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Primitive Divisors of Lucas Sequences in Polynomial Rings

Joaquim Cera Da Conceição

TL;DR

The paper addresses primitive divisors of Lucas sequences in polynomial rings, seeking a polynomial-ring analogue of Zsigmondy/BHV results and a corrected version of Sha's theorem. It develops Lucas theory in integral domains and UFDs, uses the cyclotomic factorization $U_n=\prod_{d\mid n, d>1}\Phi_d(a,b)$ with associated $Q_n=\Phi_n(a,b)$ to analyze primitive divisors via valuations and ranks of appearance. The main result identifies that for $p\nmid n$ the terms $U_n$ typically possess primitive divisors, with explicit finite exceptional cases for $n\in\{2,3,4,6\}$ governed by simple relations between $P$ and $Q$. The work thereby extends classical Zsigmondy-type phenomena to function-field-like settings and clarifies conditions under which a primitive divisor can fail to exist in polynomial rings, contributing to a corrected framework for Lucas sequences over $A[T]$.

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.

Primitive Divisors of Lucas Sequences in Polynomial Rings

TL;DR

The paper addresses primitive divisors of Lucas sequences in polynomial rings, seeking a polynomial-ring analogue of Zsigmondy/BHV results and a corrected version of Sha's theorem. It develops Lucas theory in integral domains and UFDs, uses the cyclotomic factorization with associated to analyze primitive divisors via valuations and ranks of appearance. The main result identifies that for the terms typically possess primitive divisors, with explicit finite exceptional cases for governed by simple relations between and . The work thereby extends classical Zsigmondy-type phenomena to function-field-like settings and clarifies conditions under which a primitive divisor can fail to exist in polynomial rings, contributing to a corrected framework for Lucas sequences over .

Abstract

It is known that all terms of a classical regular Lucas sequence have a primitive prime divisor if . In addition, a complete description of all regular Lucas sequences and their terms , , 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.
Paper Structure (5 sections, 17 theorems, 40 equations)

This paper contains 5 sections, 17 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Basic properties of general Lucas sequences
  3. Integral domains
  4. Unique factorization domains
  5. The main theorem

Key Result

Theorem 1

Suppose the characteristic $p > 0$ and let $U'$ be the sequence obtained from $(U_n)_{n\geq 1}$ by deleting the terms $U_n$ with $p\mid n$, then each term of $U'$ beyond the second has a primitive prime divisor. If $p = 0$, then each term of $(U_n)_{n\geq 1}$ beyond the second has a primitive prime

Theorems & Definitions (32)

  • Theorem 1
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Lemma 2: Divisibility sequence
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 22 more