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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 nearest-neighbor-spacing distribution of primes. These oscillations have the very profound period of length six. We also calculate the spectral rigidity Δ(3) for prime numbers by two methods. After suitable averaging one of these methods gives the Poisson dependence Δ(3)(L) = L/15.

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