Primitive element pairs with one prescribed trace over a finite field

In this article, we establish a sufficient condition for the existence of a primitive element α∈Fqn such that the element α+α−1 is also a primitive element of Fqn, and TrFqn|Fq(α)=a for any prescribed a∈Fq, where q=pk for some prime p and positive integer k. We prove that every finite field Fqn(n≥5), contains such primitive elements except for finitely many values of q and n. Indeed, by computation, we conclude that there are no actual exceptional pairs (q,n) for n≥5.