On Diophantine equations involving sums of Fibonacci numbers and powers of $2$

TL;DR

This paper completely solves specific Diophantine equations involving sums of Fibonacci numbers and powers of two, establishing bounds on the indices involved and providing a complete classification of solutions.

Contribution

The paper provides a complete solution to certain Fibonacci and power-of-two Diophantine equations, including explicit bounds on the indices and all solutions.

Findings

Maximum index for the first equation is 18.

Maximum index for the second equation is 16.

All solutions are explicitly characterized within these bounds.

Abstract

In this paper, we completely solve the Diophantine equations $F_{n_1} + F_{n_2} = 2^{a_1} + 2^{a_2} + 2^{a_3}$ and $ F_{m_1} + F_{m_2} + F_{m_3} =2^{t_1} + 2^{t_2} $, where $F_k$ denotes the $k$-th Fibonacci number. In particular, we prove that $\max \{n_1, n_2, a_1, a_2, a_3 \}\leq 18$ and $\max \{ m_1, m_2, m_3, t_1, t_2 \}\leq 16$.