By Zhou M (2022).
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Greener Journal of
Applied Mathematics and Statistics Vol. 2(1), pp. 8-9, 2022 Copyright ©2022, the
copyright of this article is retained by the author(s) http://gjournals.org/GJAMS |
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Lifting Chandra matrix to get three number
Theory conclusions
Suqian Economy
and Trade vocational School
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ARTICLE INFO |
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Article
No.: 122821162 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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1. INTRODUCTION
This
paper shows a very simple method of lifting Chandra matrices to explain a
conjecture. A man found that all even numbers can be expressed as a prime minus
a prime, and this conjecture is still not solved, I found a way to solve it.
2. MAIN
DERIVATIONS
Lets
look at a matrix: in 1934, one ground breaking mathematician from the Indian/
Bangladesh Region Harish Chandra (19231983), in the field of number theory,
has 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 has also
become a basic tool in analytic number theory, via the theory of auto orphic
forms, leading to Langlands program eventually. It became one of the major
mathematical edifices of the Second half of the twentieth century [9]. 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 become 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
1
7 12 17 22 27 32 37 42
2
10 17 24 31 38 45 52 59
3
13 22 31 40 49 58 67 76
4
16 27 38 49 60 71 82 93
5
19 32 45 58 71 84 97 110
...........
The
secret of this square sieve is: If a natural number N appear 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 divisor 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 arithmetical range of the numbers. Based on above
observations, we made a few similar matrices accordingly (lifting by x):
4+x
7+x 10+x 13+x
7+x 12+x 17+x 22+x
10+x
17+x 24+x 31+x
...........
Lifting
by any positive integer 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 a number N is not in the matrix beginning with 4 and 4+x,then
2N+1and 2N-(2x-1) are primes. If so,(2N+1)-(2N-(2x-1))=prime-prime=2x,x
is all numbers, so 2x is all even numbers! Now, we have to found the N which
not in the two matrixes. All the numbers not in the matrix of
4, and if these numbers all appear in the matrix of 4+x, is not set up.
Because Numbers in 4 and not in 4.is all numbers, now we assume not in 4 is
equally within 4+x, so numbers in 4 and numbers in 4+x
are all numbers, obviously is not set up! Exist infinite N not in the two
matrix! We can get a conclusion: all even numbers can possibly be expressed as
a prime minus a prime. And we can also get two conclusions: 1: The are possibly
infinite forms that all even numbers can expresses as a prime minute a prime 2：There are
possibly has infinite twin primes
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Cite this Article: Zhou M (2022). |