(a) Any even number ≥6 can be expressed as the sum of two prime numbers.

(b) Any odd number ≥9 can be expressed as the sum of no more than three prime numbers.

This is the famous Goldbach conjecture. Euler wrote back to him on June 30th, saying that he believed the conjecture was correct, but he couldn't prove it. Describing such a simple problem, even a top mathematician like Euler can't prove it. This conjecture has attracted the attention of many mathematicians. Since Goldbach put forward this conjecture, many mathematicians have been trying to conquer it, but they have not succeeded. Of course, some people have done some specific verification work, such as:

6 = 3 + 3, 8 = 3 + 5, 10 = 5 + 5 = 3 + 7, 12 = 5 + 7, 14 = 7 + 7 = 3 + 1 1, 16 = 5

+ 1 1, 18 = 5 + 13,

..... and so on. Someone checked the even numbers within the 8th power of 33× 10 and greater than 6, and Goldbach conjecture (a) was established. But strict mathematical proof requires the efforts of mathematicians.

Since then, this famous mathematical problem has attracted the attention of thousands of mathematicians all over the world. 200 years have passed and no one has proved it. Goldbach conjecture has therefore become an unattainable "pearl" in the crown of mathematics.

People's enthusiasm for Goldbach conjecture lasted for more than 200 years. Many mathematicians in the world try their best, but they still can't figure it out.

significant development

It was not until the 1920s that people began to approach it. 1920, the Norwegian mathematician Brown used an ancient screening method to prove and conclude that every even number greater than 6 can be expressed as (9+9). This method of narrowing the encirclement is very effective, so scientists gradually reduce the prime factor in each number from (99) until each number is a prime number, thus proving Goldbach's conjecture.

At present, the best result is proved by China mathematician Chen Jingrun in 1966, which is called Chen Theorem: "Any sufficiently large even number is the sum of a prime number and a natural number, while the latter is only the product of at most two prime numbers." [1] This result is usually called a big even number and can be expressed as "1+2".

Before Chen Jingrun, the progress of even numbers can be expressed as the sum of the products of at most S prime numbers and at most T prime numbers (referred to as the "s+t" problem) as follows:

1920, Brown of Norway proved "9+"

9"。

1924, Latmach of Germany proved "7+7".

1932, Esterman proved "6+6".

1937, Lacey in Italy successively proved "5+7", "4+9", "3+ 15" and "2+366".

1938, Bukit Tiber of the Soviet Union proved "5+5".

1940, Bukit Tiber of the Soviet Union proved "4

+ 4"。

1948, Rini of Hungary proved that the existence of C made "1+C" hold.

1956, Wang Yuan of China proved "3+"

4"。

1957, Wang Yuan of China proved "3+3" and "2+3".

1962, Pan Chengdong of China and Barba of the Soviet Union proved "1+5", and Wang Yuan of China proved "1+4".

1965, Buchwitz and George W. vinogradov of the Soviet Union and Pompeii of Italy proved "1+3".

1966, certified by China Chen Jingrun.

" 1+2"。

It took 46 years from Brown's proof of 1920 of "9+9" to Chen Jingrun's capture of 1966 of "+2". For more than 40 years since the birth of "Chen Theorem", people's further research on Goldbach's conjecture has been in vain.

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Can anyone prove that 1+ 1=2? Because 2- 1= 1, 1+ 1=2.

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Why doesn't anyone understand 1+ 1? Because someone has already

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Has anyone proved that 1+ 1=2? have

Jingrun Chen

First, the proof method

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.

1742 On June 7th, Goldbach wrote to Euler, a great mathematician at that time, and formally put forward the following conjecture: A. Any even number greater than 6 can be expressed as the sum of two prime numbers. B Any odd number greater than 9 can be expressed as the sum of three prime numbers.

This is Goldbach's conjecture. Euler wrote back that he believed the conjecture was correct, but he couldn't prove it.

Since then, this mathematical problem has attracted the attention of almost all mathematicians. Goldbach conjecture has therefore become an unattainable "pearl" in the crown of mathematics.

China mathematician Chen Jingrun proved in 1966 that any sufficiently large even number is the sum of a prime number and a natural number, which can be expressed as the product of two prime numbers. "Usually this result is expressed as 1+2. This is the best result of this problem at present.

If you want to understand Chen Jingrun's strict proof, I'm afraid most friends who have no foundation in number theory can't do it at all.

Briefly describe:

In 194 1, P. Kuhn put forward the weighted screening method, which can strengthen the effect of other screening methods. Today, many important results about screening methods are related to this idea.

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