On sums of two Fibonacci numbers that are powers of numbers with limited Hamming weight

Abstract

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 $a\ge 2$ 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 $y=F_{n_1}+\cdots +F_{n_k}$ and equation \eqref{eq:abstract} is satisfied by $y$ and some non-negative integers $n,m$ and $a\ge 2$, then \[ y^a \leq \exp\left(C{(\varepsilon)} \cdot k^{(3+\varepsilon)k^2} \right). \] Here, $\epsilon >0$ can be chosen arbitrarily and $C(\epsilon )$ is an effectively computable constant.

14 pages