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Mathematics, Number Theory
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 0.459% 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.
