Primitive Divisors of Lucas Sequences in Polynomial Rings

TL;DR

This paper extends the theory of primitive divisors from classical Lucas sequences to Lucas sequences in polynomial rings, providing new results and correcting previous theorems in this generalized setting.

Contribution

It develops Lucas sequence theory in polynomial rings and proves new results on primitive divisors, correcting earlier incomplete theorems.

Findings

All terms $U_n$ in polynomial Lucas sequences have primitive divisors for large $n$

Complete characterization of small $n$ cases without primitive divisors in polynomial rings

Correction of previous theorems on primitive divisors in polynomial Lucas sequences

Abstract

It is known that all terms $U_n$ of a classical regular Lucas sequence have a primitive prime divisor if $n>30$. In addition, a complete description of all regular Lucas sequences and their terms $U_n$, $2\leq n\leq 30$, which do not have a primitive divisor is also known. Here, we prove comparable results for Lucas sequences in polynomial rings, correcting some previous theorem on the same subject. The first part of our paper develops some elements of Lucas theory in several abstract settings before proving our main theorem in polynomial rings.