  It has been known since Vinogradov that, for irrational $\alpha$, the
sequence of fractional parts $\{\alpha p\}$ is equidistributed in
$\mathbb{R}/\mathbb{Z}$ as $p$ ranges over primes. There is a natural
second-order equidistribution property, a pair correlation of such fractional
parts, which has recently received renewed interest, in particular regarding
its relation to additive combinatorics. In this paper we show that the primes
do not enjoy this stronger equidistribution property.
