On arithmetic progressions in finite fields

TL;DR

This paper investigates the existence of arithmetic progressions with primitive and normal elements in finite fields, providing asymptotic and concrete results for different lengths, and identifying exceptions.

Contribution

It introduces new results on arithmetic progressions in finite fields with primitive and normal elements, combining theoretical and computational methods.

Findings

Asymptotic existence results for m ≥ 4

Concrete results for m = 2, 3

Complete list of exceptions for common difference in _q^*

Abstract

In this paper, we explore the existence of $m$-terms arithmetic progressions in $\mathbb{F}_{q^n}$ with a given common difference whose terms are all primitive elements, and at least one of them is normal. We obtain asymptotic results for $m \ge 4$ and concrete results for $m \in \{2,3\}$, where the complete list of exceptions when the common difference belongs to $\mathbb{F}_{q}^*$ is obtained. The proofs combine character sums, sieve estimates, and computational arguments using the software SageMath.