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 F-q. 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 > q(0)( 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.