The gaps between the gaps: some patterns in the prime number sequence

Abstract

It has long been known that the gaps between consecutive prime numbers cluster on multiples of 6. Recently it was shown that the frequency of the gaps between the gaps is lower for multiples of 6 than for other values (P. Kumar et al., Information entropy and correlation in prime numbers, arXiv: cond-mat/0303110). This paper investigates “higher moments” of the prime number series and shows that they exhibit certain peculiarities. In order to remove doubts as to whether these peculiarities are related to the known clustering of the gaps on multiples of 6, the results are compared to a benchmark series of “simulated gaps”.