What does the jewel in the crown of mathematics mean?

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 can be expressed as the sum of two odd prime numbers;

■2. Every odd number not less than 9 can be expressed as the sum of three odd prime numbers.

■ Goldbach correlation [

Goldbach C.,1690.3.18 ~1764.5438+01.20) is a German mathematician. Born in Konigsberg (now Kalinin); Studied at Oxford University in England; I originally studied law, and I became interested in mathematical research because I met the Bernoulli family when I visited European countries. I used to be a middle school teacher. /kloc-arrived in Russia in 0/725, and was elected as an academician of Petersburg Academy of Sciences in the same year. 1725 to 1740 as conference secretary of the Academy of Sciences in Petersburg; From 65438 to 0742, he moved to Moscow and worked in the Russian Foreign Ministry.

[Edit this paragraph] The source of Goldbach's conjecture

From 1729 to 1764, Goldbach kept correspondence with Euler for 35 years.

In the letter 1742 to Euler on June 7th, Goldbach put forward a proposition. He wrote:

"My question is this:

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

77=53+ 17+7;

Take an odd number, such as 46 1,

46 1=449+7+5,

It is also the sum of these three prime numbers. 46 1 can also be written as 257+ 199+5, which is still the sum of three prime numbers. In this way, I found that any odd number greater than 7 is the sum of three prime numbers.

But how can this be proved? Although the above results are obtained in every experiment, it is impossible to test all odd numbers. What is needed is a general proof, not another test. "

Euler wrote back: "This proposition seems correct, but he can't give a strict proof. At the same time, Euler put forward another proposition: any even number greater than 2 is the sum of two prime numbers, but he failed to prove this proposition. "

It is not difficult to see that Goldbach's proposition is the inference of Euler's proposition. In fact, any odd number greater than 5 can be written in the following form:

2N+ 1=3+2(N- 1), where 2(N- 1)≥4.

If Euler's proposition holds, even number 2(N- 1) can be written as the sum of two prime numbers, and odd number 2N+ 1 can be written as the sum of three prime numbers, so Goldbach conjecture holds for odd numbers greater than 5.

But the establishment of Goldbach proposition does not guarantee the establishment of Euler proposition. So Euler's proposition is more demanding than Goldbach's proposition.

Now these two propositions are collectively called Goldbach conjecture.

[Edit this paragraph] A brief history of Goldbach's 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: every even n greater than it (not less than 6) can be expressed as the product of nine prime numbers plus the product of nine prime numbers, which is called 9+9 for short. 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] hershey 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 of England 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".

[Edit this paragraph] 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 about 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. ]

[Edit this paragraph] Reportage: Goldbach conjecture

One,

Let px( 1, 2) be the number of prime numbers p suitable for the following conditions: x-p=p 1 or x-p=P2p3, where p 1, p2 and p3 are all prime numbers. [This is not easy to understand; You can skip these lines when you don't understand. X represents a sufficiently large even number.

p- 1 1

Life CX = II-II 1-

p \ x p-2p & lt； 2 (p- 1)2

p & gt2

For any given even number H and sufficiently large X, XH (1, 2) is used to represent the number of prime numbers P that meet the following conditions: p≤x, p+h=p 1 or h+p=p2p3, where p 1, P2 and p3 are all prime numbers. The purpose of this paper is to prove and improve all the results mentioned by the author in [10], as follows.

Second,

The above is quoted from a paper on analytic number theory. This passage is quoted from its "introduction" and raises this question. Followed by "(2) several lemmas", which are full of various formulas and calculations. Finally, "(3) Result" proves a theorem. This paper is extremely difficult to understand. Even a famous mathematician may not understand this branch of mathematics unless he specializes in it. However, this paper has been recognized by the international mathematics community and enjoys a good reputation all over the world. The theorem it proves is now called "Chen Theorem" by all countries in the world, because its author's surname is Chen and his name is Jingrun. Now he is a researcher at the Institute of Mathematics, China Academy of Sciences.

Chen Jingrun, 1933 was born in Fujian. When he was born in this real world, his family and social life did not show him the gorgeous colors of roses. His father is a post office clerk, always running around. If he had joined the Kuomintang, he would have made a fortune long ago, but his father refused to join. Some colleagues said that he was really out of touch with the times. His mother is a kind and overworked woman. She gave birth to twelve children. Only six survived, of which Chen Jingrun was the third. There are brothers and sisters in the world; And brothers and sisters. If you have more children, you won't be loved by your parents. They are becoming more and more a burden to parents-redundant children, redundant people. From the day he was born, he came to this world like a person who was declared persona non grata.

He didn't even enjoy much childhood happiness. Mother has been working hard all day to love him. As long as he can remember, a fierce war broke out. Japan invaded Fujian province. He is too young, so life is on tenterhooks. Father went to a post office in Sanming City, Sanyuan County as the director. A small post office is located in an ancient temple in the mountainous area. This place used to be a revolutionary base. But at that time, Mao Yushi mountain forest has become a miserable world. All the men were slaughtered by the Kuomintang bandit troops, and no one was spared. There are not even old people. There are only women left. Their lives are particularly bleak. Flower yarn is too expensive; I can't afford to wear clothes, and the big girls are still naked. After Fuzhou was occupied by the enemy, more people fled to the mountains. The planes are not bombed here, and the mountains are a little busy. But was transferred to a concentration camp. In the middle of the night, the whip often echoes painfully; From time to time, there were gunshots shooting martyrs. The next day, those who came out to work in chains looked even more gloomy.

Chen Jingrun's young mind was greatly traumatized. He is often conquered by panic and confusion. He didn't have any fun at home, and he was always bullied in primary school. He thinks he is an ugly duckling. No, it's human. He still feels lonely. It's just that he is thin and weak. It's impossible to be likable just by being so timid. Accustomed to being beaten, he never asks for forgiveness. This made the other side beat him hard, and he was tougher and more endurance. He is too sensitive to feel the cannibalism of those people in the old society too early. He was portrayed as an introverted person with an introverted personality. He fell in love with mathematics. Not because he is oppressed, but because he loves mathematics, and calculating mathematical exercises takes up most of his time.

Mathematically, there is also a very famous "(1+ 1)", which is the famous Goldbach conjecture. Although it sounds amazing, its title is not difficult to understand. As long as you have the mathematics level in the third grade of primary school, you can understand its meaning. It turns out that this is the18th century. The German mathematician Goldbach accidentally discovered that every even number not less than 6 is the sum of two prime numbers. Such as 3+3 = 6; 1 1+ 13=24。 He tried to prove his discovery, but failed many times. 1742, Goldbach had to turn to Euler, the most authoritative Swiss mathematician in the world at that time, and put forward his own guess. Euler quickly wrote back that this conjecture must be established, but he could not prove it.

Someone immediately checked even numbers greater than 6 until it reached 330000000. The results show that Goldbach's conjecture is correct, but it can't be proved. Therefore, the conjecture that every even number not less than 6 is the sum of two prime numbers [(1+ 1)] is called "Goldbach conjecture" and becomes an elusive "pearl" in the crown of mathematics.

19 In the 1920s, Norwegian mathematician Brown proved that every even number greater than 6 can be decomposed into a product of no more than 9 prime numbers and another product of no more than 9 prime numbers, which is called "(9+9)" for short. Since then, mathematicians all over the world have adopted screening method to study Goldbach conjecture.

At the end of 1956, Chen Jingrun, who had written more than 40 papers, was transferred to the Academy of Sciences and began to concentrate on the study of number theory under the guidance of Professor Hua. 1966 In May, he rose to the sky of mathematics like a bright star and announced that he had proved it (1+2).

1973, the simplified proof of (1+ 1) was published, and his paper caused a sensation in mathematics. "(1+2)" refers to the internationally recognized "Chen Jingrun theorem" that even numbers can be expressed as the sum of the products of one prime number and no more than two prime numbers.

Chen Jingrun (1933.5~ 1996.3) is a modern mathematician in China. 1933 was born in Fuzhou, Fujian on May 22nd. 1953 graduated from the Mathematics Department of Xiamen University. Because of his improvement in problems, Hua attached great importance to it. He was transferred to the Institute of Mathematics of China Academy of Sciences, first as an intern researcher and assistant researcher, and then promoted to a researcher by leaps and bounds, and was elected as a member of the Department of Mathematical Physics of China Academy of Sciences.

1in late March, 1996, Chen Jingrun collapsed just a stone's throw from the glorious peak of Goldbach's conjecture, leaving endless regrets for the world.

When I was in junior high school, Jiangsu University moved to this mountainous area from the enemy-occupied area in the distance. Professors and lecturers from that college also come to local junior high schools to take part-time classes, which can improve their lives in exile in different places to some extent. These teachers are very knowledgeable. There is a Chinese teacher with the highest level. Everyone worships him. But Chen Jingrun doesn't like Chinese. He likes two math and science teachers from other places. Foreign teachers also like him. These teachers often boast about saving the country through science. He doesn't believe that science can save the country. But saving the country can't be without science, especially without mathematics. Moreover, mathematics is indispensable for everything. People's discrimination against him, punching and kicking, can only make him fall in love with mathematics more. Boring algebraic equations filled him with happiness and became the only pleasure.

At the age of thirteen, my mother died. Died of tuberculosis; From then on, my son dreamed of his mother, his father got married, and his stepmother was worse than his mother. After the victory of the Anti-Japanese War, they returned to Fuzhou. Chen Jingrun entered Trinity Middle School. After graduation, I went to Huaying College to attend high school. There is a math teacher who used to be the head of the Aviation Department of the National Tsinghua University.

Third,

The teacher is knowledgeable and tireless in teaching. He told his classmates a lot of interesting math knowledge in math class. Students who don't love math can be attracted to him, let alone those who love math.

Mathematics is divided into two parts: pure mathematics and applied mathematics. Pure mathematics deals with the relations and spatial forms of numbers. In the part dealing with the relationship between numbers, an important branch of discussing the properties of integers is called "number theory" Fermat,/kloc-a great French mathematician in the 7th century, and the founder of western number theory. But China made a special contribution to number theory in ancient times. Zhou Jie is the oldest classical mathematical work. There is an earlier book, The Art of War. One of the remainder theorems was initiated by China. Later, it spread to the west and was called Sun Tzu's theorem, which is a famous theorem in number theory. Until the Ming Dynasty, China made great contributions to mankind in number theory. The pi calculated by Zu Chongzhi in the 5th century was earlier than that calculated by German Otto 1000 years ago. Scientists led by Joseph named a valley on the moon "Zu Chongzhi". /kloc-The second half of the third century is the climax of ancient mathematics in China. Qin, a great mathematician in the Southern Song Dynasty, is the author of Nine Chapters and Counting Books. He solved linear equations more than 500 years earlier than the great Italian mathematician Euler. Zhu Shijie, a great mathematician in Yuan Dynasty, wrote "Four Yuan Jade Sword". His solution of multivariate higher-order equations was more than 400 years earlier than that of the great French mathematician Zhu Bi. After the Ming and Qing Dynasties, China fell behind. However, China people seem to have a special talent for mathematics. China should be a great mathematician. China is a good hotbed of mathematics.

Once, the teacher told these high school students a famous problem in number theory. He said that Peter the Great of Russia built Petersburg and hired a large number of great European scientists. Among them, there is the great Swiss mathematician Euler (his works have more than 800 kinds); There is also a German middle school teacher named Goldbach, who is also a mathematician.

1742, Goldbach found that every big even number can be written as the sum of two prime numbers. He has tested many even numbers, all of which show that it is true. But it needs to be proved. Because it has not been proved, it can only be called a guess. He couldn't prove it himself, so he wrote to the famous mathematician Euler and asked him to help him prove it. Until his death, Euler could not prove it. Since then, it has become a difficult problem, attracting the attention of thousands of mathematicians. For more than 200 years, many mathematicians have tried to prove this conjecture, but all failed.

Speaking of which, the classroom has become boiling water. Young students, like the first flowers, chatter endlessly.

The teacher added that the queen of natural science is mathematics. The crown of mathematics is number theory. Goldbach conjecture is the jewel in the crown.

The students all opened their eyes in surprise.

The teacher said that you all know even and odd numbers. We all know prime numbers and composite numbers. We taught this in the third grade of primary school. Isn't this the simplest? No, this question is the most difficult. This question is very difficult. If anyone can do it, it will be amazing, amazing!

The young people are quarrelling again. What's the big deal? Let's get started. We can do it. They brag about Haikou.

The teacher also smiled. He said, "Really, I had a dream last night. I dreamed that there was a classmate among you. He's amazing. He proved Goldbach's conjecture. "

High school students burst into laughter.

But Chen Jingrun didn't laugh. He was also shocked by the teacher's words, but he couldn't laugh. If he smiles, some classmates will stare at him with white eyes. He has become more and more lonely since he entered high school. The students ignored him because he was eccentric, dirty and sick. They looked at him contemptuously and sarcastically. He became a lonely, lonely, talking to himself and lonely weirdo. In the sky, a lonely goose.

The next day, classes began again. Several hard-working students excitedly gave the teacher some answer sheets. They said that they succeeded, which can prove the German conjecture. It can be proved in many ways. No big deal. Ha! Ha!

"Forget it!" The teacher smiled and said, "Forget it! Forget it! "

"We forget it, forget it. We figured it out! "

"You forget it! Okay, okay, I mean, forget it. What are you wasting your energy on? I won't read any of your papers. I don't need to read them. Is it that easy? You want to go to the moon by bike. "

There was another burst of laughter in the classroom. Those students who didn't hand in their papers laughed at those who did. They laughed themselves, stamped their feet and burst into laughter. Only Chen Jingrun didn't laugh. He frowned. He was excluded from all these pleasures.

The following year, the teacher returned to Tsinghua. Shen Yuan, vice president of Beijing Institute of Aeronautics and Astronautics and chairman of the National Aviation Society. He should have forgotten these two math classes long ago. How did he know how deeply he was engraved in the memory of student Chen Jingrun? Teachers are easy to forget because of many classmates, but students often remember their teachers when they were young.