On the Diophantine equation F_{n}-F_{m}=2^{a}

TL;DR

This paper solves a specific Diophantine equation involving Fibonacci numbers and powers of two, using advanced number theory techniques to find all solutions in nonnegative integers.

Contribution

It provides a complete solution to the equation F_n - F_m = 2^a, applying lower bounds for linear forms in logarithms and the Baker-Davenport reduction method.

Findings

All solutions in nonnegative integers are determined.

The equation has finitely many solutions, explicitly characterized.

Method demonstrates effectiveness of logarithmic bounds and reduction techniques.

Abstract

In this paper, we solve Diophantine equation in the tittle in nonnegative integers m,n, and a. In order to prove our result, we use lower bounds for linear forms in logarithms and and a version of the Baker-Davenport reduction method in diophantine approximation.