On the Diophantine equation $\displaystyle \sum _{k=1}^{5}F_{n_k}=2^a$

TL;DR

This paper completely characterizes all solutions where the sum of five Fibonacci numbers equals a power of two, with some exceptions, and discusses related open problems.

Contribution

It determines all solutions to the equation involving five Fibonacci numbers summing to a power of two, extending previous results and identifying specific exceptions.

Findings

All solutions to the equation are characterized with few exceptions.

The paper identifies specific solutions where the sum equals a power of two.

An open problem related to the number of solutions is proposed.

Abstract

Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence given by $F_0 = 0, F_1 = 1$ and $F_{n+2} = F_{n+1}+F_n$ for $n \geq 0$. In this paper, we have determined all the powers of 2 which are sums of five Fibonacci numbers with few exceptions that we characterize. We have also stated an open problem relating to the number of solutions of equations like those studied in this paper.