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 ${\mathbb{F}}_q$. Indeed, there are precisely eleven values of $q\le 169$ for which this is false. For $4\le n\le 8$ we present conjectures on the size of $q_0(n)$ such that $q>q_0(n)$ guarantees the existence of $n$ consecutive primitive elements in ${\mathbb{F}}_q$, provided that ${\mathbb{F}}_q$ has characteristic at least~$n$. Finally, we improve the upper bound on $q_0(n)$ for all $n\ge 3$.