In the history of mathematics, after Euclid and on the basis of Diophantine equations, detailed studies of the natural series of numbers and its subsets - which have not yet been mentioned in any mathematical literature - are presented for the first time in this book.

As, for example, numbers of twins of composite numbers, transition numbers (or numbers of chameleons), the exact distribution of composite and prime numbers on a given interval [1 - N), the definition of twin primes and their distribution algorithm, including their infinity.

An optimal and fast algorithm for identifying prime numbers is obtained, in comparison with the well-known algorithms of Eratosthenes, Suntaram and Artin.

Factorization and recognition of the simplicity of natural numbers, alternative solutions of some hypotheses - titans
in number theory, such as the binary Goldbach - Euler problem, the infinity of twin numbers, the Legendre problem,
E. Landau on the infinity of almost squares of primes, problems on the infinity of primes in numerical sequences:
M. Mersenne, P. Fermat, S. Germain and the exact law of distribution of primes, among others, will be presented and further discussed.