﻿ The Distribution of Prime Numbers in an Interval

American Journal of Mathematics and Statistics

p-ISSN: 2162-948X    e-ISSN: 2162-8475

2015;  5(6): 325-328

doi:10.5923/j.ajms.20150506.01

### The Distribution of Prime Numbers in an Interval

Jian Ye

Department of Mathematics Sichuan University, Chengdu, China

Correspondence to: Jian Ye, Department of Mathematics Sichuan University, Chengdu, China.
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Copyright © 2015 Scientific & Academic Publishing. All Rights Reserved.

This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/

Abstract

The Goldbach theorem and the twin prime theorem are homologous. The paper from the prime origin, derived the equations of the twin prime theorem and the Goldbach theorem, and new prime number theorem.

Keywords: the Goldbach Theorem, the Twin Prime Theorem, Prime Number Theorem

Cite this paper: Jian Ye, The Distribution of Prime Numbers in an Interval, American Journal of Mathematics and Statistics, Vol. 5 No. 6, 2015, pp. 325-328. doi: 10.5923/j.ajms.20150506.01.

### 1. Notation

a prime number.
p: an odd prime.
the number of primes in the open interval .
the number of twin prime pairs in the open interval .
the number of prime in the open interval . p is the largest prime number less than , and is prime number, is a large even integer.
denotes not more than p of prime numbers.
divides.
denotes equialence relation. , namely: , when tends to infinity.
mean big O notation describes the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions.
express the logarithmic integral function or integral logarithm is a special function such as .

### 2. Prime Number Theorem [1] [2]

Let is the number of primes in the open interval . p is an odd prime,
 (1)
where is a prime number.
is not more than p of prime numbers.
Lemma 1
Let is the number of primes in the open interval . p is an odd prime,
Let is the number of odd between to
and is a prime number, p is an odd prime,
Let
 (2)
where is not more than p of prime numbers.
The proof of lemma 1
Reduction to absurdity.
The proof of prime number theorem
Proof
By lemma 1 and Chinese remainder theorem, it can be derived
 (3)
Hence proving
where is a prime number. is not more than p of prime numbers.

### 3. The Twin Prime Theorem

Let is the number of twin prime pairs in the open interval , p is an odd prime,
 (4)
where is a prime number.
is not more than p of prime numbers.
Among which
 (5)
Lemma 2
Let is the number of twin prime pairs in the open interval is a prime number, p is an odd prime.
Let is the number of the odd between to and is a prime number.p is an odd prime.
Let
 (6)
Where is not more than p of prime numbers.
The proof of lemma 2
Reduction to absurdity.
The proof of twin prime theorem
Proof
By lemma 2 and Chinese remainder theorem,it can be derived
 (7)
Hence proving
 (8)
or
where is a prime number.
is not more than p of prime numbers.
Among which

### 4. The Goldbach Theorem

Let is the number of prime in theopen interval is the largest prime number less than and is prime number, is alarger even integer.
 (9)
where is a prime number.
is not more than p of prime numbers.
Since
 (10)
when
 (11)
Lemma 3
Let is the number of prime in the open interval is the largest prime number less than and is prime number, is a large even integer.
Let is the number of odd between to and is a prime number, p is the largest prime number less than is a large even integer.
Let
 (12)
where is not more than p of prime numbers.
The proof of lemma 3
Reduction to absurdity.
The proof of the Goldbach theorem
Proof
By lemma 3 and Chinese remainder theorem, it can be derived
 (13)
where is a prime number.
or
 (14)
Hence proving
 (15)
or
where is a prime number.
is not more than p of prime numbers.
Since
when ,

### References

 [1] J.B. Rosser and L. Scloenfeld, approximate formulas for some functions of prime numbers. Lllinois J. Math. Volume 6, Issue 1(1962), 64-94. [2] G.H. Hardy and E.M. Wright, An Introduction to The Theory of Numbers, section 22.8 and 22.19. The Oxford University Press, 4ed, 1959.