By Zhou M (2022).
|
Greener Journal of
Applied Mathematics and Statistics Vol. 2(1), pp. 3-5, 2022 Copyright ©2022, the
copyright of this article is retained by the author(s) http://gjournals.org/GJAMS |
|
All primes plus or minus all prime has a
high probability to get all evens
Suqian Economy
and Trade vocational School
|
ARTICLE INFO |
|
|
|||||||
|
Article
No.: 122821160 Type: Short Comm. |
Accepted: 10/12/2021 Published: 31/12/2021 |
*Corresponding
Author Zhou Mi E-mail: zhoumi19920626@ 163.com |
Keywords:
|
|
|||||
|
|
|
|
|
|
|
||||
Look
at a matrix: in 1934, one from the East Indies (now Bangladesh) ordinary
scholar - Chandra, in the field of number theory has made a brilliant achievement, this achievement makes him leaving a legacy,
immortal. Chandra square sieve the first row is the square sieve is the first
of 4, the difference between two adjacent numbers is 3 arithmetic sequence:
4,7,10, ... The second line, third line,
...... any subsequent row are also arithmetic sequence, but the
difference between two adjacent numbers gradually become larger, respectively,
5,7,9,11,13, ... and they are all odd .
......................................................
4
7 10 13 16 19 22 25 ……
7
12 17 22 27 32 37 42 ……
10
17 24 31 38 45 52 59 ……
13
22 31 40 49 58 67 76 16 27 38 49 60 71 82 93 ……
19
32 45 58 71 84 97 110 …….
This
square sieve secret is: If a natural number N appear in the table, then 2N +1
certainly not a prime number, if N does not appear in the table, then 2N +1 is
definitely a prime number. Let's look at a few examples. Beginning with 4,
skipped three numbers 1,2,3, of course, they will
never appear in the table. Then, 2 × 1 +1 = 3,2 × 2 +1
= 5, 2 × 3 +1 = 7. You see, 3,5,7 are prime numbers.
Look at the number 17, 17 is in the symmetric matrix .
17*2+1=35 and 35 is a prime number. Almost all primes can be launched from the
table to the inverse. I made a few similar matrix accordingly (simplify): 5 8
11 14 17 20………. 8 13 18 23 28 33………. 11 18 25 32 39 46………. If natural number N
in the matrix ,then 2N-1 is certainly not a prime number, if not , 2N-1 is a
prime number, because in the first matrix 5 is not appear, and 6 does not
appear in the second matrix ,2 * 6-1 =2* 5 + 1 = 11,so it was established.
Similarly, then set out a matrix: 6 9 12 15 18………. 9 14 19 24 29………. 12 19 26
33 40………. If the natural number N can be drawn in this matrix
,2 * N-3 is certainly not a prime number, if not 2N-3 is a prime number,
the reason is above. Also 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 ………. Can be obtained if the natural number N in the
matrix ,2 * N-(2x-1) is certainly not a prime number, if not 2 * N-(2x-1) must
be a prime number, if the natural number N does not appear in it,2 *
N-(2x-1)=k, return to the beginning of the matrix 5, if N is not appear in this
matrix ,2N-1must be a prime number, then 2x-1 if x does not appear in the
matrix beginning with 5 then 2x-1 must be a prime! Then 2 * N-(2x-1) = k, that is
k + (2x-1) = 2N, you may found 2N is even, and this time it's two addends k and
2x-1 are prime numbers, That is an even number can be expressed as a sum of two
primes! ! ! However, even this does not explain all set so, there is a
restriction, this time for N must not appear in the matrix beginning with 4 + x , and the x must not appear in the matrix beginning with 5
, which is to limit the condition that they meet these two constraints then all
the even number can be expressed as a sum of two prime numbers, so it is a part
of the proof. There is a better proof the following: Beginning of 5 matrix,
natural number N does not appear then 2N-1 is a prime number, matrix beginning
with 6, natural number N does not appear then 2N-3 is a prime number, then the
beginning of 7,8,9,,,, the natural number N does not appear in these matrix ,
then 2N-5, 2N-7, 2N-9, 2N-11,,,, are prime numbers, minued
is 2N, and the subtrahend are all odd numbers, when the odd is prime number, 2
N can be expressed as a sum of two prime numbers, as subtrahend are continuous
adds, so odds include all odd prime number, then the 2N which meet the
requirement "survival" is greatly improved! In addition, a number of
unresolved issues proof in a part: "Arbitrary an even number greater than
2 can be expressed as sum of two prime numbers," which is the famous Goldbach conjecture, Canadian Guy in the " unresolved
issues in number theory, "the book mentioned that a similar but opposite
conjecture "Any even number can be expressed as a sum of two primes
subtraction," I made a proof of this, the method is based on Chandra
symmetric matrix too, as follows： For the above matrix can also be listed to the similar
matrix as follows, 3 6 9 12 15 18………. 6 11 23 30 37 44………. ………. In this matrix
if the natural number N in it, the 2N +3 must be a composite number, if not, 2N
+3 is a prime number, because the matrix beginning of 4 in which 6 does not appear ,the matrix at beginning of 3 in which 5 is not
appear, and the 2 * 6 + 1 = 2*5+3, and so set. And then set out a matrix: 2 5 8
11 14 17………. 5 10 15 20 25 30………. 8 15 22 29 36 43………. ………. If a natural number
N appear in it, 2 * N +5 must be a composite number, otherwise it will be a
prime number, the reason is above. Then find a formula as follows: 4—x 7—x 10—x
13—x 16—x………. 7—x 12—x 17—x 22—x 27—x………. 10—x 17—x 24—x 31—x 38—x………. ………. If
the natural number N appear in it , 2N + (2x +1) must be
a composite number otherwise 2N +(2x +1) must be a prime number. Let a natural
number N does not appear in it, the 2N + (2x +1) =k, k is the prime number, so
2N = k-(2x +1), 2N is even, then k is a prime number, now just let 2x+1 is
prime number then 2N can be expressed as two prime numbers subtraction, when x
does not appear in the matrix beginning with 4 ,2x +1
is a prime number! At this point 2N can be expressed as sum of two primes.
However, there is a restriction that x must not appear in the matrix beginning
with 4, and N must not appear in the matrix beginning with 4 + x ,so it just is
a proof in part. like Goldbach conjecture proof, in
the matrix beginning with 3,2,1,0, -1,,,,, if the natural number N does not
appear in it, then 2N +3,2 N +5,2 N +7,2 N + 9,,,, the results are prime
numbers,3,5,7,9…….. when the addends are prime , then these 2N can be expressed
as a sum of two primes subtraction, because addend are continuous number, so
includes all primes, then the 2N which meet the requirement
"survival" also greatly improved! ! ! In fact, there is a problem,
the proof of the Goldbach conjecture, the twice of
all natural numbers which are not present in the matrix which minus 3 can be
represented as the sum of two prime numbers, in fact, it is 3 plus all the
primes just! The twice of all natural numbers which are not appear in the
matrix which minus 5 can be expressed as sum of two prime numbers, in fact, it
is 5 plus all primes just! The following are 7, 11,13,17......
plus all primes ..... .. that
is the evens which are all primes plus primes can be expressed as the sum of
two prime numbers together, it looks like no meaning. In fact, it has great
meaning, because the matrixes which minus primes number are numerous, the
natural numbers are not present in numerous matrix are very likely to contain
all natural numbers. But I do not have to complete the proof of why they contain
all natural numbers. However I found a good way to proof evidence Goldbach's Conjecture and I have a very full reason, I hope
someone finish it in my foundation. The second conjecture is the same. So my
article is valuable.
|
Cite this Article: Zhou M (2022). |