The rank of apparition of powers of Lucas sequence

Abstract

This note is devoted to studying the divisibility relation $u_{n}^{k+1}|u_{m}$ for a least positive integer $m$, where $\{u_{n}\}_{n \geq 0}$ is a nondegenerate Lucas sequence with characteristic polynomial $x^{2}-ax - b,$ for some relatively prime integers $a$ and $b$.