A
C++ source program for calculating the Polignac-Xu’s
numbers with recursive method
Wan-Dong Xu Heng Xu
School of Science, Tianjin University, Tianjin 300072, China
E-mail :
xwandong@sohu.com
Abstract: In this paper we have presented two source programs with C++ to calculate the distribution of prime
numbers in the sequence of odd numbers and the Polignac-Xu’s 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 deny
the Apostol’s question: Is there an even number >2 which is not the difference of two primes?.
Keywords: Apostol’s question, prime; distribution of primes; Polignac-Xu’s number; C++ source program.
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, furthermore, conjectured that there are infinitely many pairs of primes
between which the difference is any even constant number [1]. In 1976, T. M. Apolstol concluded
twelve outstanding unsolved problems concerning prime numbers. One of them [2], Is there an even
number >2 which is not the difference of two primes? This is a weaker conjecture than 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 a program for calculating them. By the results of this calculating, we can know that
the number of rest difference formulae, or say,
Polignac
-Xu’s number, is oscillatingly increased as an
even number increases such that the Apolstol’s question is verified to be deny.
2. Calculating the distribution of primes in the sequence of odd numbers
We have written a program, named prime.cpp, with C++ 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
Oprim0101.dat led on disk G:, in which an odd prime number is denoted 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-Xu’s numbers
There is a program for calculating with a recursive method, named Polignac1.cpp in Appendix B,
with C++ language, to calculate the numbers of rest difference formulae of two odd prime numbers, or
say, the
Polignac
-Xu’s numbers, for every even number >2. There is an input file named Oprim0101.tex
on disk G:, which is the output file in the section 2 above. And there is an output file named Polig1.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
-Xu’s numbers for every even numbers starting at 2 and ending
up to 18000. And we can know that the
Polignac
-Xu’s number is oscillatingly increased as an even
number increases such that the Apostol’s question can be denied.
There are two source programs in the author’s hands, which could be sent readers if they need them
and connect to the author.
1
http://www.paper.edu.cn