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 …