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

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 $\max \{n_1,n_2,a_1,a_2,a_3\}\le 18$ and $\max \{m_1,m_2,m_3,t_1,t_2\}\le 16$.