### Singularity as esoteric side of the natural numbers structure

#### Abstract

The article concerns the structure of natural numbers as a discrete dynamical system based on the billiard graph and the research of the introduced singularity concept and the set of singular circles. It allows to consider the structure of natural numbers from wider positions of geometrical and topological constructions and to establish a number of interesting properties. The properties of the sets of natural numbers pairs in singular circles are proved and the sets consisting of the Goldbach primes pairs are distinguished from them. The general geometry of the singular circles embeddings into each other is presented. The constructive methods for arbitrarily large singular numbers generation are offered. It was shown that the number of singular natural numbers is infinite and singular vertices play an important role in inducing cluster-type automorphisms in the structure of natural numbers. The interval interdependence of the arrangement of prime twins and composite twins was investigated, which became the basis for the hypothesis of equal power of these sets.

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