Primitive normal progressions in finite fields

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 $5\le m\le 10$. Finally, we provide a general bound which outperforms previous results in the range $m\ge 53$.

arXiv comment21 pages