Powers of two as sum of two generalized Fibonacci numbers

TL;DR

This paper investigates when sums of two generalized Fibonacci numbers equal powers of two, extending previous work on the classical Fibonacci sequence to the broader family of k-generalized Fibonacci sequences.

Contribution

It generalizes the problem of representing powers of two as sums of two Fibonacci numbers to the k-generalized Fibonacci sequences, exploring new cases and extending prior results.

Findings

Identifies specific conditions for powers of two to be expressed as sums of two k-generalized Fibonacci numbers.

Extends known results from classical Fibonacci sequences to generalized sequences for k ≥ 2.

Provides new insights into the structure and properties of k-generalized Fibonacci sequences.

Abstract

For $k\geq 2$, the $k$-generalized Fibonacci sequence $(F_n^{(k)})_{n}$ is defined by the initial values $0,0,\ldots,0,1$ ($k$ terms) and such that each term afterwards is the sum of the $k$ preceding terms. In this paper, we search for powers of two of the form $F_n^{(k)}+F_m^{(k)}$. This work is related to a recent result by Bravo and Luca concerning the case $k=2$.