Primitive Divisors of some Lehmer-Pierce Sequences

TL;DR

This paper investigates primitive divisors in Lehmer-Pierce sequences derived from units in real quadratic fields, identifying which sequence terms lack primitive prime divisors using specialized methods.

Contribution

It introduces new techniques to determine the terms of Lehmer-Pierce sequences that do not possess primitive prime divisors.

Findings

Identified sequence terms without primitive prime divisors.

Developed methods applicable to real quadratic fields.

Enhanced understanding of primitive divisor distribution.

Abstract

We study the primitive divisors of the terms of $(\Delta_n)_{n \geq 1}$, where $\Delta_n=N_{K/ \mathbb{Q}}(u^n-1)$ for $K$ a real quadratic field, and $u>1$ a unit element of its ring of integers. The methods used allow us to find the terms of the sequence that do not have a primitive prime divisor.