Arithmetic progression in a finite field with prescribed norms

Abstract Given a prime power q and a positive integer n, let 𝔽 q n Fₐ^{₍} represent a finite extension of degree n of the finite field 𝔽 q {Fₐ}. In this article, we investigate the existence of m elements in arithmetic progression, where every element is primitive and at least one is normal with prescribed norms. Moreover, for n ≥ 6 n 6, q = 3 k q=3^{k}, m = 2 m=2 we establish that there are only 10 possible exceptions.