Greener Journal of
Science, Engineering and Technological Research Vol. 9(1), pp. 12, 2019 ISSN: 22767835 Copyright ©2019, the
copyright of this article is retained by the author(s) DOI Link: http://doi.org/10.15580/GJSETR.2019.1.040919065
http://gjournals.org/GJSETR 

The equivalent proposition of an
unsolved number theory problem "is the difference between two
primes"
Zhou Mi ^{1*}, Honwei Shi ^{2}, Delong Zhang ^{3},
Xingyi Jiang ^{3}, Songting
He ^{3}
^{1} Suqian Economy and
Trade Vocational School
^{2} Suqian
College, Industrial technology
research institute of Suqian college Jiangsu, Email: 19744090@
qq.com
^{3} Suqian
College
^{ }
ABSTRACT 



This
paper uses chandra symmetric matrix to find the
equivalent proposition of a worldly unsolved problem. 


ARTICLE INFO 



Article
No.: 040919065 Type: Review Article DOI: 10.15580/GJSETR.2019.1.040919065 
Submitted: 09/04/2019 Accepted:
15/04/2019 Published: 19/04/2019 
*Corresponding
Author Zhou Mi Email: zhoumi19920626@
163.com 
Keywords:










BACKGROUND
Let
us look at a matrix: in 1934, one ground breaking mathematician from the Indian/
Bangladesh region, Harish Chandra (19231983), in the field of number theory,
had made a founding contribution of harmonic analysis on semi simple Lie
groups. This subject is equally important for an engineer, as a synthesis of
Fourier analysis, special functions and invariant theory etc., and it had also
become a basic tool in analytic number theory, via the theory of automorphic forms,
leading to modern Langlands program
eventually. It became one of the major mathematical edifices of the second half
of the twentieth century.
Chandra
matrix is a square sieve, where the first row of the square sieve consists of
the first element of 4, the difference between next every two adjacent numbers
is 3, forms an arithmetic sequence: 4,7,10, ... The first column equals to the first row. The second
row, third row, ...... any subsequent rows are also arithmetic sequence, but
the difference between two adjacent numbers gradually becomes larger, and they
are 5,7,9,11,13, ...... respectively, and they are all odd numbers, and the
matrix is symmetrical, as shown below with modulo equivalent representation:
4 7 10 13 16 19 22 25
i.e. mod(n,3)=1
7 12 17 22 27 32 37 42
i.e. mod(n,5)=2
10 17 24 31 38 45 52 59
i.e. mod(n,7)=3
13 22 40 40 49 58 67 76
i.e. mod(n,9)=4
16 27 49 49 60 71 82 93
i.e. mod(n,11)=5
19 32 58 58 71 84 97 110
i.e. mod(n,13)=6
The
secret of this square sieve is: If a natural number N appears in the table,
then 2N+1 certainly is not a prime number, because 2 times of remaining plus 1
is at least one of the divisors in the above modulo operations. If N does not
appear in the table, then 2N+1 is definitely a prime
number. Because 2 times of remaining plus 1 is not any of the divisors in the
above modulo operation. Primes are left out. Almost all primes can be launched
from this table, assume that the number of primes follow the prime number
theorem: x/ln(x), in the arithmetical range of the
numbers.
Proof of matrix properties：
4 7 10 13 16
4+3(n1)
7 12 17 22 27
7+5(n1)
10 17 24 31 38
10+7(n1)
13 22 31 40 49
13+9(n1)
16 27 38 49 60
16+11(n1)
3m+1 5m+2 7m+3 9m+4 11m+5
2mn+m+n
In fact, if mn
+ m + N, N = 2, 2 N + 1 = 2 (2 mn + m + n)+ 1 = 4 mn + m + 2 N + 1 = 2 (2 m
+ 1)
(2
n + 1), it is not a prime number. If 2N+1 is not a
prime number, then 2N+1 must be the product of two odd Numbers.
2N+1=(2m+1)(2n+1)=4mn+2m+ 2N+1, you get N=2mn+m+ n, which
appears in the table, contradicting the hypothesis. So 2N+1 must be a prime
when N doesn't show up in the matrix.
Based
on above observations, we made a few similar matrices accordingly (lifting it
by x):
4+x
7+x 10+x 13+x
i.e. mod(n,3)=1+ mod(x,3)
7+x
12+x 17+x 22+x
i.e. mod(n,5)=2+ mod(x,5)
10+x
17+x 24+x 31+x
i.e. mod(n,7)=3+ mod(x,7)
Lifting
by any positive integer can be obtained if the natural number N in the matrix,
2N(2x1) is certainly not a prime number, otherwise
2N(2x1) must be a prime number.
So
we've made some matrices like this
5
8 11 14 17 20
.
8
13 18 23 28 33
.
11
18 25 32 39 46
.
If
N in it, 2N－1 is not prime, if not, it is
6
9 12 15 18
.
9
14 19 24 29
.
12
19 26 33 40
.
If
N in it, 2N－3 is not prime, if not, it is:
List：
4
2n+1
5
2n1 (2n+1)2
6
2n3 （2n+1）4
7
2n5 (2n+1)6
8
2n7 (2n+1)8
9
2n9 (2n+1)10
10
2n11 (2n+1)12
11
2n13 （2n+1)14
4+x
2n－(2x－1) （2n+1）－2x
If
N not in 4 and 5,2N+1－（2N－1）=prime －prime =2.
If
N not in 4 and 6,2N+1－（2N－3）=prime －prime=4
If
N not in 4 and 4+x, 2N+1 －（2N－2x+1）=prime－prime=2x
As
long as find that the matrix starts with 4 + x and 4 has an n that doesn't show
up at the same time, this conjecture sets up. It can be stated as 2mn+m+n, 2mn+m+n+x,
the two cant
express all the number bigger than 4+x, this
conjecture sets up.
CONCLUSION
The
unsolved number theory problem "even is the difference between two
primes" has an equivalent proposition:
2mn+m+n,
2mn+m+n+x, the two cant express all the numbers bigger than 4+x，this conjecture sets
up.
CONFERENCE
1.
Guy, Unsolved problems in number theory, https://www.springer.com/cn/book/9780387208602
2.Chandrahttps://www.baidu.com/link?url=he22R1y3hlW58exQFObNED3awVSZaPwuEiEsIZv7gapjyCEIIutFnktY8oSHmnFxG0plAtcyGXbXXMZjn6G1Mq&wd=&eqid=d4594794000075f2000000035c2315c5