Least Prime Primitive Roots

This note presents an upper bound for the least prime primitive roots $g^{\ast }(p)$ modulo $p$, a large prime. The current literature has several estimates of the least prime primitive root $g^{\ast }(p)$ modulo a prime $p\ge 2$ such as $g^{\ast }(p)\ll p^c,c>2.8$. The estimate provided within seems to sharpen this estimate to the smaller estimate $g^{\ast }(p)\ll p^{5\mathrm{/}\log \log p}$ uniformly for all large primes $p\ge 2$.