On The Diophantine equation F_n +F _m =2 ^a

Abstract

In this paper, we find all the solutions of the title Diophantine equation in positive integer variables (n, m, a), where F _k is the kth term of the Fibonacci sequence. The proof of our main theorem uses lower bounds for linear forms in logarithms (Baker's theory) and a version of the Baker-Davenport reduction method in diophantine approximation.