The composition of billiard graphs in the representation of the natural numbers structure as a discrete dynamical system and singular mega-intervals

Abstract

The article is a continuation of the natural numbers structure research from the position of representation in the form of six infinite arithmetic progressions and addition and multiplication semi-group actions on this set. Such representation leads to the dynamic discrete system on the basis of the billiard graph (BG) creation. The regular and singular vertices, and corresponding natural numbers are defined on the graph. The singular vertices induce the singular circles- cluster type subsets of natural numbers with a special property of pair additivity (prime-prime and prime-composite pairs). Further consideration leads to the introduction of the composite twins (CT) by analogy with the twin primes (PT), and the hypothesis of the equivalence of these subsets. Methods and examples of constructing arbitrarily large composite twins are given. The BG composition represents the entire phase space of the shortest paths between vertices. The union of BG compositions into a single, geometric-topological form (infinite triangular prism) is proposed. BG features close to the quantum-mechanical are shown.