By Zhou M (2022).
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Greener Journal of
Applied Mathematics and Statistics Vol. 2(1), pp. 1-2, 2022 Copyright ©2022, the
copyright of this article is retained by the author(s) http://gjournals.org/GJAMS |
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The salutation to 1+2
Suqian Economy
and Trade vocational School
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ARTICLE INFO |
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Article
No.: 122821159 Type: Short Comm. |
Accepted: 10/12/2021 Published: 31/12/2021 |
*Corresponding
Author Zhou Mi E-mail: zhoumi19920626@ 163.com |
Keywords:
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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 ,2N-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 ,2N-(2x-1) is certainly not a
prime number, if not2N-(2x-1) must be a prime number, if the natural number N
does not appear in it, 2N-(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 su m of two primes! ! ! When x does
not appear at the matrix beginning of 5 , the matrix beginning of 4 + x is the
matrix beginning of the number which not appear in the matrix beginning of 9 ,
and the N is the number doesn't appear in the matrix does not appear in the
matrix beginning with 9, two negatives make a positive., that is to say at this
time of the N contains all the numbers appear in the matrix beginning of 9,
then all the numbers appear in the matrix beginning of 9 two times of them can
all be expressed as two prime numbers add up, that is1 + 1! Besides all the
even numbers not appear in the matrix beginning with 9 can be expressed as a
prime plus 9 ( beginning with 9, not appear in it 2N minus 9 is a prime
number), and 9 = 3 * 3, that is to say, the numbers can be expressed as 1 + 2.
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Cite this Article: Zhou M (2022). |