Primitive Root Bias for Twin Primes

Abstract

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 ⩽ p ⩽ 10, 000). Assuming the Bateman–Horn conjecture, we prove that at least 0.459% 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.