Distributions of the Integers

'The Distributions of Integers' has considerable novel material. It begins with the established idea that sets of pairs of primes, that are separated by composites and defined by gap size, have densities on the number line in proportion to the square of the density of primes. Span, a new parameter, in contrast to gap, is introduced and defined as the difference between two primes with other primes allowed in the interval of separation. Empirical evidence shows that spans have more exciting properties than gaps. The above results are expanded to include (i) 'triplets' and higher numbers of primes. (ii) a theory developed to show where gaps of each allowed magnitude should first appear in the number line (iii) the Goldbach primes and (iv) an examination of the composites in a similar manner to the primes, but knowing that composites are divided into non-interacting sets, each distinguished by the number of prime factors of its members... 44 equations, 26 tables, 31 figures, 16,443 words.