"In the past 20 years, there has been no substantial progress in proving Goldbach's conjecture." Chen Mufa, a professor of mathematics at Beijing Normal University who will give a 45-minute report at this international congress of mathematicians, said, "Its proof is only the last step. If the research makes essential progress, then the conjecture will eventually be solved. "
According to Chen Mufa, in 2000, an international organization listed seven Millennium problems in the field of mathematics, and offered a reward of10 million dollars to solve them, but it did not include Goldbach's conjecture.
"In recent years or even more than ten years, Goldbach's conjecture is still difficult to prove." Gong Fuzhou, a researcher at the Institute of Mathematics and System Science, Chinese Academy of Sciences, analyzed this, and now conjecture has become an isolated problem, which is not closely related to other mathematics disciplines. At the same time, researchers also lack effective ideas and methods to finally solve this famous conjecture. "Mr. Chen Jingrun has used the existing methods to the extreme before his death."
Becker, a professor at Cambridge University and a Fields Prize winner, also said that the progress made by Chen Jingrun in this work is the best verification result so far, and there is no greater breakthrough at present.
"It may be difficult to make progress in solving this kind of mathematical problems for one or two hundred years, or it may make significant progress in a short time." In Gong Fuzhou's view, there is a certain contingency in mathematical research, which may make people make progress in conjecture proof in advance.
Conjecture and verification require new ideas.
In order to solve the "challenging problems in core mathematics", the Institute of Mathematics and System Science of Chinese Academy of Sciences has set up a special international research team. Li Fuan, head and researcher of the institute, said: "We expect to make breakthroughs in areas such as Riemann conjecture. This research team did not take Goldbach conjecture as the direction of efforts. "
Chen Jingrun, the mathematician closest to "the jewel in the crown", left us on 1996. His achievements once aroused people's enthusiasm for Goldbach's conjecture. In March 2000, two British and American publishing companies offered a million dollars for the final solution of Goldbach's conjecture, which once again made it a hot spot of social concern. Two years have passed, and no one has come to receive this bonus until the deadline.
It is estimated that there are about twenty or thirty people in the world who have the ability to test the conjecture. For the final solution of this famous conjecture, Pan Chengdong once wrote that it is impossible to solve this conjecture in the way people imagine. We must make major improvements or propose new methods to the related methods, so as to achieve further research results on conjecture. Wang Yuan's judgment is basically similar to this: "The further study of Goldbach's conjecture must have a brand-new idea." Wang Yuan and Pan Chengdong, as famous mathematicians in China, have made great contributions to the proof of conjecture.
"Mathematical research is not just about solving difficult problems. I don't approve of one-sided hype about these issues. In my opinion, people who study these mathematical problems are less than 1% of mathematicians all over the world. " Chen Mufa believes that "mathematical research does not have to answer questions raised by others. It is necessary to do more original research and pay attention to the improvement of the overall research strength. "
How far is the "folk mathematician" from the "Pearl"?
On the eve of the opening of the International Congress of Mathematicians, some "folk mathematicians" came to Beijing one after another, claiming that they "completely proved" Goldbach's conjecture and aroused social concern.
In fact, in recent years, the Chinese people have been visiting many mathematicians in turn with the "final proof result" of conjecture, and from time to time, there have been "explosive news" such as "the farmer successfully proved Goldbach conjecture" and "the tractor hand won the" crown jewel ".
"With the approach of the conference, more and more manuscripts about conjecture research results have been received by mathematics." Li Fuan, a researcher at China Academy of Sciences, said: "Over the past 20 years, there have been thousands of amateurs, and I have received more than 200 letters. Their topic mainly focuses on Goldbach's conjecture. Because the conjecture is very concise and most people can understand it, many people want to solve this problem. "
"People's enthusiasm for science should be protected, but we don't advocate people to attack the world's mathematical problems. They can use this enthusiasm to do more appropriate things. " Li Fuan said, "It can be seen from the submissions that many authors lack basic mathematical literacy and don't look at other people's mathematical papers, and the results are all wrong."
"Foreign countries also have this phenomenon. For example, during the International Congress of Mathematicians in Berlin, someone posted a paper at the venue, claiming that he had proved it (1+ 1). " Wu Wenjun, winner of the first National Science and Technology Award and chairman of the current International Congress of Mathematicians, said: "Some amateurs know a little mathematics and have a little arithmetic foundation, so they will verify it (1+ 1) and send me the so-called proof papers. In fact, problems like Goldbach's conjecture should be left to' experts' to solve and should not become' mass movements'. "
To this end, many mathematicians give advice to math lovers: "If you really want to make achievements in the proof of Goldbach's conjecture, you'd better master the corresponding mathematical knowledge systematically first and avoid taking unnecessary detours."
News background: There is one last step to the "jewel in the crown".
Xinhuanet Beijing, August 20th (Reporter Li Bin, Zhang Jingying, Zou Wensheng) Xu Chi's famous reportage made hundreds of millions of ordinary people know that "the queen of natural science is mathematics; The crown of mathematics is number theory; Goldbach guessed that it was the jewel in the crown ",and he also knew that Chen Jingrun was the closest person to that jewel in the world-just the last step. But after more than 20 years, no one can cross this step.
Goldbach conjecture has been speculated by human beings for 260 years. 1742, the German mathematician Goldbach wrote to the great mathematician Euler, proposing that every even number not less than 6 is the sum of two prime numbers (referred to as "1+ 1"). For example, 6 = 3+3, 24 = 1 1+ 13 and so on. Euler wrote back that he believed the conjecture was correct, but he couldn't prove it.
Since then, in the recent 170 years, many mathematicians have made great efforts to conquer it, but they have not made a breakthrough. Until 1920, the Norwegian mathematician Brown finally got closer to it, and proved that every big even number is the product of nine prime factors plus nine prime factors, that is, (9+9).
Since then, the "encirclement circle" of conjecture has been shrinking. 1924, German mathematician Rad mahar proved (7+7). 1932, the British mathematician eissmann proved (6+6). 1938, the Soviet mathematician Buchstaber proved (5+5), and proved (4+4) two years later. 1956, the Soviet mathematician vinogradov proved (3+3). 1958, China mathematician Wang Yuan proved (2+3) again. 1962 China mathematician pan chengdong proved (1+5) and Wang Yuan proved (1+4); 1965, Buchstaber and others proved it again (1+3). The "encirclement circle" is getting smaller and smaller and closer to the final goal (1+ 1).
1966, China mathematician Chen Jingrun became the closest person to this pearl in the world-he proved it (1+2). His achievements are in a leading position in the world and are called "Chen Theorem" by the international mathematics community. Due to his outstanding achievements in the study of Goldbach's conjecture, 1982, Chen Jingrun won the first prize of the National Natural Science Award together with Wang Yuan and Pan Chengdong.
Since Chen Jingrun's proof (1+2), the last step of Goldbach's conjecture-proof (1+ 1) has not made essential progress. Relevant experts believe that the original method has been used to the extreme, and it is necessary to put forward new methods and adopt new ideas, so as to get further research results in conjecture. (End)
Attached:
Introduction to Goldbach conjecture
In Xu Chi's reportage, China people know the conjectures of Chen Jingrun and Goldbach.
So, what is Goldbach conjecture?
Goldbach conjecture can be roughly divided into two kinds of conjecture:
■ 1. Every even number not less than 6 is the sum of two odd prime numbers;
■2. Every odd number not less than 9 is the sum of three odd prime numbers.
■ Goldbach correlation
Goldbach is a German middle school teacher and a famous mathematician. He was born in 1690, and was elected as an academician of Russian Academy of Sciences in 1725.
A brief history of Goldbach conjecture
1742, Goldbach found in his teaching that every even number not less than 6 is the sum of two prime numbers (numbers that can only be divisible by 1 and itself). For example, 6 = 3+3, 12 = 5+7 and so on. 1742 On June 7th, Goldbach wrote to the great mathematician Euler at that time. In his reply on June 30th, Euler said that he thought this conjecture was correct, but he could not 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+/kloc. Someone checked the even numbers within 33× 108 and above 6 one by one, 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.
It was not until the 1920s that people began to approach it. 1920, the Norwegian mathematician Brown proved by an ancient screening method, and reached a conclusion that every even number with a large ratio can be expressed as (99). 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 large enough even number is the sum of a prime number and a natural number, while the latter is only the product of two prime numbers." This result is often called a big even number and can be expressed as "1+2".
■ Goldbach conjecture proves the relevance of progress
Before Chen Jingrun, the progress of even numbers can be expressed as the sum of the products of S prime numbers and T prime numbers (referred to as the "s+t" problem) as follows:
1920, Norway Brown 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 "1+ c", where c is a large natural number.
1956, Wang Yuan of China proved "3+4".
1957, China and Wang Yuan successively 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 Taber and vinogradov Jr. of the Soviet Union and Pemberley of Italy proved "1+3".
1966, China Chen Jingrun proved "1+2".
It took 46 years from Brown's proof of 1920 of "9+9" to Chen Jingrun's capture of 1966 of "+2". Since the birth of Chen Theorem for more than 40 years, people's further research on Goldbach conjecture is futile.
■ Brownian sieve correlation
The idea of Brownian screening method is as follows: any even number (natural number) can be written as 2n, where n is a natural number, and 2n can be expressed as the sum of a pair of natural numbers in n different forms: 2n =1+(2n-1) = 2+(2n-2) = 3+(2n-3) = 2i and 2i. 3j and (2n-3j), j = 2, 3, ...; And so on), if it can be proved that at least one pair of natural numbers is not filtered out, such as p 1 and p2, then both p 1 and p2 are prime numbers, that is, n=p 1+p2, then Goldbach's conjecture is proved. The description in the previous part is a natural idea. The key is to prove that' at least one pair of natural numbers has not been filtered out'. No one in the world can prove this part yet. If it can be proved, this conjecture will be solved.
However, because the big even number n (not less than 6) is equal to the sum of odd numbers of its corresponding odd number series (starting with 3 and ending with n-3). Therefore, according to the sum of odd numbers, prime+prime (1+ kloc-0/) or prime+composite (1+2) (including composite+prime 2+ 1 or composite+composite 2+2) (Note:/kloc) That is, the occurrence "category combination" of 1+ 1 or 1+2 can be derived as 1+ 1 and 1+2. Because 1+2 and 2+2 and 1+2 do not contain1+. So 1+ 1 does not cover all possible "category combinations", that is, its existence is alternating. So far, if the existence of 1+2 and 1+2 can be excluded, it is proved that 1+ 1 But the fact is that 1+2 and 2+2, and 1+2 (or at least one of them) are some laws revealed by Chen's theorem (any large enough even number can be expressed as the sum of two prime numbers, or the sum of the products of one prime number and two prime numbers), such as the existence of 1+2 and the coexistence of 6542. Therefore, 1+2 and 2+2, and 1+2 (or at least one) "category combination" patterns are certain, objective and inevitable. So 1+ 1 is impossible. This fully shows that the Brownian sieve method cannot prove "1+ 1".
Because the distribution of prime numbers itself changes in disorder, there is no simple proportional relationship between the change of prime number pairs and the increase of even numbers, and the value of prime number pairs rises and falls when even numbers increase. Can the change of prime pairs be related to the change of even numbers through mathematical relations? Can't! There is no quantitative law to follow in the relationship between even values and their prime pair values. For more than 200 years, people's efforts have proved this point, and finally they choose to give up and find another way. So people who used other methods to prove Goldbach's conjecture appeared, and their efforts only made progress in some fields of mathematics, but had no effect on Goldbach's conjecture.
Goldbach conjecture is essentially the relationship between an even number and its prime number pair, and the mathematical expression expressing the relationship between even number and its prime number pair does not exist. It can be proved in practice, but the contradiction between individual even numbers and all even numbers cannot be solved logically. How do individuals equal the average? Individuals and the general are the same in nature, but opposite in quantity. Contradictions will always exist. Goldbach conjecture is a mathematical conclusion that can never be proved theoretically and logically.
The significance of Goldbach conjecture
"In contemporary languages, Goldbach conjecture has two contents, the first part is called odd conjecture, and the second part is called even conjecture. Odd number conjecture points out that any odd number greater than or equal to 7 is the sum of three prime numbers. Even conjecture means that even numbers greater than or equal to 4 must be the sum of two prime numbers. " (Quoted from Goldbach conjecture and Pan Chengdong)
I don't want to say more about the difficulty of Goldbach's conjecture. I want to talk about why modern mathematicians are not interested in Goldbach conjecture, and why many so-called folk mathematicians in China are interested in Goldbach conjecture.
In fact, in 1900, the great mathematician Hilbert made a report at the World Congress of Mathematicians and raised 23 challenging questions. Goldbach conjecture is a sub-topic of the eighth question, including Riemann conjecture and twin prime conjecture. In modern mathematics, it is generally believed that the most valuable is the generalized Riemann conjecture. If Riemann conjecture holds, many questions will be answered, while Goldbach conjecture and twin prime conjecture are relatively isolated. If we simply solve these two problems, it is of little significance to solve other problems. So mathematicians tend to find some new theories or tools to solve Goldbach's conjecture "by the way" while solving other more valuable problems.
For example, a very meaningful question is: the formula of prime numbers. If this problem is solved, it should be said that the problem of prime numbers is not a problem.
Why are folk mathematicians so obsessed with Kochi conjecture and not concerned about more meaningful issues such as Riemann conjecture?
An important reason is that Riemann conjecture is difficult for people who have never studied mathematics to understand its meaning. Goldbach guessed that primary school students could watch it.
It is generally believed in mathematics that these two problems are equally difficult.
Folk mathematicians mostly use elementary mathematics to solve Goldbach conjecture. Generally speaking, elementary mathematics cannot solve Goldbach's conjecture. To say the least, even if an awesome person solved Goldbach's conjecture in the framework of elementary mathematics that day, what's the point? I'm afraid this solution is almost as meaningful as doing a math exercise.
At that time, brother Bai Dili challenged the mathematical world and put forward the problem of the fastest descent line. Newton solved the steepest descent line equation with extraordinary calculus skills, John Parker tried to solve the steepest descent line equation skillfully with optical methods, and Jacob Parker tried to solve this problem in a more troublesome way. Although Jacob's method is the most complicated, he developed a general method to solve this kind of problems-variational method. Now, Jacob's method is the most meaningful and valuable.
Similarly, Hilbert once claimed to have solved Fermat's last theorem, but he did not announce his own method. Someone asked him why, and he replied, "This is a chicken that lays golden eggs. Why should I kill it? " Indeed, in the process of solving Fermat's last theorem, many useful mathematical tools have been further developed, such as elliptic curves and modular forms.
Therefore, modern mathematics circles are trying to study new tools and methods, expecting Goldbach's conjecture to give birth to more theories.
Wrong examples of proof of Goldbach's conjecture
Goldbach's conjecture formula and Gao Chi's proof of Goldbach's conjecture: Let the even number be m and the prime number deletion factor be √M≈N, then the even number's odd prime number deletion factor is: 3,5,7, 1 1…N, 1, even number (65438). 2. If the even number can be divisible by the odd prime number deletion factor L ... the even number of a prime number pair is the smallest prime number pair *(L- 1)/(L-2), for example, the even number can be divisible by prime number 3, that is ≥(3- 1) /(3-2)*N/4=N/2. When the even number is greater than 6 and less than 14, everyone knows that Goldbach conjecture (1+ 1) has a solution. According to the positive formula of Goldbach conjecture above, even numbers (1+1) greater than1all hold.
Guess: Goldbach conjecture 1: any one >; Even number =6 can be expressed as the addition of two prime numbers.
I guess: the final number of any odd prime number must be 1, 3,5,7,9 (where 1, 9 is at least two digits, such as1,19).
So there are: 1+ 1, 1+3, 1+5, 1+7, 1+9,
3+3,3+ 1,3+5,3+7,3+9,
5+5,5+ 1,5+3,5+7,5+9,
7+7,7+ 1,7+3,7+5,7+9,
9+9,9+ 1,9+3,9+5,9+7,
(Can be added to prime numbers with multiple digits)
The sum obtained must end with 0, 2, 4, 6, 8 (both must be even numbers greater than or equal to 6).
This sum must be an even number greater than or equal to 6.
But this may not fill all even numbers, so this method is wrong! The conditions are not sufficient!
Seeking adoption is a satisfactory answer.