Artin twin primes (Q2112730)

A prime \(p\) is called an Artin prime for \(g\) if \(g\pmod p\) generates the finite cyclic group of order \(p-1\). The famous Artin primitive root conjecture states that if \(g\) is neither a perfect square nor \(-1\), then the set of Artin primes for \(g\) contains infinitely many primes. The authors present a conjecture for the asymptotic number of primes \(p\le x\) such that both \(p\) and \(p+d\) are Artin primes for \(g\), under certain conditions on \(d\) and \(g\). In particular, a class of pairs \((d,g)\) is found such that the above number is zero. The obtained results suggest that, the distribution of Artin prime pairs, amongst the ordinary prime pairs, is governed mainly by a Poisson binomial distribution. The exact statements of results are too complicated to be stated here.