1742 On June 7, the German mathematician Goldbach put forward two bold conjectures in a letter to the famous mathematician Euler:

1. Any even number not less than 6 is the sum of two odd prime numbers;

2. Any odd number not less than 9 is the sum of three odd prime numbers.

This is the famous Goldbach conjecture in the history of mathematics. Obviously, the second guess is the inference of the first guess. So it is enough to prove one of the two conjectures.

On June 30th of the same year, Euler made it clear in his reply to Goldbach that he was convinced that both Goldbach's conjectures were correct theorems, but Euler could not prove them at that time. Because Euler was the greatest mathematician in Europe at that time, his confidence in Goldbach's conjecture influenced the whole mathematics field in Europe and even the world. Since then, many mathematicians are eager to try and even devote their lives to proving Goldbach's conjecture. However, until the end of 19, there was still no progress in proving Goldbach's conjecture. The proof of Goldbach's conjecture is far more difficult than people think. Some mathematicians compare Goldbach's conjecture to "the jewel in the crown of mathematics".

1900, Hilbert, the greatest mathematician in the 20th century, listed Goldbach conjecture as one of the 23 mathematical problems at the International Mathematical Congress. Since then, mathematicians in the 20th century have "joined hands" to attack the world's "Goldbach conjecture" fortress, and finally achieved brilliant results.

The main methods used by mathematicians in the 20th century to study Goldbach's conjecture are screening method, circle method, density method, triangle method and so on. The way to solve this conjecture, like "narrowing the encirclement", is gradually approaching the final result.

1920, the Norwegian mathematician Brown proved the theorem "9+9", thus delineating the "great encirclement" that attacked "Goldbach conjecture". What is this "9+9"? The so-called "9+9", translated into mathematical language, means: "Any large enough even number can be expressed as the sum of two other numbers, and each of these two numbers is the sum of nine odd prime numbers." Starting from this "9+9", mathematicians all over the world concentrated on "narrowing the encirclement", and of course the final goal was "1+ 1".

1924, the German mathematician Redmark proved the theorem "7+7". Soon, "6+6", "5+5", "4+4" and "3+3" were captured. 1957, China mathematician Wang Yuan proved "2+3". 1962, China mathematician Pan Chengdong proved "1+5", and cooperated with Wang Yuan to prove "1+4" in the same year. 1965, Soviet mathematicians proved "1+3".

1966, Chen Jingrun, a famous mathematician in China, conquered "1+2", that is, "any even number large enough can be expressed as the sum of two numbers, one of which is an odd prime number and the other is the sum of two odd prime numbers." This theorem is called "Chen Theorem" by the world mathematics circle.

Thanks to Chen Jingrun's contribution, mankind is only one step away from the final result of Goldbach's conjecture "1+ 1". But in order to achieve this last step, it may take a long exploration process. Many mathematicians believe that to prove "1+ 1", new mathematical methods must be created, and the previous methods are probably impossible.

Goldbach's letter and conjecture

1742 On June 7th, the mathematician Goldbach (1690- 1764), who was born in Germany and later worked and settled in Russia, wrote a letter from Moscow to Euler, a famous Swiss mathematician who was working at the Berlin Academy of Sciences at that time. The full text of the letter reads as follows:

Euler, my dear friend!

I am deeply encouraged by your ingenious and simple way to solve the problem of "Seven Bridges", which has been dumped by thousands of people and is puzzling. He has been urging me to advance on the road of mathematics. After full deliberation, I want to take a risk and make a guess. Now I am writing to you for your advice. My questions are as follows:

Take any odd number, such as 77, which can be written as the sum of three prime numbers:

77=53+ 17+7

Then, take an odd number 46 1

46 1=449+7+5

It is also the sum of three prime numbers. 46 1 can also be written as

257+ 199+5

It is still the sum of three prime numbers.

In this way, I found that any odd number greater than 5 is the sum of three prime numbers.

But how to prove it? Although the above results can be obtained in any experiment, it is impossible to test all odd numbers. What is needed is general proof, not individual inspection. Can you help?

Goldbach, June 1 day

After reading Goldbach's letter, Euler was attracted by the genius conjecture in the letter, and at the same time, he admired his old friend more.

Goldbach, an East Prussian, 1690 was born in Konigsberg, the hometown of "Seven Bridges". He was an envoy to Russia in his early years. He has been an academician of Petersburg Academy of Sciences since 1725. Two years later, when Euler came to the Academy of Sciences in Petersburg, they became friends. They have kept in correspondence for more than 30 years.

Goldbach mainly studies differential equations and series theory. He likes to communicate with others and discuss math problems.

On June 30th of the same year, Euler said in his reply to Goldbach:

Goldbach, my old friend, hello!

Thank you for praising me in your letter!

I have carefully scrutinized and studied your proposition, and it seems to be correct. However, I can't give a strict proof. Here, based on you, I think:

Any even number greater than 2 is the sum of two prime numbers.

However, this proposition cannot give a general explanation, but I am sure it is completely correct.

June 30, Euler

Later, Euler published their Letter to the World, calling on mathematicians all over the world to help solve this number theory problem. At that time, the mathematical circles called the problems involved in their communication "Goldbach conjecture".

Because western mathematicians are used to treating 1 as a prime number, 4= 1+3, 7= 1+3+3 is also considered as the correct decomposition. Today, this conjecture is roughly summarized as follows:

(1) Even numbers greater than 6 can be expressed as the sum of two odd prime numbers.

(2) Odd numbers greater than 9 can be expressed as the sum of three odd prime numbers.

Goldbach conjecture has been published for more than 250 years. Although countless mathematicians have tried to solve this conjecture, it is still a "conjecture" that has not been proved or refuted so far.

/kloc-Cantor, a famous mathematician in the 9th century, patiently tested all even numbers below 1000, and Hoppeler tested all even numbers between 1000 and 2000, and the conjecture was established.

1900, david hilbert included Goldbach's conjecture in 23 difficult problems and introduced it to mathematicians in the 20th century.

19 12 at the fifth international congress of mathematicians, the famous mathematician Landau said: "Even if Goldbach's problem is replaced by a weaker proposition (3), it is beyond the power of modern mathematicians."

The content of proposition (3) is that no matter whether there are no more than 3 or 30, as long as you want to prove the existence of such a positive number C, every integer greater than 2 can be expressed as the sum of no more than C prime numbers.

192 1 year, Thomas Hardy, a famous British mathematician, said at the international mathematical conference held in Copenhagen that Goldbach's conjecture was as difficult as any unsolved mathematical problem.

1930, the 25-year-old mathematician Snelman of the Soviet Union created the "density method". Combined with the "screening method" founded by Norwegian mathematician Braun in 1920, the proposition (3) is successfully proved, and it is estimated that this number will not exceed k, and k

The success of sneering at Lerman was a major breakthrough in the research history of Goldbach conjecture at that time, which greatly encouraged mathematicians to attack Goldbach conjecture. The value of k also shrinks with the attack of warriors:

1935 k

1936 k

1937 k

1950 k

1956 k

1976 k

1937, vinogradov of the Soviet Union proved that an odd number large enough can be expressed as the sum of three odd prime numbers by applying Hardy's and Litowood's "circle method" and his own "triangle sum method", which is equivalent to Snelmann's K.

On the route of attacking Goldbach's conjecture, people have also come up with a way to write even numbers as the sum of two natural numbers, and then try to reduce the number of prime factors of these two natural numbers. If these two numbers become 1 and 1, it is the sum of two prime numbers called 1+ 1. This proposition is called the factor Goldbach problem.

China's famous mathematician Hua proved in 1930s that almost all even numbers are the sum of two prime numbers.

1920, the Norwegian mathematician Braun created the "screening method" and used him to prove 9+9.

1924 7+7 (Rademacher)

1932 6+6 (Hysmans, UK)

1937 5+7, 4+9, 3+5 (Richie, Italy)

1938 5+5 (Buch Hita, USSR)

1940 4+4 (same as above)

1956 3+4 (Wang Yuan, China)

1956 3+3 (Hita, Soviet Union)

1957 2+3 (Wang Yuan, China)

1962 1+5 (China Pan Chengdong)

1963 1+4 (Wang Yuan, China)

1965 1+3 (vinogradov, Soviet Union, BuchHita, Bambier)

1966 1+2

Chinese mathematician Chen Jingrun proved that "1+2" is the best result so far. It is only one step away from "1+ 1", but after more than 30 years, there is no obvious progress so far. We experienced the excitement that Fermat's Last Theorem was broken, and we are more eager for China mathematicians to continue to create more brilliant achievements in the international mathematics field.