Artin Twin Primes

Abstract

We say that a prime number $p$ is an $\text{Artin prime}$ 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 $p\le x$ 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.

29 pages; Revised version. Appendix with relevant Mathematica code included. Accepted for publication in the Journal of Number Theory

References in corpus (3)

• Bounded gaps between primes with a given primitive root
• Character sums for primitive root densities
• Vinogradov's three primes theorem with primes having given primitive roots