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arXiv · 2610.07538

Primitive normal progressions in finite fields

Gustav Kjærbye Bagger·Matthew Fernando
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Abstract

Consider an arithmetic progression inside the finite field $\mathbb{F}_{q^n}$. We study the existence of progressions whose terms are all primitive and at least one of them is normal. Using a combination of sieve estimates, ramified bounds, and computational arguments, we improve significantly upon previous literature for progressions of length $m=4$ and obtain partial results whenever $5\leq m\leq 10$. Finally, we provide a general bound which outperforms previous results in the range $m\geq 53$.

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Gustav Kjærbye Bagger, Matthew Fernando. 2026-10-06. Primitive normal progressions in finite fields. https://arxiv.org/abs/2610.07538

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