Prescribed Primitive Roots And The Least Primes

TL;DR

This paper investigates the distribution of primes for which a fixed integer is a primitive root, providing bounds close to a long-standing conjecture and proving the existence of small such primes.

Contribution

It establishes the existence of small primes where a fixed integer is a primitive root, approximating the conjectured upper bounds.

Findings

Proves the existence of small primes p where q is a primitive root.

Provides bounds p  (\u221a) ( ext{log} x)^c, close to conjectured bounds.

Advances understanding of primitive roots and prime distribution.

Abstract

Let $q\ne \pm1,v^2$ be a fixed integer, and let $x\geq 1$ be a large number. The least prime number $p \geq3 $ 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.