Bounded gaps between primes with a given primitive root, II

Let $m$ be a natural number, and let $\mathcal{Q}$ 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 $q\in \mathcal{Q}$ as a primitive root, all lying in an interval of length $O_{\mathcal{Q}}(\exp (C^{\mathrm{\prime }}m))$. 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 $E\mathrm{/}\mathbb{Q}$ 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 $E({\mathbb{F}}_p)$ is cyclic, all lying an interval of length $O_E(\exp(C’’ m))$\mathrm{.}If$E$hasCM,thentheGRHassumptioncanberemoved\mathrm{.}Here$C$,$C’$, and $C^{\mathrm{\prime }\mathrm{\prime }}$ are absolute constants.