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1、A FORTRAN source program for calculating the Polignac-Xus numbers with recursive method Wan-Dong Xu School of Science, Tianjin University, Tianjin 300072, China E-mail : Abstract: In this paper we have presented two source programs with FORTRAN to calculate the distribution of prime numbers in the s
2、equence of odd numbers and the Polignac-Xus numbers for every even number, respectively. Form the result we can know that this number is oscillatingly increased as an even number increases. And it is possible to verify the Apostols question: There isnt an even number which isnt the difference of two
3、 primes. Keywords: Problem of Goldbachs type; prime; distribution of primes; Polignac-Xus number. MSC: 11P32; 11A41; 11N05; 11N35 1. Introduction In 1849, A. de Polignac conjectured that there are infinitely many pairs of primes between which the difference is 2, and he conjectured, furthermore, tha
4、t there are a number of pairs of primes between which the difference is any constant number 1. In 1976, T. M. Apostol concluded twelve outstanding unsolved problems concerning prime number. One of them 2, Is there an even number 2 which is not the difference of two primes? This is weaker question th
5、an the former. Recently, we have advanced a recursive method to calculate the number of rest difference formulae of two odd prime numbers, which are to express every even number in natural sequence 3. Now we will list some programs for calculating them. By the results of calculating for them, we can
6、 know that the number of rest difference formulae, or say, Polignac-Xus number, is oscillatingly increased as an even number increases such that the Apostols question is denied. 2. Calculating the distribution of primes in the sequence of odd numbers We have written a program, named Primes.for, with
7、 FORTRAN language to calculate the distribution of odd prime numbers in the sequence of odd numbers in Appendix A. There is a input number in that program is “nupto”, it means to calculate the primes up to “nupto”, and there is a output file named prim0101.dat, in which an odd prime number is denote
8、d by symbol “1” and an odd composite number by symbol “0”, and in which there are 100 figures in every row to indicate 100 odd numbers in order. This is completely the same as ref. 4-5. 3. Calculating the Goldbach-Xus numbers There is a program for calculating with a recursive method, named lrb1.for
9、 in Appendix B, with FORTRAN language, to calculate the numbers of rest difference formulae of two odd prime numbers, or say, the Polignac-Xus numbers, for every even number. There is an input file named prim0101.dat, which is the output file in the section 2 above. And there is an output file named
10、 polig2f.dat led on disk G:, which could be opened by the written-board in Windows XP. In that program many of variant names are the same as symbols in ref. 3. In the Table 1 we listed some Polignac-Xus numbers for every even numbers starting at 2 and ending up to 18000. And we can know that the Pol
11、ignac-Xus number is oscillatingly increased as the even number increases such that the Apostols question can be be denied. There are two source programs in the authors hands, which could be sent readers if they need them and connect to the author. 1 http:/ Table 1. The oscillatingly increasing chara
12、cteristic of the Polignac-Xus numbers n Lr(n) 1 30 1 1 2 2 2 3 2 3 4 3 2 6 4 3 7 3 4 7 4 5 8 4 4 7 6 4 9 8 4 11 31 60 5 5 11 6 8 9 4 7 12 7 4 13 7 5 15 7 8 14 8 9 11 7 7 13 10 5 13 7 7 19 61 90 9 8 17 9 10 16 9 9 16 12 7 19 9 7 19 9 12 18 8 14 18 8 10 20 13 8 20 11 8 25 91 120 11 10 20 10 12 17 11 1
13、2 19 15 10 22 10 12 30 9 9 21 11 16 23 13 12 22 14 12 25 12 15 28 121 150 11 12 24 12 18 27 11 11 24 17 11 26 18 12 33 13 12 25 12 19 22 14 15 23 18 14 29 14 11 34 151 180 11 12 30 19 17 27 14 14 26 18 15 27 18 15 41 14 13 37 14 19 31 15 18 29 24 16 26 17 13 40 181 210 16 18 31 16 22 30 19 18 35 22
14、16 32 16 15 44 18 17 33 17 17 34 17 20 34 20 19 35 18 17 50 211 240 16 17 35 18 22 32 20 20 35 24 19 36 17 21 46 16 18 35 19 25 41 20 17 36 23 16 35 24 18 50 241 270 16 20 36 19 30 35 22 20 36 24 17 45 22 17 53 18 21 37 26 25 32 19 19 46 27 21 42 20 19 48 271 300 22 22 51 22 28 43 21 22 39 34 21 40 24 18 56 26 25 42 22 28 45 23 19 49 31 17 51 23 22 58 301 330 28 25 40 24 28 43 24 28 40 30 21 48 23 18 72 23 22 43 26 30 42 26 28 43 33 21 46 26 24 68 331 360 25 26 49 23 29 53 25 23 42 33 25 50 28 25 61 21 20 48 23 31 48 27
