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arXiv · 2208.02876

On arithmetic progressions in finite fields

Abílio Lemos·Victor Neumann·Sávio Ribas
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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.

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Abílio Lemos, Victor Neumann, Sávio Ribas. 2022-08-04. On arithmetic progressions in finite fields. https://arxiv.org/abs/2208.02876

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