Greener Journal of
Science, Engineering and Technological Research Vol. 9(1), pp. 811, 2019 ISSN: 22767835 Copyright ©2019, the
copyright of this article is retained by the author(s) DOI Link: http://doi.org/10.15580/GJSETR.2019.1.040919068
http://gjournals.org/GJSETR 

Even numbers are the sum of two prime
numbers
Honwei
Shi ^{1}, Zhou Mi^{2*}, Delong Zhang ^{3},
Xingyi Jiang ^{3}, Songting He ^{3}
^{1} Suqian College, Industrial technology research institute of Suqian college Jiangsu, Email: 19744090@ qq.com^{}
^{2} Suqian Economy and Trade Vocational School
^{3} Suqian College
^{ }
ABSTRACT 



"Even numbers are the sum of two prime
numbers." This is the description of Goldbach's conjecture, and in
Canadian Gaye's book "unsolved problems in number theory", it is an
open question to put forward a contrary conjecture that "even numbers
are the difference between two prime numbers". In this paper, Chandra
sieve is used to deduce that the sum of large and even numbers is the sum of
two prime numbers, and that "even numbers are the difference between two
prime numbers" is a great possibility. At the same time, it is possible
to guess the possibility of twin prime conjecture. 


ARTICLE INFO 



Article
No.: 040919068 Type: Review article DOI: 10.15580/GJSETR.2019.1.040919068 
Submitted: 09/04/2019 Accepted:
13/04/2019 Published: 19/04/2019 
*Corresponding
Author Zhou Mi Email: zhoumi19920626@
163.com 
Keywords:










INTRODUCTION:
1.
Goldbach conjecture
In
Goldbach's letter to Euler in 1742, Goldbach put forward the following
conjecture: any even number greater than 2 can be written as the sum of two
prime numbers. But Goldbach could not prove it himself, so he wrote to consult
the famous great mathematician, Euler, to help prove it, but until death, Euler
could not prove it. [1] because modern mathematics has no use of "1 is
also a prime number" this agreement, the original conjecture of modern
statement is: any integer greater than 5 can be written as the sum of three
prime numbers. Euler also put forward another equivalent version in reply, that
is, any even number greater than 2 can be written as the sum of two prime
numbers. The common conjecture today is represented by the version of Euler.
Put the proposition "any sufficiently large, even number" can be
expressed as a prime factor, the number of which does not exceed a, and the sum
of the number of the other prime factors not more than B.
"A+b"". In 1966, Chen Jingrun proved that "1+2" was
established, namely, that any sufficiently large even number can be expressed
as the sum of two prime numbers, or the sum of a prime number and 1.5 prime
number". In this paper, the majority of even numbers are obtained by using
the properties of Chandra sieves and their derived sieves.
2. An unsolved
problem in number theory
Canada
Gaye in "theory did not solve the problem of" the book "even for
the two prime numbers" conjecture, also by Chandra a deduction, that there
was this conjecture very large possibility of conclusion.
3. Twin prime
conjecture
A
twin prime is a pair of primes with a difference of 2, such as 3 and 5, 5 and
7, 11 and 13.... The conjecture was formally proposed by Hilbert in the eighth
report of the 1900 International Congress of mathematicians, which can be
described in this way:
There
are infinitely many prime numbers P, so that P + 2 is prime.
The
prime pair (P, P + 2) is called the twin prime.
In 1849, Alphonse de Polignac proposed
general conjecture: all natural number k, there are infinitely many primes of
(P, P + 2K). The case of k = 1 is the prime twins conjecture.
In May 14, 2013, the "nature"
(Nature) magazine reported online that Zhang Yitang proved that there are infinitely
many primes less than 70 million, of the difference, this study was a major
breakthrough in the twin prime conjecture of this ultimate theory, some people
even think that its impact on the academic circles will exceed Chen Jingrun
"1+2". [1] in the new study, Zhang Yitang does not depend on the
premise without proof of corollary, that there are infinitely many primes less
than 70 million of the difference, and on this important issue of the twin
prime conjecture, it was a big step forward.
Zhang Yitang's paper was published online in
May 14th, and two weeks later, in May 28th, the constant dropped to 60 million.
Only two days later, in May 31st, it dropped to 42 million. Three days later,
in June 2nd, it was 13 million. The next day, 5 million. June 5th, 400
thousand. In the Polymath project sponsored by British mathematician Tim Gowers
and others, the twin prime problem became a typical example of collaboration
among global mathematical workers using the internet. People continue to
improve the proof of Zhang Yi Tang, further closer to the final solution of the
twin prime conjecture. As of October 9, 2014 (20141009) [update], the prime
difference is reduced to less than 246[4].
Through the Chandra sieve, we can get: twin
prime conjecture is very likely to be established.
Text:
In
1934, Chandra of East India (now Bangladesh) presented a square sieve:
The
first line is the first of 4, 3 of the tolerance of arithmetic sequence 4, 7,
10,... 4+3 (n1),...
The
second line is the first of 7, 5 of the tolerance of arithmetic sequence 7, 12,
17,... 7+5 (n1),...
The
third line is the first of 10, 7 of the tolerance of arithmetic sequence 10,
17, 24,... 10+7 (n1),...
The
fourth line is the first of 13, 9 of the tolerance of arithmetic sequence 13,
22, 31,... 13+9 (n1),...
...
Line
m is the first 3m+1, tolerance of arithmetic sequence 2m+1 3m+1, 5m+2, 7m+3,...
Its entry n is 3m+1+ (n1) (2m+1) =2mn+m+n,...
Write
the following array:

First
columns 
second
columns 
third
columns 
fourth
columns 
fifth
columns 

Column
n 

First
lines 
4 
7 
10 
13 
16 

4+3(n1) 

Second
lines 
7 
12 
17 
22 
27 

7+5(n1) 

Third
lines 
10 
17 
24 
31 
38 

10+7(n1) 

Fourth
lines 
13 
22 
31 
40 
49 

13+9(n1) 

Fifth
lines 
16 
27 
38 
49 
60 

16+11(n1) 










Line
m 
3m+1 
5m+2 
7m+3 
9m+4 
11m+5 

2mn+m+n 










The secret of this square screen is that if a
natural number N appears in the table, then 2N + 1 is certainly not prime; if N
does not appear in the table, then 2N + 1 is definitely prime.
In fact, if N=2mn+m+n, then 2N+1=2 (2mn+m+n)
+1=4mn+2m+2n+1= (2m+1) (2n+1), is not prime. On the other hand, if N is not
present in the table, if 2N+1 is not prime, then 2N+1 must be the product of
two odd numbers, write 2N+1= (2m+1) (2n+1) =4mn+2m+2n+1, and get N=2mn+m+n,
which appears in the table and conflicts with the hypothesis. So when N does
not appear in the matrix, 2N+1 must be the prime number.
A number of similar matrices (Simplified) are
then deduced, and numerous matrices of the same nature are found. After the
deduction of this matrix, there will be many wonderful phenomena, and there are
many effective applications.
5811141720.........
.
81318232833.........
.
111825323946.........
.
.........
.
This matrix in some natural number N that appears
in the matrix of 2N  1 is certainly not prime, if there is 2N  1, it must be
prime, because the first 5 matrix does not appear, the second matrix 6 does not
appear and also the 2 * 5+1=2 * 6  1.
Similarly, we can list a matrix:
69121518.........
.
914192429.........
.
1219263340.........
.
.........
.
It can be concluded that if the natural
number N appears in this matrix, the 2N  3 is certainly not prime. If it does
not appear, the 2N  3 must be the prime number, and the truth is the same as
above.
It may also be listed:
4+x,
7+x, 10+x, 13+x......... .
7+x
12+x 17+x 22+x......... .
10+x
17+x 24+x 31+x......... .
.........
.
It can be concluded that if the natural
number N appears in the matrix, then 2N  (2x  1) must not be a prime number.
If not, then 2N  (2x  1) must be prime.
"Any one even greater than 2 can be
expressed as the sum of two prime numbers", this is in the famous Goldbach
conjecture, Canada Gaye in the book "unsolved problems in number
theory". In this book, a similar conjecture is obtained, " instead of
any even number can be expressed as two primes, for subtraction" for
example: 2=53,4=73,6=115,8=113,10=133......
The moments corresponding to the matrix at
the beginning of the "" are 2N1, 2n3, 2n5, 2n7, 2n9, 2n11,
2n13...
When n does not appear in the matrix at the
beginning of 4, 2n+1 is the prime number; when n does not appear in the matrix
at the beginning of 4 and 5, 2n+1 and 2N1 are prime numbers, so 2= (2n+1)  (2n1)
= prime  prime number.
When n does not appear in the matrix at the
beginning of 4, 2n+1 is the prime number; when n does not appear in the matrix
at the beginning of 4 and 6, 2n+1 and 2n3 are prime numbers, 4= (2n+1) 
(2n3) = prime number  prime number.
At the same time and so on, if it does not
appear in the matrix and 4 4+x at the beginning of the N, 2n+1 and 2n (2x1) are
prime numbers, this time 2x= (2n+1)  (2n2x+1), apparently 2x can be all even.
As long as any sieve and the beginning of the
4 screen has a n at the same time, this conjecture was established, each one
has an infinite number of screen and does not appear, so at least the same
possibility is very large, so this conjecture has established the possibility
of a very large occurrence!
If it does not appear in 6, 7, 8, 10,
11...... The natural numbers n, 2n, minus 3, 5, 7, 11, 11, and 13 (must be
minus prime numbers) results in prime numbers, and these 2n are all prime
numbers plus 3, 5, 7, and two...... It is a prime number with a prime number. All
natural numbers do not appear in the screen twice and contain the great
majority of even numbers. Even number is the sum of two prime numbers, so the
great majority of even number is the sum of two prime numbers.
In addition:
The natural number N of the matrix at the beginning of 5 corresponds to
2N  1, and the natural number N of the matrix at the beginning of 6
corresponds to 2N  3, then 7, 8, 9...... The natural number N of the initial
matrix corresponds to 2N  5,2N  7,2N  9, 2N  11......
Now, there are 4 cases:
A 2N Prime＝prime
B 2N prime＝Composite
C 2N compotation＝Prime
D 2N compotation＝compotation
All
2N A, B includes all even odd prime number, prime number + + get prime;
All
2N C in D, including all the even odd prime number, odd number + + odd number
obtained.
The
2N in A and B includes all the odd numbers plus 3, 5, 7, 11,...... The even
numbers obtained by prime numbers, 2N in C and D, include all odd numbers plus
9, 15, 21...... Odd numbers are even,
All
odd numbers +3, all odd numbers +9, even numbers, large, even numbers.
All
odd numbers +5, all odd numbers +15, even numbers, large, even numbers.
All
odd numbers +7, all odd numbers +21, even numbers, large, even numbers.
......
.
That
is to say, the 2N in A, B, and 2N in C and D are the same in large and even
numbers.
The
2N in A and B removes the 2N in B, the 2N in C and D, removes the 2N in C, and
the remaining large, even numbers of 2N are the same, because the 2N in B and C
are the same, the reason:
B:
2N= + C:2N= + prime number, prime number.
So
A, B and C, D in 2N, remove the same part of even numbers, the remaining large,
even number, or the same.
D
2N is the number + even number, all less than 40 even can be expressed as odd
number with odd number, reason:
The
number of N bits must be 0, 2, 4, 6, 8, and now the n is numerically
classified:
(1)
if a digit n is 0, n=15+5k (k = 5 for odd);
(2)
if a digit n is 2, n=27+5k (k = 3 for odd);
(3)
if a digit n is 4, n=9+5k (k = 7 for odd);
(4)
if a digit n is 6, n=21+5k (k = 5 for odd);
(5)
if a digit n is 8, n=33+5k (k = 3 for odd);
To sum up, not less than any even number 40,
can be expressed as two and the odd number.
Therefore D 2N contains all the even numbers greater
than 40, including all the large even, all A in 2N also includes all the even numbers
greater than 40, the A 2N is a prime + prime, so all the large even number is
the sum of two prime numbers. Twin prime number conjecture is a famous unsolved
problem in number theory. The conjecture was formally proposed by Hilbert in
the eighth report of the 1900 International Congress of mathematicians, it
ranked 23 in the "Hilbert problem", and it describes "the
existence of infinite primes P, and for
each P, p+2 this is a prime number".
If there is no other interference between the
two adjacent natural numbers in the first row of the Chandra sieve, there are
countless adjacent natural numbers, and their 2 and 1 are twin prime numbers.
But the second and third lines......, it is difficult to completely cover the
first line, so the twin prime conjecture is very likely to be established.
CONCLUSION:
1.
For Goldbach's conjecture, even numbers are the sum of two prime numbers, and
most even numbers are the sum of two prime numbers;
2.
"Even numbers are the difference between two prime numbers" is a
great possibility;
3.
Twin prime conjecture is very likely to be established.
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Cite this
Article: Honwei S;
Zhou M; Delong Z; Xingyi J; Songting H (2019). Even numbers are the sum of
two prime numbers. Greener Journal of Science, Engineering and
Technological Research, 9(1): 811, http://doi.org/10.15580/GJSETR.2019.1.040919068. 