On solutions of the Diophantine equation $\mathcal{P}_m-L_n=c$

TL;DR

This paper characterizes all integers with multiple representations as differences of two linear recurrent sequences, advancing understanding of a variant of Pillai's equation through advanced Diophantine approximation techniques.

Contribution

It provides a complete classification of integers with multiple representations in a specific linear recurrence difference equation, using sophisticated number theory methods.

Findings

Identifies all such integers with multiple representations

Employs bounds for linear forms of logarithms and continued fractions

Uses Baker-Davenport reduction method for Diophantine approximation

Abstract

\noindent In this article, we determine all the integers $c$ having at least two representations as difference between two linear recurrent sequences. This is a variant of the Pillai's equation. This equation is an exponential Diophantine equation. The proof of our main theorem uses lower bounds for linear forms of logarithms, properties of continued fractions, and a version of the Baker-Davenport reduction method in Diophantine approximation.