,m,k and s, if 3 <= k <= min{m, log s}. (C) 2015 Published by Elsevier Inc.