On sums of Fibonacci numbers with few binary digits

TL;DR

This paper completely solves a specific Fibonacci sum equation involving powers of two, using advanced number theory techniques including linear forms in logarithms and p-adic methods.

Contribution

It provides a complete solution to a Fibonacci sum Diophantine equation with constraints on binary digit sums, employing novel p-adic approaches.

Findings

Explicit solutions to the equation are identified.

The methods extend existing techniques with p-adic analysis.

The results contribute to understanding Fibonacci sums with binary digit restrictions.

Abstract

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.