Prescribed Primitive Roots And The Least Primes

Abstract

Let $q\ne \pm 1,v^2$ be a fixed integer, and let $x\ge 1$ be a large number. The least prime number $p\ge 3$ such that $q$ is a primitive root modulo $p$ is conjectured to be $p\ll (\log q)(\log \log q{)}^3),$ where $\gcd (p,q)=1$. This note proves the existence of small primes $p\ll (\log x{)}^c$, where $c>0$ is a constant, a close approximation to the conjectured upper bound.

Eight Pages. Keywords: Primitive root, Least prime number, Artin primitive root conjecture

References in corpus (1)

• Simple Proof of the Primitive Root Conjecture