Information Entropy and Correlations in Prime Numbers

The difference between two consecutive prime numbers is called the distance between the primes. We study the statistical properties of the distances and their increments (the difference between two consecutive distances) for a sequence comprising the first $5\times 10^7$ prime numbers. We find that the histogram of the increments follows an exponential distribution with superposed periodic behavior of period three, similar to previously-reported period six oscillations for the distances.