Bounded gaps between primes with a given primitive root, II

TL;DR

This paper proves the existence of infinitely many bounded gaps between primes with a specified primitive root and explores an elliptic curve analogue of Artin's conjecture, under GRH and unconditionally for CM curves.

Contribution

It establishes bounded prime gaps with prescribed primitive roots and extends results to elliptic curves, including an elliptic analogue of Artin's conjecture.

Findings

Infinitely many strings of m consecutive primes with a given primitive root within bounded intervals.

Existence of infinitely many m-length prime strings where the elliptic curve group is cyclic, within bounded intervals, under GRH.

Unconditional results for CM elliptic curves.

Abstract

Let $m$ be a natural number, and let $\mathcal{Q}$ be a set containing at least $\exp(C m)$ 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 $O_{\mathcal{Q}}(\exp(C'm))$. 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 $E/\mathbb{Q}$ 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 $O_E(\exp(C'' m))$. If $E$ has CM, then the GRH assumption can be removed. Here $C$, $C'$, and $C''$ are absolute constants.