An equation related to k-generalized Fibonacci numbers

Abstract

For k ≥ 2, the k-generalized Fibonacci sequence (F_n^(k))_n 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 shall prove that the only solutions of the Diophantine equation F_n ^(k)=k2^m + 1 in positive integers m, n and k ≥ 2, are (n, k,m) = (5,2,1), (5,3,1) and (6,3,2). For that, we shall use lower bounds for linear forms in logarithms together with a computational approach using Mathematica software.