A DIOPHANTINE EQUATION RELATED TO THE SUM OF SQUARES OF CONSECUTIVE k-GENERALIZED FIBONACCI NUMBERS

Let (F-n)n >= 0 be the Fibonacci sequence given by Fn+2 = Fn+1 Fn, for n >= 0, where F-0 = 0 and F-1 = 1. There are several interesting identities involving this sequence such as F-n(2) + F-n+1(2) = F2n+1, for all n >= 0. One of the most known generalizations 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 prove that contrarily to the Fibonacci case, the Diophantine equation (F-n((k)))(2) (vertical bar) (F-n+1((k)))(2) = F-m((k)) has no any solution in positive integers n, m and k, with n > 1 and k >= 3.