Structure of Prime Numbers' Distribution

1.    Groups of Prime Numbers' Differences

    This paper reports some results of experimental investigation on computer prime numbers in the range until 1 billion.

    It has been detected that gaps between consecutive gaps (second differences of primes), named variations, constitute regular and repeating irregular groups. Maximal size of such a group of every type of them depends on 4 special characteristics of each type. All numbers constituting maximal sizes of such groups exist in 2 geometrical figures - hexagram and pentagram.

    Besides, as a range of primes increased, the ratio of the number of repeating groups of 2 consecutive variations to the number of primes in the range aspires to 100%.


Chart of gaps in the range of primes until 10,000


Lists of variations' groups:

    Regular (symmetrical) groups in the range of primes until 1,000,000

        Symmetriads S2, S3, S4, S5, S6, S8, and S10

        Curls            C3, C4, C5, C6, C7, C8, C9, and C10

        Stairs            S2, S3, S4, S5, and S6

        Zigzags         Z3, Z4, and Z5

        Flats             F2 and F3

    Repeating irregular (asymmetrical) groups in the range of primes until 100,000

       Any               Q2, Q3, Q4, Q5, Q6, Q7, Q8, Q9, Q10, Q11, and Q12

       Independent  Qi2, Qi3, Qi4, Qi5, Qi6, Qi7, Qi8, Qi9, Qi10, and Q12


2.     Classes of Prime Numbers and Their Differences
    In this paper, the author discusses primes' and their differences' classes on the base of 6. These classes are of importance in distribution values of differences of primes, as the first ones, gaps, as the second ones, variations.
    The main reason of six period oscillations in distributions both gaps and variations and the peculiarity of each of them is the distribution of primes themselves in groups of the same class. Besides, in such an oscillation in exponent distribution of variations values, the numbers determining the class of each a variation on the same base play an essential role.
    An analytical research of a problem of independent appearing of a next prime of the same class as of a current one or of the opposite class proves it possible in the in infinite general set of prime numbers only.