Primitive root bias for twin primes
Stephan Ramon Garcia, Elvis Kahoro, Florian Luca

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 of pairs of twin primes, has more primitive roots than does . If this occurs, we say that is exceptional (there are only two exceptional pairs with ). Assuming the Bateman-Horn conjecture, we prove that at least of twin prime pairs are exceptional and at least are not exceptional. We also conjecture a precise formula for the proportion of exceptional twin primes.
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Taxonomy
TopicsAnalytic Number Theory Research · Finite Group Theory Research · Coding theory and cryptography
