### On natural numbers structure on the basis of six arithmetical progressions

#### Abstract

In the article the fundamental direction of Dirichlet about the number of primes in arithmetical progressions on the basis of infinitary structures composition ("k-dimensional shortest paths in n-cube, global k-ary tree and the set of natural numbers") representation in the form of six infinite arithmetic progressions is developed. The theorem on two progressions from these six containing all prime numbers is proved. The geometric-topological construction and the two-dimensional numbering of natural numbers induced on it are considered on the same basis.Examples of natural numbers properties as consequences of the offered constructions are given.** **

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