  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.
