1. Introduction
In the long process of human understanding the world, there are many problems that attract many scholars and confuse many wise men.
Mathematicians found that the interval between two adjacent prime numbers is 2. This is a prime number, which is called twin prime number [1] [2] [3] [4].
Example 1
Twin prime numbers
11 and 13,
59 and 61,
That is, 11 and 11 + 2,
59 and 59 + 2,
In this way, there are infinite primes, which are called twin prime conjecture.
The twin prime distribution problem is the corresponding prime distribution.
Set the integer k, let’s look at the sequence,
In k + 2, integers
2, 3, 4, 5, ...
are called the corresponding number of k.
In k + 2, prime numbers
2, 3, 5, 7, ...
are called the corresponding prime numbers of k. That is, the prime number corresponding to an integer.
In k, the composite numbers
4, 6, 8, 9, ...,
correspond to the prime numbers
11, 17, ...,
which are called the corresponding prime numbers of the composite numbers.
The key to the distribution of twin prime numbers is to study the corresponding prime numbers of integers and composite numbers.
Before studying the twin prime distribution problem, let’s look at the prime distribution problem.
Chebyshev function is the focus of mathematicians’ research on prime distribution. Let’s look at the mathematician’s method.
In 1852, the Russian mathematician Chebyshev proposed the function [3] [4] [5]:
Get
Set x > y, for large y,
, Obtained by Chebyshev function
Obviously
, can get
Set y = xλ, we get [4] [6]:
Obviously estimate π(xλ) < xλ, get
(1.1)
By (1.1) get
Can get
λ < 1, can confirm
By π(x) > xλ, we can get
λ ~1, can get:
By Chebyshev function
The prime theorem is proved
(1.2)
Example 2
x = 1018, calculated by (1.2)
The method of proof is to construct unequal forms, carry out unequal transformation, delete
by limit, and then obtain the prime theorem.
However, some scholars can’t understand it. We can improve the traditional proof method.
2. Improve the Traditional Proof Method
We improve the proof method of (1.1).
Might as well set
Obviously, x tends to infinity, so we can confirm
, and we get [4] [7]
(2.1)
By (1.1) and (2.1) can get
Now, prove π(x) > xλ, You can delete it xλ/x, however, It’s complicated.
We use another method to prove the prime theorem.
Obviously xλ < 2xλ, by (1.1) can get
We get
and
Set positive number a and b, can get
and
We can get
Obviously
We get the arithmetic mean of
Can get
x tends to infinity,
, and (2.1) get
Can get
The prime number theorem can be obtained from Chebyshev function
We get:
(2.2)
Example 3
x = 1018, calculated by (2.2)
The prime theorem can also be obtained in another way.
3. New Research on Prime Theorem
We use a new method to study the prime theorem.
Set prime number p ≤ x,
(3.1)
Set e = 2.718…, by (3.1) can get
Can get [6] [7]
Can get
We get
Arithmetic mean of
:
Can get
Get
We can get
Obviously prove
From this, we can get a new prime theorem
We can get:
Can get
That is
(3.2)
Example 4
x = 1018, calculated by (3.2)
The problem of prime distribution was discussed earlier. We continue to discuss the distribution of twin prime numbers.
4. Corresponding Prime Distribution
The twin prime distribution problem is the corresponding prime distribution.
Set Integer x = 16, let’s look at the sequence of k and k + 2
k + 2 is called the corresponding number of k
In k + 2, prime
2, 3, 5, 7, 11, 13,
a total of 6. The table shows
and the proportion of prime numbers is:
Generally [6] [8]:
(4.1)
Here (4.2) is called the prime distribution rate of integers.
Let’s look at the prime number corresponding to the composite number.
Set the composite number c. for the convenience of discussion, temporarily set 0 as the composite number to see the sequence of composite numbers c and c + 2
In c, there are 8 composite numbers, and the table is F = 8.
In c + 2, there are 2 primes, 2, 11. The table is c (16) = 2.
The proportion of prime numbers is:
Generally:
(4.2)
Here (4.2) is called the prime distribution rate of composite numbers.
Example 5
(4.1) and (4.2) Partial calculation:
A composite number is part of an integer. The greater the proportion of the composite number in the integer, the closer the distribution rate of the corresponding prime number of the composite number is to the distribution rate of the corresponding prime number of the integer.
Now let’s discuss the corresponding prime of prime.
Let the prime number p. for the convenience of discussion, temporarily set 1 as the prime number to see the sequence of prime numbers p and p + 2
In p + 2, there are four primes: 3, 5, 7, 13. The table is
Represents the number of primes in p + 2. Can get
According to statistics, if x is larger, then L(x) is more.
If we can prove that L(x) is infinite, we can prove the twin prime conjecture.
5. Principle of Composite Number Corresponding to Prime Number
You can get from the front
Generally:
(5.1)
By (5.1) can get
Set number of set composite numbers
By (4.2) we can get
We get
Set x tends to infinity, so we can confirm [4] [8]:
and
We can get
Set λ, can get
Example 6
By λ Partial calculation:
we get:
(5.2)
Here (5.2) is called the corresponding prime number principle of composite numbers.
6. Twin Prime Distribution Theorem
We continue to discuss the distribution of twin prime numbers.
The number of twin prime numbers is obtained from (5.1)
(6.1)
By (5.2) and (6.1), we get [6] [7]:
(6.2)
Here (6.2) is called the twin prime distribution theorem [6] [7].
Example 7
Calculated by part (6.2):
7. Discuss the Distribution of Twin Prime Numbers
We confirm the number of twin prime numbers
From (6.2), we can get:
(7.1)
Can confirm
Example 8
By (7.1) Partial calculation:
x tends to infinity,
, can get [6] [8]
Set
, we get
Set
, and
, we get:
(7.2)
From (7.2), L(x) can be confirmed there is an infinite number.
The twin prime conjecture is correct.
8. Conclusion
We discuss the prime distribution
The twin prime conjecture is discussed
The twin prime conjecture is correct.