Bounded gaps between primes with a given primitive root, II

Let m be a natural number, and let Q be a set containing at least exp(Cm) primes. We show that one can find infinitely many strings of m consecutive primes each of which has some q ∈ Q as a primitive root, all lying in an interval of length OQ(exp(C ′m)). This is a bounded gaps variant of a theorem of Gupta and Ram Murty. We also prove a result on an elliptic analogue of Artin’s conjecture. Let E/Q be an elliptic curve with an irrational 2-torsion point. Assume GRH. Then for every m, there are infinitely many strings of m consecutive primes p for which E(Fp) is cyclic, all lying an interval of length OE(exp(C ′′m)). If E has CM, then the GRH assumption can be removed. Here C, C ′, and C ′′ are absolute constants. 2010 Mathematics subject classification. Primary: 11A07, Secondary: 11G05