  Although the prime numbers are deterministic, they can be viewed, by some
measures, as pseudo-random numbers. In this article, we numerically study the
pair statistics of the primes using statistical-mechanical methods, especially
the structure factor $S(k)$ in an interval $M \leq p \leq M + L$ with $M$
large, and $L/M$ smaller than unity. We show that the structure factor of the
prime-number configurations in such intervals exhibits well-defined Bragg-like
peaks along with a small "diffuse" contribution. This indicates that the primes
are appreciably more correlated and ordered than previously thought. Our
numerical results definitively suggest an explicit formula for the locations
and heights of the peaks. This formula predicts infinitely many peaks in any
non-zero interval, similar to the behavior of quasicrystals. However, primes
differ from quasicrystals in that the ratio between the location of any two
predicted peaks is rational. We also show numerically that the diffuse part
decays slowly as $M$ and $L$ increases. This suggests that the diffuse part
vanishes in an appropriate infinite-system-size limit.
