Artin Twin Primes

TL;DR

This paper investigates the distribution of primes called Artin primes, introduces a conjecture for their asymptotic count in prime pairs, and explores the influence of specific parameters on their occurrence.

Contribution

It presents a new conjecture on the asymptotic behavior of prime pairs where both are Artin primes, and identifies conditions leading to zero counts.

Findings

Conjecture for the asymptotic count of Artin prime pairs.

Identification of parameter classes with zero Artin prime pairs.

Distribution of Artin prime pairs resembles a Poisson binomial distribution.

Abstract

We say that a prime number $p$ is an $\textit{Artin prime}$ for $g$ if $g$ mod $p$ generates the group $(\mathbb{Z}/p\mathbb{Z})^{\times}$. For appropriately chosen integers $d$ and $g$, we present a conjecture for the asymptotic number $\pi_{d,g}(x)$ of primes $p \leq x$ such that both $p$ and $p+d$ are Artin primes for $g$. In particular, we identify a class of pairs $(d,g)$ for which $\pi_{d,g}(x) =0$. Our results suggest that the distribution of Artin prime pairs, amongst the ordinary prime pairs, is largely governed by a Poisson binomial distribution.