Artin prime producing polynomials

TL;DR

This paper explores the generation of polynomials that produce long strings of primes which are Artin primes for a fixed integer, extending Artin's conjecture and providing explicit examples with record lengths.

Contribution

It introduces a general method for finding polynomials that generate many Artin primes for a fixed integer, including explicit examples with record lengths.

Findings

Generated polynomials with over 6000 consecutive Artin primes.

Established a connection between polynomial prime generation and Artin's conjecture.

Provided explicit examples for linear, quadratic, and cubic polynomials.

Abstract

We define an Artin prime for an integer $g$ to be a prime such that $g$ is a primitive root modulo that prime. Let $g\in \mathbb{Z}\setminus\{-1\}$ and not be a perfect square. A conjecture of Artin states that the set of Artin primes for $g$ has a positive density. In this paper we study a generalization of this conjecture for the primes produced by a polynomial and explore its connection with the problem of finding a fixed integer $g$ and a prime producing polynomial $f(x)$ with the property that a long string of consecutive primes produced by $f(x)$ are Artin primes for $g$. By employing some results of Moree, we propose a general method for finding such polynomials $f(x)$ and integers $g$. We then apply this general procedure for linear, quadratic, and cubic polynomials to generate many examples of polynomials with very large Artin prime production length. More specifically, among many other examples, we exhibit linear, quadratic, and cubic (respectively) polynomials with 6355, 37951, and 10011 (respectively) consecutive Artin primes for certain integers $g$.