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

摘要

Let (F_n)_(≥0) be the Fibonacci sequence given by F_(n+2) = F_(n+1) + F_n, 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 = F_(2n+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 + (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.