  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 $q \leq 169$ for which this is false. For
$4\leq n \leq 8$ 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 $n\geq 3$.
