孪生素数：相关介绍和链接 《中国大百科全书》第二版 中文名称：孪生素数 外文名称：twin primes 正 文: 相差为2的一对素数 。例如{3,5}，{5,7}，{11,13}，及{17,19}等。猜想有无限多对这样的素数，这就是著名的孪生素数猜想，至今未被证明。最好的理论结果属于陈景润(1973)，他证明：存在无穷多个素数p，使得p＋2是素数或是两个素数的乘积。设不超过正数x的孪生素数个数为π2(x)。G.H.哈代和J.E.李特尔伍德(1923)还进一步猜测渐近公式是， 式中C2为一常数。大量数值计算均支持这一猜测。已经找到的最大孪生素数是有51,780位的100,314,512,544,015·2171,960±1(2006)。人们还知道，所有孪生素数的倒数组成的级数是收敛的。 一般地，设b是任意给定的大于1的正整数，相差为2b的一对素数称为广义孪生素数。例如，当素数p=3,7,13,19,37,43,67,79时，p＋4均为素数；当素数p=5,7,11,13,17,23,31,37时，p＋6均为素数；当素数p=3,5,11,23,29, 53,59,71时，p＋8均为素数；当素数p=3, 7,13,19,31,37,43,61时，p＋10均为素数；当素数p=5,7,11,17,19,29,31,41时，p＋12均为素数。对给定的b，猜想有无限多对这样的素数，这就是广义孪生素数猜想，至今尚未被证明。同样，若猜想成立，哈代和李特尔伍德 (1922)猜测也将有相应的渐近公式成立。大量的数值计算均支持这一猜测。 http://dbk2.chinabaike.org/indexengine/entry_browse.cbs?valueid=%C2%CF%C9%FA%CB%D8%CA%FDdataname=dbk2%40C%3A%5CProgram+Files%5Cdbk%5Cdbkdms%5Cdata%5Cbook2%5Cdbk2.tbfindexvalue=%B3%C2%BE%B0%C8%F3 Twins - Encyclopedia of Mathematics Two primes the difference between which is 2. http://www.encyclopediaofmath.org/index.php/Twins The twin prime conjecture: There are infinitely many primes p such that p + 2 is also prime. http://www.princeton.edu/~achaney/tmve/wiki100k/docs/Twin_prime_conjecture. html Twin Primes - Wolfram It is conjectured that there are an infinite number of twin primes (this is one form of the twin prime conjecture) http://mathworld.wolfram.com/TwinPrimes.html Twin prime - Wikipedia, the free encyclopedia Are there infinitely many twin primes? http://en.wikipedia.org/wiki/Twin_prime 孪生素数猜想 - 维基百科，自由的百科全书 存在无穷多个素数p，与p + 2都是素数。 http://zh.wikipedia.org/wiki/%E5%AD%AA%E7%94%9F%E7%B4%A0%E6%95%B0%E7%8C%9C%E6%83%B3 请您提供更多链接！ 以便大家更准确地理解张益唐老师的成果。
Matrix Formula of Non Prime Odd Number ---- Distribution of Twin Primes on the Axis. Ke-An Feng Institute of Physics, Chinese Academy of Science, Beijing 100080 Abstract: The distribution of the Twin Primes is an interesting problem. In this short paper, all of primes and Twin primes can be found in terms of a set of matrix of non prime odd number. Key words: primes; Twin primes, MR-11A Odd numbers can be separated two parts: whole prime except 2 and non prime odd number. Although there is no distributed regularity of the prime on the number axis, but the distributed regularity of the non prime odd number can be written as a set of matrix: 7+12m1,2; 11+12m3,4; 1+ 12m5,6; 5+12m7,8; 3+12(n+1); 9+12n. where n= 0,1,2,3,4,5. . . ; m1,2 indicate the matrix m1 and m2. The expression of the eight matrix m1,m2,. . .m8 are m1 4 9 14 19 24 29 34 15 26 37 48 59 70 81 32 49 66 83 100 177 134 55 78 101 124 147 170 193 84 113 142 171 200 229 258 m3 2 7 12 17 22 27 32 11 22 33 44 55 66 77 26 43 60 77 94 111 128 47 70 93 116 139 162 185 74 103 132 161 190 219 248 m5 2 7 12 17 22 27 32 10 21 32 43 54 65 76 24 41 58 75 92 109 126 44 67 90 113 136 159 182 70 99 128 157 186 215 244 m7 5 10 15 20 25 30 35 17 28 39 50 61 72 83 35 52 69 86 103 120 137 59 82 105 128 151 174 197 89 118 147 176 205 234 263 m2 7 14 21 28 35 42 49 20 33 46 59 72 85 98 39 58 77 96 115 134 153 64 89 114 139 164 189 214 95 126 157 189 219 250 281 m4 9 16 23 30 37 44 51 24 37 50 63 76 89 102 45 64 83 102 121 140 159 72 97 122 147 172 197 222 105 136 167 198 229 260 291 m6 4 11 18 25 32 39 46 14 27 40 53 66 79 92 30 49 68 87 106 125 144 52 77 102 127 152 177 202 80 111 142 173 204 235 266 m8 6 13 20 27 34 41 48 18 31 44 57 70 83 96 36 55 74 93 112 131 150 60 85 110 135 160 185 210 90 121 152 183 214 245 276 1 In terms of these matrixes, we can know the position of whole prime except 3. For example: 5+12m7,8. 5, 6, 10, 13, 15, 17, 18, 20. . . ..are the matrix elements in m7 or m8, then the non prime odd numbers (npon) are 5+12(5, 6, 10, 13, 15, 17, 18, 20. . . ) = 65, 77, 125, 161, 185, 209, 221, 245. . . ..But 0, 1, 2, 3, 4, 7, 8, 9, 11, 12, 14, 16, 19. . . are not the matrix elements in m7 and m8 , then the primes are 5+12(0, 1, 2, 3, 4, 7, 8, 9, 11, 12, 14, 16, 19. . . ) = 5, 17, 29, 41, 53, 89, 101, 113, 137, 149, 173, 197, 233 . . . .Definition: the prime holes in m7 and m8 are the common non matrix elements in m7 and m8. Obviously, any prime can correspond to the prime hole in one of the four pair of matrix The properties of the matrix m(i,j)( m7(0,0)= 5) are 1) For m1, m3, m5, m7,the interval of two nearest column in row i is 5+6i 2) For m2, m4, m6, m8,the interval of two nearest column in row i is 7+6i 3) For any matrix, the common properties in column 0 is ( i+2, 0) ?( i+1, 0) = ( i+1,0) ?( i,0) + 6, i ?0 From these matrixes, the existence of the twin primes can be explained. For example, If the integer A is not only the prime hole in matrix m7 and m8, but also is the prime hole in m1 and m2, then the twin primes are 5+12A and 7+12A. when A = 0, the twin primes are 5 and 7; when A=1, the twin primes are 17 and 19; when A= 2, the twin primes are 29 and 31; when A=3, the twin primes are 41 and 43; when A= 8, the twin primes are 101 and 103. In the same manner, when A is a prime hole in m3,4and A+1 is another prime hole in m5,6,then the twin primes are 11+12A and 1+12(A+1). When A=0, the twin primes are 11 and 13; when A=4, the twin primes are 59 and 61; when A=5, the twin primes are 71 and 73; when A=8, the twin primes are 107 and 109. Obviously, from these matrixes, all twin primes can be found. Appendix The the origin of the 8 matrices A) As we know, the prime is a part of odd number except 2, that is fall of oddng = fnpong + fprimeg B) a npon = a npon (or prime) ?a npon (or prime) a npon O = (2o+1)=(2n+1)(2p+1)= 2(2np+n+p)+1 where, integer n,p?. Example: 39 = (2 ?19) + 1 = (2 ?1 + 1)(2 ?6 + 1) = 2(2 ?6 ?1 + 6 + 1) + 1 a prime P=(2o+1)=2(2np+n+p)+1 where n=0, p0. o6=2np+n+p, if n,p? Example: 37=(2?8)+1, where 18 can not be written by 2np+n+p, where n,p? C) a npon=2o+1, where o=2np+n+p can be written by a square matrix 4 7 10 13 16 19 22 25 28 ??? 7 12 17 22 27 32 37 42 47 ??? 10 17 24 31 38 45 52 59 66 ??? 13 22 31 40 49 58 67 76 85 ??? 16 27 38 49 60 71 82 93 104 ??? 19 32 45 58 71 84 97 110 123 ??? 22 37 52 67 82 97 112 127 142 ??? 25 42 59 76 93 110 127 144 161 ??? 28 47 66 85 104 123 142 161 180 ??? 2 a triangle matrix can be discussed for p竛 owing to n is equivalent to p D) Properties of the triangle matrix 47 12 10 17 24 13 22 31 40 16 27 38 49 60 19 32 45 58 71 84 22 37 52 67 82 97 112 ??? 25 42 59 76 93 110 127 144 28 47 66 85 104 123 142 161 31 52 73 94 115 136 157 178 34 57 80 103 126 149 172 195 37 62 87 112 137 162 187 212 40 67 94 121 148 175 202 229 all of the element are concerned with npon. Example 4,7,10,12,13,16,17 are related to the npon 9,15,21,25,27,33,35. Non-element in the matrix 1,2,3,5,6,8, 9,11,14 are corresponded with the prime 3,5,7,11,13,17,19,23,29. Definition: the hole in first column of the triangle matrix. The sum of the matrix element and matrix hole in first column are all of positive integer. The even elements in first column: 4+3?m The odd elements in first column: 4+3?2m?) 1) odd and even expression of the hole in the first column up and down positions of even element are corresponded with the odd hole, 4+3?m??) up and down positions of odd element are corresponded with the even hole, 4+3(2m?)? Example: m=1. even element: 10, odd holes: 9 and 11. Odd element: 7, even holes: 6 and 8. When m!m+1, the number of even and odd hole is varied by 6. Definition: the hole of non-prime (or prime) in first column. If the element in the column ni ? equal to the hole in first column, then the hole is called non-prime hole(nph), otherwise, the hole is prime hole(ph). Obviously, the set of ph in first column is related to the set of all prime. Example: 17, 24 in first column are nph, 35, 49 are npon. But the ph 11, 14, 20 relate to the prime 23 29 41. 2) odd and even expression of element in the column ni ?. The even elements 2ni(ni+1)+(2ni+1)2n The odd elements 2ni(ni+1)+(2ni+1)?2n+1) 3) The relationship between nph in first column and element in column ni ?. The odd nph equal to odd element 4 + 3 ?2m ?(?) = 2ni(ni + 1) + (2ni + 1) ?(2n + 1) The even nph equal to even element 4 + 3(2m ?1) ?1 = 2ni(ni + 1) + (2ni + 1)2n for up odd np hole: 4+3?m?=2ni(ni+1)+ (2ni+1)?2n+1) (ni?)?ni+2)+(2ni+1)n+ni=3m?. The solutions are a) ni=2+3i, n=1+3j, m1 = 3i2+6ij+8i+5j+4 3 b) ni=3+3i, n=1+3j, m2=3i2+6ij+10i+7j+7 up odd nph in first column are 3+6m1, 3+6m2 for down odd nph, down odd nph in first column equal to the odd element in the column ni ?, 5+6m=2ni(ni+1)+(2ni+1)?2n+1), (ni-1)?ni+2)+(2ni+1)n+ni=3m, the solutions are a) ni=2+3i, n=3j, m3=3i2+6ij+6i+5j+2 b) ni=3+3i, n=2+3j, m4=3i2+6ij+12i+7j+9 down odd nph in first column are 5+6m3, 5+6m4 for up even nph, 3+3(2m+1)=2ni(ni+1)+(2ni+1)2n (ni-1)?ni+2)+(2ni+1)n=3m+1, the solutions are a) ni=2+3i, n=3j, m50=3i2+6ij+5i+5j+1 b) ni=3+3i, n=3j, m60=3i2+6ij+7i+7j+3 up even nph in first column are 6+6m50=6m5, 6+6m60=6m6, where m5 = m50 +1, m6 = m60 + 1 for down even nph, 5+3(2m+1) = 2ni(ni +1) + (2ni+1)2n (ni?)?ni+2) + (2ni+1)n = 3m+2, the solutions are a) ni = 2+3i, n= 2+3j, m70=3i2+6ij+9i+5j+4 b) ni = 3+3i, n= 1+3j, m80= 3i2 +6ij + 9i+ 7j + 5 down even nph in first column are 8+6m70=2+6m7, 8+6m80=2+6m8, where m7 = m70 +1, m8 = m80 + 1 For the 4 nph sets in first column, the elements in the 8 matrices m1,???m8 are corresponding to all of the element in the triangle matrix, which are concerned with the npon. (see Table 1, 8 matrices, (0,0) is exist). E) Six types relate to odd number in first column of triangle matrix. There are: up odd hole: 3+6m1,2, even element: 4+6m down odd hole: 5+6m3,4 , up even hole: 0+6m5,6 odd number: 1+6m, down even hole: 2+6m7,8 Six types of odd number can be obtained: 7+12m1,2, 9+12m, 11+12m3,4, 1+12m5,6, 3+12m, 5+12m7,8 Obviously, 3+12m and 9+12m are not prime. So that, the four hole sets are concerned with the prime. The author is a professor of math-phys emeritus. 4
标签: 孪生素数
