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This work is devoted to the theory of prime numbers. Firstly it introduced the concept of matrix primes, which can help to generate a sequence of prime numbers. Then it proposed a number of theorems, which together with theorem of Dirichlet, Siegel and Euler allow to prove the infinity of twin primes.

A problem of twin prime numbers infinity formulated at the 5^{th} International Mathematical Congress is one of the main problems of the theory of prime numbers that has not been solved for 2000 years. It has been known that twin prime numbers are pairs of prime numbers which differ from each other by 2. For example, numbers 11 and 13, and numbers 17 and 19 are twin primes, but next adjacent prime numbers 37 and 41 are not twin primes. This problem is also known as the second Landau’s problem.

In the year 2013 American mathematician Zhang Yitang from the University of New Hampshire has proved that there are an infinite number of pairs of prime numbers, separated by a fixed distance is greater than 2 but less than 70 million. That is, the number of pairs of twin primes

A goal of this paper is to prove infinity of twin primes.

To solve this problem, we have proposed a new method, which allows to empirically estimate infinity of twin primes [

First, for convenience we introduce the following notation. As is known, a sequential

multiplication of all integers up to a certain number n is called as factorial:

Hereafter a sequential multiplication of prime numbers will be occurred frequently, therefore for such cases we use the following notation:

Here

In this paper we try to prove the infinity of number of twin primes. The proof will be on the basis of properties matrix of prime numbers.

The development of these matrices is implemented as follows.

Let we represent a set of natural numbers in a form of

Here, a maximum number of rows of matrix

Here and further it is supposed that we don’t know any prime number. Prime numbers will be generated in the course of creating matrices

First, we show how matrix

The numbers located in the second row of new matrix, also form an arithmetic progression. The first term and common difference of this progression are 3 and 2 respectively, i.e.

As previously defined, number 2 is a prime number. Therefore, all numbers divisible by 2 are composite numbers. In view of this, all numbers, except 2, which are located in the first row of the considered matrix (

This implies that number 1 is not a prime number, otherwise all numbers divisible by 1 would be composite numbers. Number 1 is also not a composite number, since it is not divided by other numbers. That is why number 1 is located separately in the upper left corner in this and other matrices.

It is seen from matrix

Thus, we transform matrix

For transforming a matrix from one into another type a simple method is used. Implementation of this method lies in a simple transposition of numbers of certain rows and columns of the original matrix into corresponding rows and columns of a new matrix. For example, for forming a first column of matrix

Note, that in the new matrix (

In the new matrix all numbers located in each row, as in the case of matrix

The common difference of this arithmetic progression is equal to

As you can see, all the numbers of this row are divisible by 3. Therefore, they (except number 3) are composite numbers. In a view of this they are dark painted in matrix

It should be pointed out that here and in all next figures, letter a denotes those matrices (e.g.

It should be noted that during the process of the second matrix transformation, all numbers divisible by 3, are finally determined and dark painted accordingly.

It is seen from matrix

For example, for forming the first column of matrix ^{st} - 6^{th}) row of new matrix^{th} - 12^{th}) row of a new matrix. After this, the numbers (14 ÷ 19) are transposed into (13^{th} - 18^{th}) row of the new matrix. Then the numbers (20 ÷ 25) are transposed into (19^{th} - 24^{th}) row of the new matrix and finally, the numbers (26 ÷ 31) are transposed into (25^{th} - 30^{th}) row of the new matrix. This completes formation of the first column of matrix

For forming a second column of matrix

A maximum number of rows in third matrix should be equal to a special factorial of the third prime number

For matrix

From this expression, we obtain that all the numbers located in the fourth row are divisible by 5, and those numbers that located in the 24^{th} row are also divisible by 5. In this context, all of them are accordingly transposed into the series of composite numbers and repainted into dark color (except a prime number 5). Here, in the case of the third matrix (

Note that in all cases, there is no any strict regularity for location of painted and unpainted numbers within one column of any considered matrix. But a picture of mutual arrangement of these numbers within one column is repeated with perfect precision in the next columns (starting from the second column). This regularity of repeating pictures by columns is appeared when each previous matrix

In matrix

In this case, carrying out a number of similar operations, as in previous cases, we can finally identify a set of all composite numbers, which should be divisible by 7. Similarly, we can build other matrices.

Here we have presented a procedure that allows to perform mechanical transformation of prime numbers matrices from one type to another. In general case an algorithm of this transformation is as follows.

Let there be given matrix

And a maximum number of matrix

Then the algorithm of building next matrix

Again, in this case a maximum number of matrix

In a similar way matrix

Here, based on the Dirichlet’s theorem on prime numbers in arithmetic progressions, it follows that if the first term and difference of the progression are not coprime numbers, then this progression will not contain any prime number or will contain only one prime number. And this prime number is the first term of the progression.

It also follows from the Dirichlet’s theorem that if the first term and difference of the progression are coprime numbers, then this progression contains prime numbers and composite numbers as well.

Therefore, in our case, first we determine if the first term and difference of the progression that consists of the numbers located in considered row of given matrix are coprime numbers. If they are not coprime numbers, then we conclude that all numbers of this row are composite numbers and they are dark painted for illustration purposes.

If the first term and difference of the considered progression are coprime numbers, then as mentioned above, this row contains both prime and composite numbers. Therefore, the numbers of these rows remain unpainted. Note, that dark painted are only those rows that contain only composite numbers.

Now, using matrices

First, we set a number of definitions:

Definition 1. If in a certain row of a matrix there are only composite numbers, then the row is dark painted for illustration purposes and for convenience we call it as a painted row.

Definition 2. If in a certain row of a matrix there are both prime and composite numbers, the row is not painted and for convenience we call such rows as not painted row.

Definition 3. If the first number of a row is not painted but the rest numbers are painted, then this not painted number is a prime number and the rest numbers are composite.

Definition 4. If a difference between index numbers of two neighbor and not painted rows is equal to 2, then such rows we call a pair of twin rows or twin rows. For the numbers located in different rows but in one column of twin rows pairs, an equation

Definition 5. If an index number of a certain painted row differs from an index number of the nearest not painted row by greater than 2, then the raw is called as a single row.

From these definitions it follows that twin prime numbers can be only in twin rows.

A goal of the paper is determine a total number of prime numbers. Therefore, hereafter we will put main emphasis on pairs of twin rows.

Theorem 1. A number of twin rows pairs in matrix

As is known, all twin prime numbers can be located in paired twin rows only. Moreover, if at some point, for example when considering matrix

We will analyze whether such case is possible and prove Theorem 1 conjointly.

Proof of Theorem 1.

Let suppose that some matrix has only one single pair of twin rows (for example, as in the case of

For example, in matrix

Here signs “−” and “+” correspond to upper and lower row of the pair of twin rows respectively.

But this only one pair of rows generates 5

A set of numbers located in each row of 5 new pairs of rows of matrix

where

From expression (3) we obtain that if at some value of

then all numbers of this row are divided by

All the numbers of each row of 3 newly formed pairs of twin rows, as stated above, form an arithmetic progression and in each of them the first term and difference of the arithmetic progression are coprimes, i.e.:

In virtue of this, it follows from Dirichlet theorem on prime numbers in arithmetic progressions, that in each row of these three pairs of twin rows there is an infinite number of prime numbers.

Now we consider a transformation of matrix

where

It can be seen from (5), that a set of numbers lying in each row of newly created 21 pairs of rows, separately forms an arithmetic progression with the difference of

We now consider which of these 21 pairs of rows of matrix

where

Let we consider divisibility of the first terms

In this case, in the same way as for case (4), we find that within an interval of

Besides, all first terms and difference of the arithmetic progression formed from the numbers lying in each row of the newly created 15 pairs of twin rows of matrix

where

In view of this, it follows from Dirichlet theorem for prime numbers in arithmetic progression, that in each row of 15 newly formed pairs of twin rows there is an infinite number of prime numbers.

If we consider further similar transformations of matrices into the next following matrices, for example, matrix

or

where-an index number of matrix

It follows from (7) that a number of twin rows pairs is monotonically increased while moving to the next matrices, i.e. with increasing of index number k of matrix

The theorem 1 is proved.

As is shown in (7), a number of twin row pairs will be progressively increasing during the process of moving to the next matrices. But a number of ordinary rows of each next matrix is increased as a special factorial

row pairs is progressively decreased along with the matrices since the ratio

is progressively reduced with rising of

Theorem 2. There are prime twin numbers in any pair of twin rows of any matrix

As shown above, all twin prime numbers can be located in twin rows only. But the question arises are there cases where in some pair of twin rows no any pair of two prime numbers is located in one column. Then, due to the asymmetry (i.e. due to the skewness of prime numbers location) there will be no any pair of prime twin numbers in this pair of twin rows. If such skewness happens in all twin rows pairs of this matrix, then this and all next matrices will no longer contain prime twin numbers. Therefore we can definitely say that a number of prime twins should be limited.

We will analyze this case now and prove the Theorem 2.

Proof of the Theorem 2.

Let consider matrix

A simple analysis shows that pairs of twin prime numbers conform to some simple rules. In particular, the last digit of any prime number (except 2 and 5) can not be an even number and it can not be equal to 5 as well. This means that last digits of the first and second number of any pair of twins should be respectively (1 and 3) and (7 and 9), and (9 and 1) as well. Therefore, a set of twin prime numbers should be divided into 3 subsets on these grounds. In fact, when we form matrix

Consequently, the fact that in each pair of twin rows of matrix

Now we consider next matrix

Here we suppose that an average distance by columns between cells, where adjacent prime numbers are located in one pair of twin rows of matrix

If this is so, then in each pair of twin rows of matrix

To verify this, first we enter a new parameter

where

On the other hand, if two adjacent prime numbers are located in two neighboring and adjoining cells lying in the same column (i.e. vertically as shown in

In this case these two prime numbers are twins.

As an example, we now consider a fragment of one pair of twin rows of any matrix. Let suppose that this fragment contains n prime numbers, as shown in

In this case, as shown above, a distance by columns between cells, where adjacent prime numbers with index numbers (1 and 3) and (4 and 6) are located, is equal 1, i.e.:

On the other hand a distance by columns between cells of adjacent prime numbers with index numbers (1 and 2), (5 and 6) and (7 and 8) is equal to zero, that is

Note, that the numbers with index numbers (1 and 2), (5 and 6), and (7 and 8) are twins (

It should be noted that the first prime number can be located in a cell that is lying not in the first column of the considered fragment (

It should be noted that a value of the considered parameter

If we also add the parameter

Here, a number of summands in the numerator of expression (9) will be greater by 1, i.e. a number of prime numbers in question is supposedly increased by 1 and becomes equal to

Now we consider a real case of matrix

Let

Then when applying (10) for the case of matrix

where

Note that here and further the first index of the parameter in question (in this case index k − 1) will correspond to index number of the considered matrix, and second index of this parameter (in this case index l) is an index number of the analyzed pair of twin rows.

Above we made an assumption that prime numbers in pairs of twin rows of each new matrix must be spaced more closely than in pairs of twin rows of previous matrix. To analyze and evaluate the assumption we will analyze a value of

From the papers of Siegel [

where

Then from this and expression (12) we obtain:

or

where

On the other side, it follows from (13) that

Here, we call attention to the following:

1) A certain selected row of any pair of twin rows in any matrix, excepting matrices

2) If two prime numbers, as shown above, are located in one column within one pair of twin rows, then they are twins. In this case a distance by columns between cells, where these prime twin numbers are located should be equal to zero.

3) Considered distance

As shown in a case of proving Theorem 1, it follows from (7), that while transforming matrix

In brief, a set of

Therefore, a number of prime numbers, located in one selected pair of twin rows of the considered fragment of new matrix

On other hand, while transforming a fragment of

Then from (14), (15) and (16) we obtain

Here we note the following. In the numerator (10) parameter

If using this expression, enter appropriate simple changes to (12) and (14), then the expression (17) eventually goes into the following form:

From the inequality (11) get that

Therefore, with this in mind, we obtain that the expression for the parameter

From this it follows that with increase of index number k of matrix

As can be seen, “a density”, i.e. closeness of prime numbers is increasing. Therefore due to infinity of prime numbers and identity of their distribution in any pair of twin rows of matrix

In particular, due to the fact that general skewness of prime numbers in pairs of twin rows of matrix

The Theorem 2 is proved.

Now we consider a problem posed in front of this paper.

Theorem 3. A number of twin primes is infinite.

Proof of the Theorem 3.

As is shown above, each matrix

It follows from the Theorem 1 and expression (7) that with rising growth of matrix

It also follows from the Theorem 2, that in any pair of twin rows there are prime twin numbers. This entire means that a number of prime twins is infinite.

This conclusion is also unavoidably followed from the expression (17).

If in (17) we express the parameter

then

Next, in the same manner, we transform parameter

where

It follows from (18), that with rising of index number

Here, the infinite series containing reciprocals of prime numbers diverges, as was shown by Euler [

Therefore,

This means that with a growth of matrix

The Theorem 3 is proved.

Study authors introduce the concept of matrix primes for researching of the properties of prime numbers. After, a number of theorems were proved in the work. Using these theorems and the theorems of Dirichlet, Siegel and Euler the proof of the infinity of twin primes was offered.

Baibekov, S.N. and Durmagambetov, A.A. (2016) Infinite Num- ber of Twin Primes. Advances in Pure Mathematics, 6, 954-971. http://dx.doi.org/10.4236/apm.2016.613073