We say that a prime number p is an Artin prime for g if g mod p generates the group (Z/pZ)^×. For appropriately chosen integers d and g, we present a conjecture for the asymptotic number π_d,g(x) of primes p≤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 π_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.