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 ,