On Pillai's Problem involving Lucas sequences of the second kind

TL;DR

This paper investigates a specific Diophantine equation involving Lucas sequences of the second kind, establishing bounds on solutions when multiple solutions exist, thus contributing to the understanding of Pillai's problem in this context.

Contribution

It provides new bounds on solutions to a Pillai-type equation involving Lucas sequences of the second kind, under certain technical conditions.

Findings

Bounded the size of solutions when at least three solutions exist

Established upper limits on coefficients in the characteristic polynomial

Extended understanding of Pillai's problem for Lucas sequences

Abstract

In this paper we consider the Diophantine equation $ V_n - b^m = c $ for given integers $ b,c $ with $ b \geq 2 $, whereas $ V_n $ varies among Lucas-Lehmer sequences of the second kind. We prove under some technical conditions that if the considered equation has at least three solutions $ (n,m) $, then there is an upper bound on the size of the solutions as well as on the size of the coefficients in the characteristic polynomial of $ V_n $.