Computation of the least primitive root

TL;DR

This paper reports extensive computations of the smallest primitive roots modulo primes and their squares up to 10^16, and proves bounds on these roots for all primes beyond small cases.

Contribution

It provides the first large-scale computation of primitive roots up to 10^16 and establishes new bounds on their sizes for all primes.

Findings

g(p)<p^{5/8} for all primes p>3

h(p)<p^{2/3} for all primes p

Computed primitive roots for all primes up to 10^16

Abstract

Let $g(p)$ denote the least primitive root modulo $p$, and $h(p)$ the least primitive root modulo $p^2$. We computed $g(p)$ and $h(p)$ for all primes $p\le 10^{16}$. Here we present the results of that computation and prove three theorems as a consequence. In particular, we show that $g(p)<p^{5/8}$ for all primes $p>3$ and that $h(p)<p^{2/3}$ for all primes $p$.