Abstract
The article considers the methods of defining and finding the distribution of composite numbers CN, prime numbers PN, twins of prime numbers Tw and twins of composite numbers TwCN that do not have divisors 2 and 3 in the set of natural numbers - ℕ based on a set of numbers like Θ = {6∙κ ± 1, κ ∈ ℕ}, which is a semigroup in relation to multiplication. There has been proposed a method of obtaining primes by using their ordinal numbers in the set of primes and vice versa, as well as a new algorithm for searching and distributing primes based on 
 a closedness of the elements of the set Θ. It has been shown that a composite number can be presented in the form of products (6x ± 1) (6y ± 1), where x, y ℕ - are positive integer solutions of one of the 4 Diophantine equations: . It has been proved that if there is a parameter λ of prime twins, then none of Diophantine equations P (x, y, λ) = 0 has positive integer solutions. There has been found the new distribution law of prime numbers π(x) in the segment [1 ÷ N]. Any even number is comparable to one of the numbers i.e. . According to the above remainders m, even numbers are divided into 3 types, each type having its own way of representing sums of 2 elements of the set Θ. For any even number in a segment [1 ÷ ν], where ν = (ζ−m) / 6, , there is a parameter of an even number; it is proved that there is always a pair of numbers that are elements of the united sets of parameters of prime twins and parameters of transition numbers , i.e. numbers of the form with the same λ, if the form is a prime number, then the form is a composite number, and vice versa.
Highlights
Gauss was the first who observed the regularity of the arrangement of primes showing that the probability of the appearance of primes in the int (1 ÷ x) is equal to x
The article considers the methods of defining and finding the distribution of composite numbers CN, prime numbers PN, twins of prime numbers Tw and twins of composite numbers TwCN that do not have divisors 2 and 3 in the set of natural numbers - N based on a set of numbers like Θ = {6·κ ± 1, κ ∈ N}, which is a semigroup in relation to multiplication
There has been proposed a method of obtaining primes p ≥ 5 by using their ordinal numbers in the set of primes p ≥ 5 and vice versa, as well as a new algorithm for searching and distributing primes based on a closedness of the elements of the set Θ
Summary
Gauss was the first who observed the regularity of the arrangement of primes showing that the probability of the appearance of primes in the int (1 ÷ x) is equal to x. By values of the parameters λ = 6xy − x − y and λ = 6xy + x + y of a composite number of the form 6λ +1∈Θ from a file Rπ by direct access to the records id = λ in the field F2 the sign “+” changes into the sign “–” and the remaining records at the end of the algorithm in the adjacent field F1 = “+” indicate the presence of primes of type:. The set ΠTwCN is parameters λ of twins of composite numbers, which lie on non – empty intersections of the values of two functions (5), one of which belongs to FN + and the other to FN −. The values of the fields F1 = “–” and F2 = “–” in Table 2 correspond to the parameters λ, the set of twins of composite numbers.
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