Distribution of prime and composite numbers and their algorhythmic appendices

The article focuses on methods defining and distributing the composite numbers, prime numbers, twins of prime numbers and composite numbers of twins that do not have divisors $2$ and $3$ in $N$, based on the set of numbers of type $\Theta = \{6k \pm 1 / k\in N \}$ where $N$ is the set of all natural numbers, which is a semigroup with respect to multiplication. The calculation the exact quantity of primes in a given interval is given. A method for obtaining prime numbers $p\geqslant 5$ by their ordinal numbers in a set of primes $p \geqslant 5$ is proposed, as well as a new algorithm for finding and distributing prime numbers on the basis of the closeness of the set $\Theta$. The article shows that any composite number $n \in \Theta$ is representable as products $(6x \pm 1) (6y \pm 1)$, where $x$, $y\in N$ are the natural solutions of one of the four Diophantine equations $P(x, y, \lambda) = 0 : 6 \cdot xy \pm x \pm y - \lambda = 0$. It has been proved that if there is a parameter $\lambda$ of twins of prime numbers, then none of the Diophantine $P(x, y, \lambda) = 0$ equations has any solutions. A new universal, deterministic, polynomial and independent verification test is provided for the simplicity of the numbers of a species $6 \cdot k \pm 1$. Algorithms of distributions of parameters of twins of prime numbers and parameters composite numbers of twins are given, they are not divisible by $2$ and $3$, and variants of proofs for their infinite number are given.