The smallest primitive root modulo a prime

TL;DR

This paper establishes new bounds on the smallest primitive root modulo a prime, demonstrating it is smaller than p^{0.68} for all primes p by refining classical inequalities.

Contribution

It introduces improved bounds on the smallest primitive root by leveraging refined applications of Pólya--Vinogradov and Burgess inequalities.

Findings

Smallest primitive root is less than p^{0.68} for all primes p

Refined use of Pólya--Vinogradov inequality

Enhanced application of Burgess inequality

Abstract

In this paper we will consider new bounds on the smallest primitive root modulo a prime. We will make more judicious use of the P\`olya--Vinogradov and Burgess inequalities, and use them to prove that the smallest primitive root is smaller than $p^{0.68}$ for all primes $p$.