On Pillai's problem involving two linear recurrent sequences : Padovan and Fibonacci

TL;DR

This paper completely characterizes integers with multiple representations as differences between Padovan and Fibonacci sequences, using advanced number theory techniques.

Contribution

It provides a complete solution to a Pillai problem involving Padovan and Fibonacci sequences, applying bounds for linear forms in logarithms and Diophantine approximation methods.

Findings

Identifies all integers with multiple representations as differences of the sequences.

Employs lower bounds for linear forms in logarithms and continued fractions.

Uses Baker-Davenport reduction method for Diophantine approximation.

Abstract

In this paper, we find all integers $c$ having at least two representations as a difference between linear recurrent sequences. This problem is a pillai problem involving Padovan and Fibonacci sequence. The proof of our main theorem uses lower bounds for linear forms in logarithms, properties of continued fractions, and a version of the Baker-Davenport reduction method in Diophantine approximation.