Arithmetic progression in a finite field with prescribed norms

Given a prime power q and a positive integer n , let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝔽</m:mi> <m:msup> <m:mi>q</m:mi> <m:mi>n</m:mi> </m:msup> </m:msub> </m:math> {F_{qⁿ}} represent a finite extension of degree n of the finite field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> </m:msub> </m:math> {{Fq}} . 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>6</m:mn> </m:mrow> </m:math> {n≥ 6} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>q</m:mi> <m:mo>=</m:mo> <m:msup> <m:mn>3</m:mn> <m:mi>k</m:mi> </m:msup> </m:mrow> </m:math> {q=3ᵏ} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {m=2} we establish that there are only 10 possible exceptions.