2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/116696In the course of studies of the measure of chaos for the distribution of the prime numbers among the positive integers N arched structures have been found. It is given a brief description of the fine structure of the positive integers sequence, including the distribution of the prime numbers. A new non-Eratostenian sieve was built to demonstrate the multi-periodicity in the structure of the positive integers sequence. This sieve presents an infinite triangular matrix, which is named by us as R-matrix (Russian matrix). The simple deterministic equations for prime numbers and the main term of asymptotic were derived with the use of this matrix.10 pages, 4 figures, 2 tablesGeneral Mathematicstext