On the Lucas and Lehmer sequences in Dedekind domains

TL;DR

The paper develops a unified framework to extend strong divisibility and Zsigmondy-type primitive divisor results for Lucas and Lehmer sequences from classical settings to Dedekind domains, including function fields. By adapting Sha’s polynomial- and ideal-theoretic methods—centered on resultants, cyclotomic polynomials $oldsymbol{\\Phi}_n$, and coprimality lemmas for ideals—the authors prove strong divisibility for Lehmer sequences $U_n$, Lucas sequences $L_n$, and related $F_n$ and $G_n$, and establish primitive divisors for $n$ satisfying a bound tied to $oldsymbol{\\varphi}(n)$ and the function-field degree $d$. In the function-field context, the primitive part of $U_n$ (and of $L_n$ and $F_n$ analogues) is shown to be generated by principal ideals $ig\\\\langle oldsymbol{\Phi}_n( abla, abla) ig angle$ (or $ig\\\\langle oldsymbol{\Phi}_n(f,g) ig angle$), with degree- and characteristic-dependent adjustments (e.g., replacing $d$ by $2d$ or $4d$ when containing the defining elements). These results recover and extend the classical Zsigmondy-type theorems to the Dedekind/function-field setting and provide explicit structural descriptions of the primitive divisors in terms of cyclotomic forms.

Abstract

In this paper, we first obtain the strong divisibility property for the Lucas and Lehmer sequences in Dedekind domains, and then establish analogues of Zsigmondy's theorem and the primitive divisor results for such sequences in function fields.

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 1.2: Zsigmondy's theorem in $R$
• Remark 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Theorem 1.8
• Theorem 1.9
• Remark 1.10