DI ST RI BU TI ON Correlations and symbol entropy of the distances between consecutive prime numbers and their increments

The difference between two consecutive prime numbers is called the distance between primes. We study the statistical properties of the distances, and also of their increments (the difference between two consecutive distances). Using a sequence comprising the first Np = 5× 10 7 prime numbers, we obtain an empirical form for the symbol entropy for the set of distances as a function of sequence length Np. We find a similar form for the symbol entropy for the sequence of corresponding increments. For a given Np, we show that the entropy of the sequence of increments is always greater than the entropy of the sequence of distances, consistent with a greater variability in the values of the increments. We find that the histogram of the increments follows an exponential distribution, with superposed periodic behavior with period-three oscillations. We observe that the magnitude of these increments follows a logarithmic trend as a function of Np, similar to the trend previously known for distances. We also investigate the correlations between both distances and increments; we find that at small and intermediate scales the distances exhibit a weak powerlaw anticorrelation followed by a crossover to strongly correlated behavior at large scales. This crossover is due to the logarithmic trend in the sequence of distances. For increments, we find that (i) the sequence of increments is strongly anticorrelated at all scales, (ii) due to the logarithmic trend, their magnitudes exhibit a crossover from weakly anticorrelated behavior at small scales to strongly correlated behavior at large scales, and (iii) their signs are anticorrelated at small scales and uncorrelated at large scales. PACS numbers: