On arithmetic progressions in finite fields

Abstract

In this paper, we explore the existence of m-terms arithmetic progressions in 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 ≥ 4 and concrete results for m ∈ {2, 3}, where the complete list of exceptions when the common difference belongs to F * q is obtained. The proofs combine character sums, sieve estimates, and computational arguments using the software SageMath.