On solutions of the Diophantine equation $F_{n_{1}}+F_{n_{2}}+F_{n_{3}}+F_{n_{4}}=2^a$

TL;DR

This paper completely characterizes all solutions to the equation where the sum of four Fibonacci numbers equals a power of two, identifying specific exceptions and providing a comprehensive solution.

Contribution

It offers a complete solution to the problem of representing powers of two as sums of four Fibonacci numbers, including the characterization of exceptions.

Findings

Identified all solutions to the equation with four Fibonacci numbers summing to a power of two.

Characterized the specific exceptions where solutions do not follow the general pattern.

Provided a comprehensive classification of solutions and exceptions.

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 solve all powers of two which are sums of four Fibonacci numbers with a few exceptions that we characterize.