On consecutive primitive elements in a finite field

For q an odd prime power with q > 169, we prove that there are always three consecutive primitive elements in the finite field Fq. Indeed, there are precisely eleven values of q ≤ 169 for which this is false. For 4 ≤ n ≤ 8, we present conjectures on the size of q0(n) such that q > q0(n) guarantees the existence of n consecutive primitive elements in Fq, provided that Fq has characteristic at least n. Finally, we improve the upper bound on q0(n) for all n ≥ 3.