On a divisibility relation for Lucas sequences

TL;DR

This paper investigates a specific divisibility relation within Lucas sequences, focusing on conditions under which certain sequence terms divide differences of powered terms, contributing to number theory and sequence divisibility understanding.

Contribution

The paper introduces new divisibility criteria for Lucas sequences involving powers and shifts, expanding the theoretical framework of sequence divisibility properties.

Findings

Established divisibility conditions for Lucas sequences involving powers.

Identified specific relations between sequence indices and divisibility.

Enhanced understanding of Lucas sequence divisibility patterns.

Abstract

In this note, we study the divisibility relation $U_m\mid U_{n+k}^s-U_n^s$, where ${\bf U}:=\{U_n\}_{n\ge 0}$ is the Lucas sequence of characteristic polynomial $x^2-ax\pm 1$ and $k,m,n,s$ are positive integers.