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

Abstract

In this paper, we consider the Diophantine equation $V_n-b^m=c$ for given integers b, c with $b\ge 2$, 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 $V_n$.

Keywords: Diophantine equations; Lucas sequences; Pillai’s Problem.