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

TL;DR

This paper establishes an explicit upper bound for perfect powers that are sums of two Fibonacci numbers, based on the Hamming weight of the number's Zeckendorf representation, advancing understanding of Fibonacci sum powers.

Contribution

It introduces a new bound depending solely on the Hamming weight of the number's Zeckendorf representation, improving previous bounds that depended on the size of the number.

Findings

Derived an explicit upper bound for y^a based on Hamming weight

Bound depends on the number of Fibonacci terms in y's Zeckendorf representation

Provides an effectively computable constant C(ε) for the bound

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 $a\geq 2$ 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.