Let n be any even number greater than 6 and Gn be a positive integer not greater than N/2, then:

N=(N-Gn)+Gn ( 1)

If n-Gn and Gn are not divisible by all prime numbers not greater than √N at the same time, then n-Gn and gn are both odd prime numbers. Let Gp(N) represent the number of odd prime numbers Gp, where N-Gp and Gp are both odd prime numbers, then only need to prove:

When n > m, there is gp (n) > 1, then Goldbach conjecture holds when n > m.

Second, the double number screening method

Let Gn be a natural number from 1 to N/2 and Pi be an odd prime number not greater than √N, then the total number of natural numbers corresponding to Gn is N/2. If any number of n-Gn and Gn can be divisible by odd prime Pi, the natural number corresponding to this Gn is filtered out, so that the number of natural numbers corresponding to the Gn filtered by odd prime Pi is not greater than INT(N/Pi), the number of natural numbers corresponding to the remaining Gn is not less than n/2-int (n/pi), and the ratio of the total number of natural numbers corresponding to Gn is r (.

r(Pi)≥(N/2-INT(N/Pi))/(N/2)≥( 1-2/Pi)×INT((N/2)/Pi)/((N/2)/Pi)(2)

Third, the estimation formula

Since all prime numbers are coprime, we can apply the cross product formula of independent events in set theory, and from formula (2), we can get the estimation formula of any number, in which an even number is the sum of two odd prime numbers:

gp(N)≥(N/4- 1)×∏R(Pi)- 1≥(N/4- 1)×∏( 1-2/Pi)×∏( 1-2Pi/N)- 1(3)

Where ∏R(Pi) represents the product of ratio formulas corresponding to all odd prime numbers not greater than √ n.

Fourth, simple proof.

When the even number N≥ 10000, it can be obtained by formula (3):

gp(N)≥(N/2-2-∑Pi)×( 1- 1/2)×∏( 1-2/Pi)- 1

≥(N-2×√N)/8×( 1/√N)- 1 =(√N-2)/8- 1≥ 1 1 > 1(4)

Formula (4) shows that every even table greater than 10000 is the sum of two odd prime numbers, and there are at least 1 1 table methods.

Experience has proved that every even number greater than 4 and not greater than 10000 can be expressed as the sum of two odd prime numbers.

Final conclusion: Every even number greater than 4 can be expressed as the sum of two odd prime numbers.

(December 24th, 1986)

Goldbach conjecture is one of the three major mathematical problems in the modern world. 1742 was first discovered by German middle school teacher Goldbach in teaching.

Or let P_x( 1, 2) be the number of prime numbers p suitable for the following conditions:

X-p=p_ 1 or x-p=(p_2)*(p_3)

Where p _ 1, p _ 2 and p _ 3 are all prime numbers.

X represents a sufficiently large even number.

Life CX = {∏ p | x, p2} (p-1)/(p-2) {∏ p2} (1-1(p-1) 2)

For any given even number H and sufficiently large X, xh( 1, 2) is used to represent the number of prime numbers P satisfying the following conditions:

P ≤ x, p+h = p _ 1 or h+p = (p _ 2) * (p _ 3),

Where p _ 1, p _ 2 and p _ 3 are all prime numbers. ：