Kochkarev B.S

ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ

The descent axiom and polynomial and non polynomial sets

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

Abstract. An overview of some of the author's main results obtained using the previously introduced descent axiom for binary mathematical statements is given. Next, we consider a countable class of undirected graph with an odd number of vertices. The problem of finding all simple paths in the graph under consideration that pass through all the vertices of the graph is formulated and solved. We prove that the number of such paths is nonpolynomial for a sufficiently large number of graph vertices.We show how to transform such graph into Hamiltonian graph with the addition of single edge, and prove that the probability of recognizing the Hamiltonian nature of such a graph tends to zero as the number of vertices of the graph increases.

Keywords: binary problem, axiom of descent, bipartite graph, Hamiltonian graph.

1. Introduction

Analisis of many open problems in number theory helped us to identify a class of binary problems and formulate for them the descent axiom, which is an algebraic interpretation of Fermat's descent method, with which He proved his hypothesis regarding diophantine equations x n + y n = z n ,n ³ 3, xyz ¹ 0 for the special case n=4. In this section, we present some of our main results obtained using the descent axiom in [1-12].

First of all, we recall the definition of a binary mathematical statement that depends on the natural parameter n and formulate the descent axiom for such statements.

Definition 1. An mathematical statement A n that depends on a natural parameter n is called binary if for any value n = a A a it has one of two values: true or false.

Axiom of descent: Let A n be a binary mathematical statement such that 1) there is an algorithm that for any value of n gives the answer to the question: "statement A n is true or false", 2) for the values of the parameter n 1 ,n 2 ,...,n k A n 1 , A n 2 ,..., A n k are true, and for any n > n k A n is false. Then the statement A n is true for an infinite set of values of n.

Pythagoras also introduced the concept of a perfect natural number n.

Definition 2. (Pythagoras) A natural number n is called perfect if, å d i = n , where d i ¹ n are the divisor d i !n of n.

Euclid discovered [14] that perfect numbers are always multiples of two numbers, one of which is equal to the power of 2, and the other is one less than the next power of 2, i.e. the perfect number is represented as 2 k (2 k 1+ -1) .

Using the descent axiom in [3], we prove that Euclid's formula exhausts all the set of perfect numbers if 2 k 1+ -1 it is a Prime number. It is also proved by the descent axiom [3, 9] that all odd numbers are not perfect. The Prime numbers of the form 2 n -1 were first noticed by Mersenne and therefore such numbers bear the name of Mersenne. In number theory, the question of finitely or infinitely [13] such numbers remained open. In [3], using the descent axiom, the infinity of the Mersenne numbers was proved. The use of high-speed calculating machines made it possible to find large Mersenne primes. The number 2 11213 -1 has 3376 digits and is generally the largest known Prime number [13, 35].

Finally, in [3] it was proved with the help of the axiom of descent that slightly redundant numbers do not exist, i.e. the question that intrigued the Pythagorean brotherhood was closed [14].

The next problem that remained open for a long time was the problem of twins, i.e. finitely or infinitely many pairs of Prime numbers with a difference between them equal to 2. In [8], it was also proved using the descent axiom that there are infinitely many twins from the residue classes 1( ,3),(7,9),(9,1) modulo 5 and their such twins there are infinitely many of each type. In [8], Euclid's theorem on the infinity of a set of Prime numbers was generalized. Namely, using the descent axiom, it was proved that there are infinitely many Prime numbers ending in 1, 3, 7, and 9.

Another open problem in number theory was the Goldbach Euler problem about the representability of any even natural number n ³ 4 as the sum of two primes. In [3] this problem was solved using the descent axiom, and in [12] we proved its generalization. Namely, using the descent axiom, it was proved that any even number n > 6 can be represented as the sum of two primes p,p', one of which is less than n , and the other is greater than 2 n . This property of even numbers makes it possible to 2 build efficient cryptographic systems, as shown in [12].

It should be noted that the complete solution of the Fermat problem [1],[4] allowed in [5] to solve the Tenth of D. Hilbert problem. The solution to this problem proposed by Matiyasevich in the 70s is not correct, since it uses the Turing thesis, but the Turing thesis was refuted by us in [2]

2. A countable class of bipartite graph with an odd number of vertices

Let G be a graph with V = 1 V È 2 V in which no pair of vertices from V i ,i =1,2 is connected by an edge, and each vertex from 1 V is connected to all vertices from 2 V . Then let 1 V = ê ë ê 2 n ú û ú , 2 V = ê ê é 2 n ú ú ù , where n is an odd number. Thus, we have defined a countable class of graphs with an odd number of vertices. For n=3 we have a bippattite graph with 1 V =1, 2 V = 2 , for n=5 a graph with 1 V = 2, 2 V = 3 etc.

We formulate a problem for such graphs: find the number of all simple paths that pass through all the vertices of the graph.Since the vertices of the graph G in 1 V and 2 V are not connected, those paths start in 2 V and end in 2 V . Let's number the vertexes in 2 V and in 1 V : 2 V = ,1{ ,...,2 },k 1 V ={k + ,...,1 2k - }1 . Let ( 1 i ,i 2 ,...,i k )

References:

some permutation of numbers 1, 2, ...,k from 2 V and ( 1 j , j 2 ,..., j k 1- ) some permutation numbers k+1,...,2k-1 from 1 V Obviously i 1 , j 1 ,i 2 , j 2 ,..., j k 1- ,i k is a simple path in graph G, that contain all the vertices of the graph. Hence the number of simple paths in graph G containing all vertices of graph G is ê ê é 2 n ! ú ú ù ê ë ê 2 n ! ú û ú (1)

Since the expression (1) is not a polynomial of the number of vertices of graph G, the set of all simple paths of graph G containing all vertices is a non -polynomial set [2].

Conclusion

This article shows that a thorough analisis of works devoted to the study of open problems sometimes, and very rarely, reveal gaps. It is especially important to find gaps in the foundation of mathematics, since such gaps hinder the development of science.

1. Kochkarev B.S. On a class of algebraic equations that do not have rational solution (Ob odnom klasse algebraicheskikh uravnenii, ne imeyutchikh rationalnykh reshenii ( in Russian) Problems of modern science and education, №4 (22), 2014, Pp.8-10 2. Kochkarev B.S. About one algorithm that does not agree with the thesis of Turing, Church, and Markov. (Ob odnom algorithme, ne soglasuyutchemsya s tezisami Turinga, Churcha i Markova in Russian) of modern science and education, 2014, №3 (21) 3. 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 4. Kochkarev B.S. About one Binary Problem in a class of algebraic equations and Her Communication with the Great Hypothesis of Fermat, IJcMS, October 2016, vol. 2, Issue 10, PP.457-459 5. Kochkarev Bagram About Tenth Problem of D.Hilbert, AJER, 2017, Volume-6, Issue-12, PP.241-242 6. Kochkarev B.S. On One Class of Undirected Graph, Control Science and Engineering, 2017; 1(1):28-30 doi: 10.11648/jcse 0101.14 7. Kochkarev B.S. Infinite Sequences of Primes of Form 4n-1 and 4n+1, ijhssi.org// Vol. 6 Issue 2// February. 2017// PP. 04-05 8. Kochkarev B.S. Regularities of Prime numbers and twin primes. (Zakonomernosti prostykh chisel i prostykh chisel bliznetsov in Russian) International Journal Chronos, 3 November 2018, PP.52-53 9. Kochkarev B.S. Axiom of Descent and Binary Mathematical Problems AJER e-ISSN: 2320-0847 p-Issn:2320- 0936, 2018 vol.-7, Issue-2, PP.117-118 10. Kochkarev B.S. Problem of Recognition of Hamiltonian Graph, IJWCMC. vol.4, №2, 2016, PP. 52-55 11. Kochkarev B.S. Algorithm of Search of Large Prime Numbers, IJ of DM, vol.1, №1, 2017 12. Kochkarev B.S. A Binary Problem of Goldbach Euler and its Generalization, International Journal of Discrete Mathematics, 2018, 3(20: 32-35 13. Buchstab A.A. Theory of Numbers, izd. Prosvetchenie Moscau 1966 P.383 14. Singh S Fermat's Last Theorem Fourth Estate London 1997 P. 288