Infinite Number of Twin Primes

Transcription

1 dvances in Pure Mathematics, 06, 6, htt://wwwscirorg/journal/am ISSN Online: ISSN Print: Infinite Number of Twin Primes S N Baibeov, Durmagambetov LN Gumilyov Eurasian National University, stana, Kazahstan How to cite this aer: Baibeov, SN and Durmagambetov, (06) Infinite Number of Twin Primes dvances in Pure Mathematics, 6, htt://dxdoiorg/06/am06607 Received: October 6, 06 cceted: December, 06 Published: December 5, 06 Coyright 06 by authors and Scientific Research Publishing Inc This wor is licensed under the Creative Commons ttribution International License (CC BY 0) htt://creativecommonsorg/licenses/by/0/ Oen ccess bstract This wor is devoted to the theory of rime numbers Firstly it introduced the concet of matrix rimes, which can hel to generate a sequence of rime numbers Then it roosed a number of theorems, which together with theorem of Dirichlet, Siegel and Euler allow to rove the infinity of twin rimes Keywords Prime Numbers, Twin Primes, Comosite Numbers, Natural Numbers, lgorithms, rithmetic Progression, Prime Numbers Matrix, Secial Factorial, Generation of Prime Numbers Introduction roblem of twin rime numbers infinity formulated at the 5 th International Mathematical Congress is one of the main roblems of the theory of rime numbers that has not been solved for 000 years It has been nown that twin rime numbers are airs of rime numbers which differ from each other by For examle, numbers and, and numbers 7 and 9 are twin rimes, but next adjacent rime numbers 7 and are not twin rimes This roblem is also nown as the second Landau s roblem In the year 0 merican mathematician Zhang Yitang from the University of New Hamshire has roved that there are an infinite number of airs of rime numbers, searated by a fixed distance is greater than but less than 70 million That is, the number of airs of twin rimes ( i, i+ = i + n) is infinite, where n is greater than but less than or equal to 70,000,000 James Maynard later imroved this result to 600 In 0, scientists led by Terence Tao (Polymath roject) this result was imroved u to 6 [] goal of this aer is to rove infinity of twin rimes To solve this roblem, we he roosed a new method, which allows to emirically estimate infinity of twin rimes [] In this aer we resent another simle roof, DOI: 06/am06607 December 5, 06

2 S N Baibeov, Durmagambetov which rovides more correct results for twin rimes infinity First, for convenience we introduce the following notation s is nown, a sequential n multilication of all integers u to a certain number n is called as factorial: i n! i = = Hereafter a sequential multilication of rime numbers will be occurred frequently, therefore for such cases we use the following notation: Here i 5 7 = =! n n i n i= is a rime number with index number i combination of symbols! means a sequential multilication of rime numbers from to only We shall call it as a secial factorial of a rime number n For examle,! is a secial factorial of rime number = 7 or! = 7! = 5 7 = 0 Matrices of Prime Numbers n In this aer we try to rove the infinity of number of twin rimes The roof will be on the basis of roerties matrix of rime numbers The develoment of these matrices is imlemented as follows Let we reresent a set of natural numbers in a form of matrices family with elements a(, I, j), where is a row index number, j is a column index number, and is an indexing number of matrix Here, a maximum number of rows of matrix must be equal to the secial factorial! ie i,max =! This means, that for every matrix with index number there is a secific set of rime number sequence:,,,, (note that the last rime number, which corresonds to this matrix is ) number of columns can be arbitrarily large u to infinity Here and further it is suosed that we don t now any rime number Prime numbers will be generated in the course of creating matrices First, we show how matrix is formed For this, we consider a series of natural numbers from to infinity (Figure, 0 ) In this series, number is followed by number So number is divided by and only Therefore, the first rime number is, ie = Then, the first matrix built with consideration of the first rime number, has only rows ( i,max = i!!,max = = = ) number of columns is infinite (Figure, -a) Numbers,, 6, are located in the first row of matrix These numbers form an arithmetic rogression The first term and common difference (ste) of this rogression are equal to, ie terms of the first row are resectively equal to: ( ) ( ) ( ) where,, ai,, j = a,, j = + j j=, The numbers located in the second row of new matrix, also form an arithmetic rogression The first term and common difference of this rogression are and resectively, ie ai,, j = a,, j = + j j=, ( ) ( ) ( ) where,, n 955

3 S N Baibeov, Durmagambetov Figure Matrices of rime numbers s reviously defined, number is a rime number Therefore, all numbers divisible by are comosite numbers In view of this, all numbers, excet, which are located in the first row of the considered matrix (Figure, -в) are dar ainted for illustrative uroses Thus, all comosite numbers which should be divisible by are defined by using matrix 956

4 S N Baibeov, Durmagambetov This imlies that number is not a rime number, otherwise all numbers divisible by would be comosite numbers Number is also not a comosite number, since it is not divided by other numbers That is why number is located searately in the uer left corner in this and other matrices lgorithm for Matrix Transformation from One to nother Tye It is seen from matrix (Figure, -в), that not ainted number next to number is number and it is not divisible by, and therefore it is the second rime number, ie = Thus, we transform matrix (Figure, -в) into next matrix (Figure, -а) maximum number of rows of this matrix must be equal to a secial factorial of the second rime number =, ie i!! 6,max = i,max = = = number of columns as in the first case can be arbitrary For transforming a matrix from one into another tye a simle method is used Imlementation of this method lies in a simle transosition of numbers of certain rows and columns of the original matrix into corresonding rows and columns of a new matrix For examle, for forming a first column of matrix firstly the numbers and located in the first column of the original matrix are transosed to the first and second rows of the new matrix, then the numbers and 5 of the first matrix are transosed to the third and fourth rows of the new matrix Then numbers 6 and 7 are also transosed to the fifth and sixth rows of the new matrix This comletes formation of the first column of matrix To form a second column of matrix, we similarly transose the numbers (8 and 9), (0 and ), and ( and ) in airs into a second column of matrix Next, we form other columns in a similar way Note, that in the new matrix (Figure, -а) all numbers located in the third and fifth rows are dar ainted as they, due to matrix, he been already defined as comosite numbers In the new matrix all numbers located in each row, as in the case of matrix, form an arithmetic rogression, which in general taes the following form: ai,, j = i+ +! j where j=,,, ; i=,,,! () ( ) ( ) ( ) The common difference of this arithmetic rogression is equal to! Exression () for matrix, articularly for its second row, aears as follows: a,, j = + 6 j, where j =,,, ; i =,,,! ( ) ( ) s you can see, all the numbers of this row are divisible by Therefore, they (excet number ) are comosite numbers In a view of this they are dar ainted in matrix (Figure, -в) In regard to the numbers located in the fifth row, they are also divided by However, these numbers, as mentioned above while considering matrix, he been already defined as comosite numbers set of numbers of fourth row (and sixth row as well) also form an arithmetic rogression But among them there are both rime and comosite numbers Therefore, numbers of these rows are not ainted 957

5 S N Baibeov, Durmagambetov yet Note, that rows containing only comosite numbers are dar ainted It should be ointed out that here and in all next figures, letter a denotes those matrices (eg -а, -a, -a), which are formed after transformation of the revious matrix Letter в denotes those matrices (eg, -в, -в, -в), which are obtained after rocessing has already transformed matrix It should be noted that during the rocess of the second matrix transformation, all numbers divisible by, are finally determined and dar ainted accordingly It is seen from matrix (Figure, -в) that next to numbers and unainted number is 5 Thus it is third rime number = 5 Therefore, the second matrix (Figure, -в) is transformed into third matrix (Figure, -а) For this we use a similar rocedure which was alied for transforming matrix into matrix For examle, for forming the first column of matrix at first the numbers ( - 7) located in the first column of the original matrix are transosed into ( st - 6 th ) row of new matrix Then, the numbers (8 - ) of the second column of matrix are transosed into (7 th - th ) row of a new matrix fter this, the numbers ( 9) are transosed into ( th - 8 th ) row of the new matrix Then the numbers (0 5) are transosed into (9 th - th ) row of the new matrix and finally, the numbers (6 ) are transosed into (5 th - 0 th ) row of the new matrix This comletes formation of the first column of matrix For forming a second column of matrix, the numbers ( - 7), (8 - ), ( - 9), (50-55) and (56-6), located in matrix, are gradually transosed into second column of matrix in a similar way fter that, other columns are similarly formed maximum number of rows in third matrix should be equal to a secial factorial of the third rime number =, ie i = i =! = 5! = 0,max,max For matrix exression () aears as follows: ( ) = ( + ) + ( ) = = a, i, j i 0 j, where j,,, ; i,,,! From this exression, we obtain that all the numbers located in the fourth row are divisible by 5, and those numbers that located in the th row are also divisible by 5 In this context, all of them are accordingly transosed into the series of comosite numbers and reainted into dar color (excet a rime number 5) Here, in the case of the third matrix (Figure, -в) it should be also noted that all numbers which must be divisible by 5, are finally defined and reainted into dar color (for examle, a row with an index number ) Note that in all cases, there is no any strict regularity for location of ainted and unainted numbers within one column of any considered matrix But a icture of mutual arrangement of these numbers within one column is reeated with erfect recision in the next columns (starting from the second column) This regularity of reeating ictures by columns is aeared when each revious matrix n with a number of rows equal to n! is transformed into next matrix n+ with a number of rows equal to 958

6 S N Baibeov, Durmagambetov +! n In matrix (Figure, -в) a number 7 which is next to the numbers,, and 5 is not ainted Therefore, it is a fourth rime number = 7 Now, nowing the fourth rime number 7, matrix can be similarly transformed into the next fourth matrix In this matrix a maximum number of rows must be equal to a secial factorial of rime number 7, ie 7! = 0 In this case, carrying out a number of similar oerations, as in revious cases, we can finally identify a set of all comosite numbers, which should be divisible by 7 Similarly, we can build other matrices Here we he resented a rocedure that allows to erform mechanical transformation of rime numbers matrices from one tye to another In general case an algorithm of this transformation is as follows Let there be given matrix with elements a(,i,j), where is an row index number, j is a column index number, and is an index number of matrix Then, as it follows from (), the elements of this matrix are determined by the following exression: ai,, j = i+ +! j, i=,,,!, j=,,, ( ) ( ) ( ) where nd a maximum number of matrix rows should be equal to a secial factorial!, ie i,max =! number of columns can be arbitrary large u to infinity On the other hand, a family of numbers located in any selected row of this matrix, creates an arithmetic rogression the first term of which is equal to ( i + ) and common difference of the rogression is equal to! Then the algorithm of building next matrix + lies in a simle calculation of values of new matrix elements using the following equation: a +, i, j = i+ +! j, where i =,,,!, j =,,, ( ) ( ) ( ) + + gain, in this case a maximum number of matrix + rows should be also equal to a secial factorial! +, ie i! +,max = + number of columns can be also arbitrary large u to infinity as in case of matrix In this case a family of numbers located in any selected row of this matrix, also creates an arithmetic rogression the first term of which is equal to (i+) while common difference of the rogression is equal to! + In a similar way matrix + is being transformed into matrix +, etc Here, based on the Dirichlet s theorem on rime numbers in arithmetic rogressions, it follows that if the first term and difference of the rogression are not corime numbers, then this rogression will not contain any rime number or will contain only one rime number nd this rime number is the first term of the rogression It also follows from the Dirichlet s theorem that if the first term and difference of the rogression are corime numbers, then this rogression contains rime numbers and comosite numbers as well Therefore, in our case, first we determine if the first term and difference of the rogression that consists of the numbers located in considered row of given matrix are corime numbers If they are not corime numbers, then we conclude that all numbers of 959

7 S N Baibeov, Durmagambetov this row are comosite numbers and they are dar ainted for illustration uroses If the first term and difference of the considered rogression are corime numbers, then as mentioned above, this row contains both rime and comosite numbers Therefore, the numbers of these rows remain unainted Note, that dar ainted are only those rows that contain only comosite numbers Now, using matrices, we will try to determine a number of twin rimes Infinite Number of Twin Primes First, we set a number of definitions: Definition If in a certain row of a matrix there are only comosite numbers, then the row is dar ainted for illustration uroses and for convenience we call it as a ainted row Definition If in a certain row of a matrix there are both rime and comosite numbers, the row is not ainted and for convenience we call such rows as not ainted row Definition If the first number of a row is not ainted but the rest numbers are ainted, then this not ainted number is a rime number and the rest numbers are comosite Definition If a difference between index numbers of two neighbor and not ainted rows is equal to, then such rows we call a air of twin rows or twin rows For the numbers located in different rows but in one column of twin rows airs, an equation ai (,, j) ai (,, j) = is always satisfied Definition 5 If an index number of a certain ainted row differs from an index number of the nearest not ainted row by greater than, then the raw is called as a single row From these definitions it follows that twin rime numbers can be only in twin rows goal of the aer is determine a total number of rime numbers Therefore, hereafter we will ut main emhasis on airs of twin rows Theorem number of twin rows airs in matrix is monotonically increased with a growth of index number of the matrix and also in each row of any twin rowsair there are an infinite number of rime numbers s is nown, all twin rime numbers can be located in aired twin rows only Moreover, if at some oint, for examle when considering matrix, all airs of twin rows are disaeared, then it obviously that they will not aear in next matrices In that case, it means that a number of twin rime numbers should be limited We will analyze whether such case is ossible and rove Theorem conjointly Proof of Theorem Let suose that some matrix has only one single air of twin rows (for examle, as in the case of Figure, -в) Here, there is reason to assume that in the course of further transformation of this matrix into the next matrices, airs of twin rows may disaear But, in fact the oosite is true When transforming the matrix into the next matrix a number of twin rows airs, as shown above, becomes larger 960

8 S N Baibeov, Durmagambetov For examle, in matrix there is only one single air of twin rows (Figure, -в) From the exression () it follows that terms of the arithmetic rogression, which are located in the rows of this single air of twin rows are defined by the exression: ( ) ( ) ( ) ( ) j =! +! j where j =,,, () Here signs and + corresond to uer and lower row of the air of twin rows resectively But this only one air of rows generates 5 ( ) 5 = new airs of rows during transformation of this matrix into matrix That is, the original unique air of twin rows is ungroued by 5 new airs of rows set of numbers located in each row of 5 new airs of rows of matrix also forms an arithmetic rogression with a constant! 0 = and is defined by the exression: ( m ) + ( j ) = ( m ) + ( j ) 6 0!!, () where j =,,,, ; m =,,, From exression () we obtain that if at some value of m=,,, the following equation is satisfied! m = int eger, then all numbers of this row are divided by exactly Therefore, the numbers are comosite In fact, it is nown that within interval of 0 0 < m < the Equation () with regard to the arameter m has unique solution [] [] and [5] For examle, equation () for the case of ( m ) is satisfied at m = and for the case of ( m+ )!! at m = That is, at m = and m = a air of rows in question is not a air of twin rows and corresonding row for which the equation () is satisfied, is dar reainted s a result only of 5 newly formed airs of rows are twin rows ll the numbers of each row of newly formed airs of twin rows, as stated above, form an arithmetic rogression and in each of them the first term and difference of the arithmetic rogression are corimes, ie: ( )! m,!, where m=, and 5, m, m In virtue of this, it follows from Dirichlet theorem on rime numbers in arithmetic rogressions, that in each row of these three airs of twin rows there is an infinite number of rime numbers Now we consider a transformation of matrix into matrix In this case, each air of twin rows of matrix generates 7 ( = 7) new airs of rows and totally new airs of rows are formed in new matrix Values of numbers located in the rows of these airs are defined by the exression: ( ) ( )! i +! m +! j, (5) where j =,,,, ; m =,,, ; i =,, It can be seen from (5), that a set of numbers lying in each row of newly created () 96

9 S N Baibeov, Durmagambetov airs of rows, searately forms an arithmetic rogression with the difference of! and the first terms defined as! i +! ( m ) We now consider which of these airs of rows of matrix are airs of twin rows For this urose we analyze divisibility of the first terms of the aforementioned arithmetic rogressions by = 7 In addition, for convenience and visualization, we consider a case where i =, but at the same time we mean the cases where i = and i = Then from (5) we find that values of numbers lying in 7 new rows of matrix generated by the last air of twin rows of matrix are determined by the exression:! m +! j, ( ) where j =,,,, ; m =,,, Let we consider divisibility of the first terms! m of the arithmetic rogression in question by, ie a satisfiability of the equation:! m = int eger In this case, in the same way as for case (), we find that within an interval of 0 < m< this equation with reference to arameter m (for the case of! m, and also for the case of! m + ) has an unique solution Therefore, in this case airs of the rows in question are no longer airs of twin rows If we additionally consider the cases when i = and i =, then we finally obtain that 6 airs of rows of newly formed airs of rows cease to be airs of twin rows, and corresonding rows, as shown above, are dar reainted s a result, a number of new airs of twin rows in matrix is equal to 5 Besides, all first terms and difference of the arithmetic rogression formed from the numbers lying in each row of the newly created 5 airs of twin rows of matrix, are corimes, ie: where ( ) ( i ( m ) )! +!,!, (6) i =,, ; m, and m 5, 6, 9,0,7,8 In view of this, it follows from Dirichlet theorem for rime numbers in arithmetic rogression, that in each row of 5 newly formed airs of twin rows there is an infinite number of rime numbers If we consider further similar transformations of matrices into the next following matrices, for examle, matrix into matrix, then every time we verify that any air of twin rows of the original matrix generates new airs of rows in new matrix In addition, airs of them will not be airs of twin rows and corresonding rows are moving into a ran of ainted and single rows From this we obtain that in any matrix a total number of rows ( i,max ) and total number of twin row airs ( m ) are resectively equal to: i = i and m = m (7-),max,max ( ) or,max ( ) i =! and m =!, (7-) 96

10 S N Baibeov, Durmagambetov where-an index number of matrix and/or rime number, and ( )! = ( )( ) ( ) == 5 9 ( ) It follows from (7) that a number of twin rows airs is monotonically increased while moving to the next matrices, ie with increasing of index number of matrix, On the other side, a set of numbers located in each row of these m airs of twin rows, forms an arithmetic rogression nd the first term and difference of each this rogression are corimes Therefore it follows from Dirichlet s theorem that in each row of any air of twin rows there is an infinite number of rime numbers The theorem is roved s is shown in (7), a number of twin row airs will be rogressively increasing during the rocess of moving to the next matrices But a number of ordinary rows of each next matrix is increased as a secial factorial! In a view of this, a density of twin ( ) row airs is rogressively decreased along with the matrices since the ratio!! is rogressively reduced with rising of Theorem There are rime twin numbers in any air of twin rows of any matrix s shown above, all twin rime numbers can be located in twin rows only But the question arises are there cases where in some air of twin rows no any air of two rime numbers is located in one column Then, due to the asymmetry (ie due to the sewness of rime numbers location) there will be no any air of rime twin numbers in this air of twin rows If such sewness haens in all twin rows airs of this matrix, then this and all next matrices will no longer contain rime twin numbers Therefore we can definitely say that a number of rime twins should be limited We will analyze this case now and rove the Theorem Proof of the Theorem Let consider matrix which contains only one unique air of twin rows (Figure, -в) On the other hand, as shown above, twin rime numbers can be located in airs of twin rows only This means that all existing twin rime numbers are located only in this unique air of twin rows simle analysis shows that airs of twin rime numbers conform to some simle rules In articular, the last digit of any rime number (excet 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 air of twins should be resectively ( and ) and (7 and 9), and (9 and ) as well Therefore, a set of twin rime numbers should be divided into subsets on these grounds In fact, when we form matrix, a single air of twin rows of matrix generates new three airs of twin rows in matrix Moreover, the last digits of each number located in the rows of individually selected air of twin rows are resectively equal ( and ) and (7 and 9) and (9 and ) This fact is readily illustrated by Figure, -в Consequently, the fact that in each air of twin rows of matrix there is a set twin 96

11 S N Baibeov, Durmagambetov rimes, which is sorted with regard to a value of the latest digit, is beyond question That is, in airs of twin rows of the matrix the sewness will not aear Now we consider next matrix The analysis we erformed shows that all members of each air of twin rows of this matrix are more ordered than in case of matrix For examle, last two digits of each number of the arithmetic rogression, located in any row of any twin row air of matrix form cyclically increasing sequence:,,,, 8, 9, 0,,,, Here we suose that an erage distance by columns between cells, where adjacent rime numbers are located in one air of twin rows of matrix, must be less than an erage distance of cells, where adjacent rimes in one air of twin rows of matrix are located If this is so, then in each air of twin rows of matrix there inevitably are twin rimes Since in this case, due to the tightness even if above-mentioned sewness of rime numbers aears in matrix, then it aears in a less degree then in case of matrix t least there will not be a general sewness of rime numbers in matrix To verify this, first we enter a new arameter d ii, + which is a distance by columns between the cells, where adjacent rime numbers with index numbers i and i + (Figure ) are located: d = M M (8) ii, + i+ i, where M i is an index number of the column in which a cell of i-th rime number is located Moreover, if two adjacent rime numbers are located in two mutually adjacent cells along one row (ie horizontally, as shown in Figure, a), then a distance between these cells is equal to : Figure Location of rime numbers in a air of twins rows 96

12 S N Baibeov, Durmagambetov d M M ii, + = i+ i = On the other hand, if two adjacent rime numbers are located in two neighboring and adjoining cells lying in the same column (ie vertically as shown in Figure, в), then a distance between these cells is equal to zero: d M M = = ii, + i i 0 In this case these two rime numbers are twins s an examle, we now consider a fragment of one air of twin rows of any matrix Let suose that this fragment contains n rime numbers, as shown in Figure (c) In this figure light circles denote cells, in which comosite numbers are located and dar circles mar cells with rime numbers Then, above mentioned erage distance by columns d between cells, where adjacent rime numbers are located, will be equal to: d = n d i= ii, + n In this case, as shown above, a distance by columns between cells, where adjacent rime numbers with index numbers ( and ) and ( and 6) are located, is equal, ie: d, = M M = ; d,6 = M6 M5 = On the other hand a distance by columns between cells of adjacent rime numbers with index numbers ( and ), (5 and 6) and (7 and 8) is equal to zero, that is d, = M M = 0; d5,6 = M6 M6 = 0; d7,8 = M9 M9 = 0 Therefore, if we consider a rime number with index number, then its nearest adjacent rime number on the right side is a rime number with index number, rather than a number with index number (Figure (c)) If we consider a rime number with index number, its nearest neighbor to the right side is a rime number with index number 5, rather than a number with index number 6 This is easily seen, if a difference between values of these numbers is calculated using formula () Similarly, it can be easily determined that the nearest neighbor of a rime number with index number 9 on the left side is a rime number with a index number 8, rather than a number with index number 7, although these two rime numbers (7 and 9) are located in the same row Note, that the numbers with index numbers ( and ), (5 and 6), and (7 and 8) are twins (Figure (c)) It should be noted that the first rime number can be located in a cell that is lying not in the first column of the considered fragment (Figure (c)) Similarly, we can say that a cell in which the last rime number with index number n is located, may be also situated not in the last column These cases are not taen into account in (9), arameters d and d are not resented in exression (9) d is a number of columns calculated from the beginning of the considered fragment to a cell containing first rime number and d is a number of columns from a cell with last rime number to the end of the considered fragment (Figure (c)) To tae this into account, we consider a sum: d = d + d (9) 965

13 S N Baibeov, Durmagambetov It should be noted that a value of the considered arameter d is comarable and most liely equal to the distance by columns from the cell of the last rime number with an index number n to the cell, where next rime number with index number n + will be located s it follows from Figure (c) this number, is located outside the analyzed fragment, but it will be the nearest adjacent rime number from the right side for n-th rime number From this it follows that: d d nn, + If we also add the arameter d into the sum in a numerator of the exression (9), then we exactly obtain a length of the considered fragment (Figure (с)): n d + d = M ii, + i= Here, a number of summands in the numerator of exression (9) will be greater by, ie a number of rime numbers in question is suosedly increased by and becomes equal to n + ccordingly, exression (9) taes the following form: d n i= ii, + ( n ) d + d M = = + n Now we consider a real case of matrix m, M Here m is a number of airs of twin rows, M is a number of columns in given fragment of matrix, which corresonds to a rime number It should be noted here that a value of M should be sufficiently large so it mae statistical sense For examle, it should be: M () (0) fragment with dimensions ( )! Let π is a total number of all rime numbers lying in all airs of twin rows of the considered fragment of matrix, π, l is a number of rime numbers contained in one selected air of twin rows with index number l In addition a total number of twin rows airs should be equal m, ie l taes values from to m, in short l = m Then when alying (0) for the case of matrix we obtain: M d, l = () π where d, l is an erage distance by columns between cells where adjacent rime numbers, lying in one air of twin rows with index number l, are located Note that here and further the first index of the arameter in question (in this case index ) will corresond to index number of the considered matrix, and second index of this arameter (in this case index l) is an index number of the analyzed air of twin rows bove we made an assumtion that rime numbers in airs of twin rows of each new matrix must be saced more closely than in airs of twin rows of revious matrix To analyze and evaluate the assumtion we will analyze a value of d, l by airs of twin, l 966

14 S N Baibeov, Durmagambetov rows of the considered fragment From the aers of Siegel [6] [7] [8] and other researchers [9] [0] [] [] it follows that if the constants (common differences) of different arithmetic rogressions are equal to each other and the first term and common difference of each arithmetic rogressions are co-rimes, then rime numbers are distributed similarly and identically in these rogressions On the other hand, as shown above, a sequence of numbers located in any row of any air of twin rows of matrix forms an arithmetic rogression with the same common difference equal to! In addition the first term and common difference of these arithmetic rogressions are co-rimes These rogressions differ from each other only by a value of the first term and they are identical in all other resects This means that within the considered fragment quantities of rime numbers in any air of twin rows are aroximately equal, ie: π π π π π, () π,,,, l, m where, is an erage quantity of rime numbers containing in a air of twin rows of the considered fragment of matrix Then from this and exression () we obtain: d d d d d,,,,, l, m or d M, = () π, ср d where, an erage distance by columns between cells where adjacent rime numbers, lying in a air of twin rows of the considered fragment of matrix, are located On the other side, it follows from () that π, m π j=, j π = = m m Here, we call attention to the following: ) certain selected row of any air of twin rows in any matrix, exceting matrices 0 and (Figure, 0 and Figure, ), could not contain rime twin numbers as shown above ) If two rime numbers, as shown above, are located in one column within one air of twin rows, then they are twins In this case a distance by columns between cells, where these rime twin numbers are located should be equal to zero ) Considered distance d, by columns between cells, where adjacent rime numbers are located, should not be confused with a difference, ie with a distance between values of adjacent rime numbers It is lain, that in one cell of considered fragment of any matrix only one number can be located In addition, difference of values of adjacent rime numbers, which are located in different, but adjacent to each other by row, cells of the fragment can tae a value of any even integer s shown in a case of roving Theorem, it follows from (7), that while transform- (5) 967

15 S N Baibeov, Durmagambetov ing matrix into matrix, each air of twin rows of matrix generates new airs of twin rows and additional single rows in new matrix s a result, each air of twin rows of matrix creates airs of new rows Totally, m airs of twin rows generate m = m ( ) new airs of twin rows and additionally m = m airs of single rows in new matrix In brief, a set of = m M d, rime numbers lying in all airs of twin π ср rows of the considered fragment of matrix new airs of rows of new fragment corresonding to new matrix of rime numbers located in all newly created fragment is defined by the exression: π m π = = m is redistributed by m = m ( ) ' + m m Then an amount m airs of twin rows of matrix m π ( ) Therefore, a number of rime numbers, located in one selected air of twin rows of the considered fragment of new matrix π,, is on the erage: π π = = m m ( ) On other hand, while transforming a fragment of matrix into a fragment of, a number of columns in new fragment, as shown above, is reduced by matrix times, ie M = M Then from (), (5) and (6) we obtain d,, an erage distance by columns between the cells, where adjacent rime numbers, lying in one air of twin rows of the considered fragment of matrix, are located M m M d, = = = d, (7) π π, Here we note the following In the numerator (0) arameter d is treated as a single term In fact this otion consists of two settings d and d So if these two settings in the numerator (0) will be counted simultaneously, then the number of summands in the numerator (0) will be equal to n + With this in mind, the exression (0) taes the following form: d n i ii, + + = + d d d M = = + n+ ( n ) If using this exression, enter aroriate simle changes to () and (), then the exression (7) eventually goes into the following form: d, M = = π, + + d M, ( ) (6) From the inequality () get that = 0 M 968

16 S N Baibeov, Durmagambetov Therefore, with this in mind, we obtain that the exression for the arameter d, will he the same aearance-what is in (7) In short, the joint consideration of arameters d and d gives the same result, which is obtained at considering only one arameter d From this it follows that with increase of index number of matrix an erage distance d, between the cells of adjacent rime numbers in any air of twin rows of matrix decreases continuously s can be seen, a density, ie closeness of rime numbers is increasing Therefore due to infinity of rime numbers and identity of their distribution in any air of twin rows of matrix, a robability of occurrence of twin numbers will be greater than in case of revious matrix In articular, due to the fact that general sewness of rime numbers in airs of twin rows of matrix does not exist, as shown above, then it will not aear in airs of twin rows of the next matrices, 5, 6,,, So, there are twin rime numbers in each air of twin rows of any matrix The Theorem is roved Now we consider a roblem osed in front of this aer Theorem number of twin rimes is infinite Proof of the Theorem s is shown above, each matrix corresonds to a certain rime number It is nown that a number of rime numbers is infinite Therefore, a number of matrix variations is also infinite On the other hand, twin rime numbers can be located in airs of twin rows only It follows from the Theorem and exression (7) that with rising growth of matrix index number, a number of airs of twin rows in this matrix is steadily increased ie lim m = lim! = ( ) It also follows from the Theorem, that in any air of twin rows there are rime twin numbers This entire means that a number of rime twins is infinite This conclusion is also unoidably followed from the exression (7) If in (7) we exress the arameter d, in terms of d,, which is a mean distance by columns between cells, where adjacent rime numbers are located in a air of matrix twin rows, then we obtain d, = d, then d d, =, Next, in the same manner, we transform arameter d, through use of d р, then d, through use of d,, etc While continuing the transformation u to d, we obtain that an erage distance by columns between cells of rime numbers in, 969

17 S N Baibeov, Durmagambetov any air of matrix twin rows can be generally determined by the exression: ( )( ) ( ) ( )! d = d = d!,,, (8) where and d, is a mean distance between cells of adjacent rime numbers lying in a single not ainted row of matrix (Figure, -в) It follows from (8), that with rising of index number of matrix an erage distance d, between cells of rime numbers in any air of twin rows of matrix is rogressively decreasing nd if we obtain that d, 0 In fact, hing done small transformation we obtain from (8) that ln d = ln d +,, m i= i i= m= m i Here, the infinite series containing recirocals of rime numbers diverges, as was shown by Euler [], ie: i= i = Therefore, lim d = 0 (9), This means that with a growth of matrix s index number an erage distance d between the cells, where adjacent rimes are located, tends to zero In addition, the infinity of twins is not only inevitable but also obvious because it follows from (7) that at a number of twin rows airs tends to infinity: lim m = On the other hand as it follows from the Theorem, in each of these airs of twin rows there are twin rime numbers The Theorem is roved 5 Conclusion Study authors introduce the concet of matrix rimes for researching of the roerties of rime numbers fter, a number of theorems were roved in the wor Using these theorems and the theorems of Dirichlet, Siegel and Euler the roof of the infinity of twin rimes was offered References [] Prime Number Wiiedia htts://ruwiiediaorg/wii/простое_число#cite_note-7 [] Baibeov, SN (06) bout Infinity of Prime Twins Pois, No, 0- [] Baibeov, SN (05) Develoment of New Method for Generating Prime Numbers, Eurasian Gumilev University Bulletin, No, - [] Baibeov, SN and ltynbe, S (05) Develoment of New Method for Generating Prime Numbers Natural Science, 7, 6- [5] Baibeov, SN (05) Generation and Test of Prime Numbers Eurasian Gumilev University Bulletin, No 6, Part,

18 S N Baibeov, Durmagambetov [6] Siegеl, СL (95) cta rithmetica, v, 8-86 [7] Numbers of rithmetic Progression universal_ru_enacademicru/копия [8] Karatsuba, (975) The Fundamentals of nalytical Number Theory M, Chater 9 [9] Linni, YuV (96) The Disersion Method in Binary dditive Problems L: LGU Publishing [0] Linni, Yu, Barban, M and Tshudaov, N (96) On Prime Numbers in an rithmetic Progression with a Prime-Power Difference cta rithmetica, 9, [] Gritseno, S and Dhevtsova, MV (0) On Distribution of rimes in rithmetic Progressions with a Difference of Secial Tye Scientific Bulletin, Mathematics and Physics Series, No (00), n, 7-7 [] German-Yevtusheno, MS (0) bout a Sum of Values of Divisor Function by Numbers Lying in rithmetic Progressions with a Difference of Secial Tye Scientific Bulletin, Mathematics and Physics Series, No (55), n, -9 [] Zagier, D (977) The First 50 Million Prime Numbers htt://ega-mathnarodru/liv/zagierhtm Submit or recommend next manuscrit to SCIRP and we will rovide best service for you: cceting re-submission inquiries through , Faceboo, LinedIn, Twitter, etc wide selection of journals (inclusive of 9 subjects, more than 00 journals) Providing -hour high-quality service User-friendly online submission system Fair and swift eer-review system Efficient tyesetting and roofreading rocedure Dislay of the result of downloads and visits, as well as the number of cited articles Maximum dissemination of your research wor Submit your manuscrit at: htt://aersubmissionscirorg/ Or contact am@scirorg 97