  We give heuristic arguments and computer results to support the hypothesis
that, after appropriate rescaling, the statistics of spacings between adjacent
prime numbers follows the Poisson distribution. The scaling transformation
removes the oscillations in the NNSD of primes. These oscillations have the
very profound period of length six. We also calculate the spectral rigidity
$\Delta_3$ for prime numbers by two methods. After suitable averaging one of
these methods gives the Poisson dependence $\Delta_3(L)=L/15$.
