  In 2018, Luca and Patel conjectured that the largest perfect power
representable as the sum of two Fibonacci numbers is $3864^2 = F_{36} +
F_{12}$. In other words, they conjectured that the equation
\begin{equation}\tag{$\ast$}\label{eq:abstract}
  y^a = F_n + F_m \end{equation} has no solutions with $a\geq 2$ and $y^a >
3864^2$. While this is still an open problem, there exist several partial
results. For example, recently Kebli, Kihel, Larone and Luca proved an explicit
upper bound for $y^a$, which depends on the size of $y$.
  In this paper, we find an explicit upper bound for $y^a$, which only depends
on the Hamming weight of $y$ with respect to the Zeckendorf representation.
More specifically, we prove the following: If $y = F_{n_1}+ \dots + F_{n_k}$
and equation \eqref{eq:abstract} is satisfied by $y$ and some non-negative
integers $n,m$ and $a\geq 2$, then \[
  y^a
  \leq \exp\left(C{(\varepsilon)} \cdot k^{(3+\varepsilon)k^2} \right). \]
Here, $\varepsilon >0$ can be chosen arbitrarily and $C(\varepsilon)$ is an
effectively computable constant.
