  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.
