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arXiv · 2010.15988

Artin Twin Primes

Magdaléna Tinková·Ezra Waxman·Mikuláš Zindulka
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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.

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Magdaléna Tinková, Ezra Waxman, Mikuláš Zindulka. 2020-10-29. Artin Twin Primes. https://doi.org/10.1016/j.jnt.2022.10.006

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