On solutions of the Diophantine equation $F_{n_1}+F_{n_2}+F_{n_3}+F_{n_4}=2^a$

Abstract

Let $(F_n{)}_{n\ge 0}$ be the Fibonacci sequence given by $F_0=0,F_1=1$ and $F_{n+2}=F_{n+1}+F_n$ for $n\ge 0$. In this paper, we solve all powers of two which are sums of four Fibonacci numbers with a few exceptions that we characterize.