Kochkarev B.S

ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ

Algorithm for generating a sequence of all prime numbers starting from 3

Kochkarev B.S., Former associate Professor Kazan state University

Abstract. In this article the author present the generation of odd primes starting with the smallest number 3. The idea of dividing natural numbers into residue classes modulo 2 is used. All odd primes and composite numbers are in the classes of residue 1 . Then the author select the primes from class . 1

Keywords: Prime number, composite number, class of residue.

1. Introduction

The riddle of Prime numbers has occupied more than one generation of mathematicians since Euclid. Euclid, in particular, was the first to prove the infinity of a set of Prime numbers. There is only one even Prime number 2, since all even numbers are divisible by 2. All other primes are odd. Odd numbers end in 1, 3, 5, 7 and 9. There is only one Prime number ending in 5 since all numbers ending in 5 are divisible by 5. Using the descent axiom [1] we generalized Euclid's result on the infinity of the set of primes and proved the infinity of the set of primes ending in 1, 3, 7 and 9 and constructed algorithms for obtening such primes [2] using splitting the set of natural numbers into residue classes modulo 5. Further, splitting the set of natural numbers into residue classes modulo four allowed us to construct algorithms for obtening Prime numbers of the form 4k+1 and 4k-1, opened by Fermat [5, 73]. It was also noticed that all odd primes are divided into numbers of form 4k+1 and 4k-1, which have certain properties namely primes of the first group are always representable as the sum of two squares whereas primes of the second group are never representable as the sum of two squares. We proved these properties [4] and splitting the set of natural numbers into residue classe modulo four allowed us to construct algorithms for obtening such primes [3].

2. Building an algorithm for obtening primes numbers odd

If in these previous works the author built algorithms generating of certain classes of primes, then in this paper we build an algorithm generating of all Prime numbers from the smallest number 3 and so on to any Prime numbe4r, using the ideas of previous works. In this case of course, we assume the use of a well-known criterion of Euclid for a Prime number n: a number n is Prime number if and only if it is not divisible by all Prime numbers not exceeding n .

We divide the set of natural numbers into residue classes modulo2. According to [6] we get two classes of residue 1,0 , where class 0 includes all natural numbers that are even, and class 1 includes all odd numbers.

Obviously, if a natural number n belong to class 1 , then n+2 remains in the same class 1 . It is also obvious that class 1 contain all odd Prime and composite numbers. We only need to choose Prime numbers. If at some step of the algorithm we have a Prime number n, then we add to n so many time 2 until we get a Prime number, which we take as the next Prime4 number. The first step of the algorithm is to choose from 1 the smallest Prime number 3. Adding the number 2 to 3, we get the next Prime number 5, then Prime number 7, and after 7 we get the composite number 9 so you have to add 2 again to get a Prime number 11 etc.

Conclusion

Thus, the algorithm for obtening a sequencial series of primes starting with the smallest Prime number 3 is defined.

References:

1. Kochkarev B.S. To the Fermat descent method (K metodu spuska Ferma in Russian) Problems of modern science and education, 2015 №11 (41), pp.7-10 2. Kochkarev B. S. Regularities of Prime numbers and twin primes. (Zakonomernosti prostych chisel i prostych chisel bliznetsov in Russian) International Journal Chronos. 3 November 2018, pp.52-53 3. Kochkarev B. S. Algorithm of generation of Prime numbers of form 4k+1 and 4k-1. International Journal Shkola nauki, Issue 1 (26) January, 2020, p.2. 4. Kochkarev B.S. Problem of twin primes and other binary problems. (Problema bliznetsov i drugie binarnye problemy In Russian) Problems of modern science and education, 2015, №11 (41), PP.10-12 5. Singh S. Fermat's Last Theorem (Velikaya teorema Ferma.in Russian) MTSHMO, 2000 p. 228 6.Buchshtab A.A. Theory of Number (Teoriya chisel izd. Prosvetchenie in Russian) 1966 p.384.