  Grosswald's conjecture is that $g(p)$, the least primitive root modulo $p$,
satisfies $g(p) \leq \sqrt{p} - 2$ for all $p>409$. We make progress towards
this conjecture by proving that $g(p) \leq \sqrt{p} -2$ for all $409<p<
2.5\times 10^{15}$ and for all $p>3.67\times 10^{71}$.
