Artin Twin Primes

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 th...

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Bibliographic Details
Published in:arXiv.org 2023-01
Main Authors: Tinková, Magdaléna, Waxman, Ezra, Mikuláš Zindulka
Format: Article
Language:English
Subjects:
Prime numbers
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2010.15988