On consecutive primitive elements in a finite field

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