  A statistical analysis of the prime numbers indicates possible traces of
quantum chaos. We have computed the nearest neighbor spacing distribution,
number variance, skewness, and excess for sequences of the first N primes for
various values of N. All four statistical measures clearly show a transition
from random matrix statistics at small N toward Poisson statistics at large N.
In addition, the number variance saturates at large lengths as is common for
eigenvalue sequences. This data can be given a physical interpretation if the
primes are thought of as eigenvalues of a quantum system whose classical
dynamics is chaotic at low energy but regular at high energy. We discuss some
difficulties with this interpretation in an attempt to clarify what kind of
physical system might have the primes as its quantum eigenvalues.
