Primitive root bias for twin primes

TL;DR

This paper investigates the distribution of primitive roots among twin primes, providing numerical evidence, conditional proofs based on the Bateman-Horn conjecture, and conjectures about their proportions.

Contribution

It introduces the concept of exceptional twin primes based on primitive root bias and offers a conditional lower bound on their frequency.

Findings

Approximately 2% of twin prime pairs are exceptional based on numerical evidence.

Assuming Bateman-Horn, at least 0.47% of twin primes are exceptional.

At least 65.13% of twin primes are not exceptional.

Abstract

Numerical evidence suggests that for only about $2\%$ of pairs $p,p+2$ of twin primes, $p+2$ has more primitive roots than does $p$. If this occurs, we say that $p$ is exceptional (there are only two exceptional pairs with $5 \leq p \leq 10{,}000$). Assuming the Bateman-Horn conjecture, we prove that at least $0.47\%$ of twin prime pairs are exceptional and at least $65.13\%$ are not exceptional. We also conjecture a precise formula for the proportion of exceptional twin primes.