A Note on the Distribution of Differences between Consecutive Prime Numbers

Abstract

The results reported in this note refer to the distribution of zn = pn - pn-1 for the first three million prime numbers (p). The analyses of the note are almost purely statistical. The difference between consecutive prime numbers is treated as a random variable, and empirical frequency distributions are examined for sets of 5000 consecutive primes through the first three million. The results reported are based upon frequency distributions (59 in total) that are calculated at intervals of 50,000 primes for π(n) between 95,000 and 3,000,000. The quantities that are investigated include the means and standard deviations of the 59 distributions, together with coefficients that are obtained from exponential functions fitted by least-squares to the "poles" of the underlying density functions. The resulting vector of 59 estimated coefficients is then in turn related (via a least-squares regression equation) to the logarithm of pn. Key results of the analyses are as follows: (1) That the mean of zn increases with the logarithm of pn is clearly confirmed. (2) The support for zn increases very slowly through the first three million primes, as the maximum zn in the "samples" of 5000 consecutive primes that have been analyzed is never found to be larger than 178. (3) "Poles" in the distribution of zn are present at values of zn divisible by six. These "poles" have an analytical basis, and appear to decline exponentially.