Sums of Fibonacci numbers close to a power of 2

TL;DR

This paper characterizes all sums of two Fibonacci numbers and all Lucas numbers that are near powers of 2, using advanced number theory techniques to extend previous research.

Contribution

It provides a complete classification of Fibonacci and Lucas sums close to powers of 2, employing bounds for linear forms in logarithms and diophantine approximation methods.

Findings

Identified all Fibonacci sums near powers of 2.

Determined all Lucas numbers close to powers of 2.

Extended previous classifications with new bounds.

Abstract

In this paper, we find all sums of two Fibonacci numbers which are close to a power of 2. As a corollary, we also determine all Lucas numbers close to a power of 2. The main tools used in this work are lower bounds for linear forms in logarithms due to Matveev and Dujella-Peth\"{o} version of the Baker-Davenport reduction method in diophantine approximation. This paper continues and extends the previous work of Chern and Cui.