On the Lucas and Lehmer sequences in Dedekind domains

TL;DR

This paper extends classical properties of Lucas and Lehmer sequences to Dedekind domains and function fields, establishing divisibility properties and primitive divisor results in these algebraic structures.

Contribution

It introduces strong divisibility and primitive divisor theorems for Lucas and Lehmer sequences within Dedekind domains and function fields, generalizing prior results.

Findings

Established strong divisibility property in Dedekind domains.

Proved analogues of Zsigmondy's theorem for these sequences.

Demonstrated primitive divisor existence in the new setting.

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.