  In this paper we completely solve the Diophantine equation
$F_n+F_m=2^{a_1}+2^{a_2}+2^{a_3}+2^{a_4}+2^{a_5}$, where $F_k$ denotes the
$k$-th Fibonacci number. In addition to complex linear forms in logarithms and
the Baker-Davenport reduction method, we use $p$-adic versions of both tools.
