Explicit upper bounds on the least primitive root

Abstract

We give a method for producing explicit bounds on g(p), the least primitive root modulo p. Using our method we show that g(p) < 2r 2 rω(p−1) p 1 4 + 1 4r for p > 10 56 where r ≥ 2 is an integer parameter. This result beats existing bounds that rely on explicit versions of the Burgess inequality. Our main result allows one to derive bounds of differing shapes for various ranges of p. For example, our method also allows us to show that

The least primitive root modulo $p^{2}$

“…While explicit upper bounds on g(p) have been given [4,5,10,15], we are unaware of any such bounds for h(p). Trivially, we have h(p) < p 2 , and, if we use an appropriate version of the Pólya-Vinogradov inequality [7], we can make the estimate h(p) < p 1+ǫ explicit.…”