Artin prime producing polynomials
Amir Akbary, Keilan Scholten

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 to be a prime such that is a primitive root modulo that prime. Let and not be a perfect square. A conjecture of Artin states that the set of Artin primes for 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 and a prime producing polynomial with the property that a long string of consecutive primes produced by are Artin primes for . By employing some results of Moree, we propose a general method for finding such polynomials and integers . 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…
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Geometric and Algebraic Topology · Analytic Number Theory Research
