What is Goldbach's conjecture?

Modern science is a splendid palace. If you set foot in this mysterious temple, you will be fascinated and dazzled. If you have the heart to explore again, you will find that the treasures in the temple are dazzling, but they are not as bright as a pearl. You must be curious to know about this pearl. Then, let's learn about science first. Science is the door for us to find that bright pearl. Modern science can be divided into natural science and social science. In the category of natural science, it is divided into basic disciplines such as mathematics, physics, chemistry, biology, astronomy and geology. Among them, mathematics is the foundation of other disciplines, and any discipline depends on mathematical methods. Science that cannot be described mathematically is not science. Therefore, the queen of natural science is mathematics. 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 integer properties is discussed, which is called "number theory". Fermat, a great French mathematician in17th century, is the founder of western number theory, but China has made special contributions to number theory since ancient times. Zhou Jie is the oldest classical mathematical work. There is also an earlier book, Sunzi Suanjing, in which a remainder theorem was initiated by China. It is said that Han Xin, a great strategist, used it to point soldiers, and later it spread to the west, and it was called Sun Tzu's concentration, which is a famous theorem in number theory. Until the Ming Dynasty, China made great contributions to mankind in number theory. /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 of Counting Books. He solved linear equations more than 500 years earlier than the great Italian mathematician Euler. Zhu Shijie, a great mathematician in the Yuan Dynasty, wrote "Four Jade Mirrors". His solution of multivariate higher-order equations was more than 400 years earlier than that of Zhu Bi, a great French mathematician. In mathematics, the most basic theory is number theory. Without number theory, the beautiful queen of mathematics is no longer the queen. The crown of mathematics is number theory. Don't worry, cover the crown first, and then explore the beautiful pearls in the crown. Let's learn math in the second grade of junior high school first. 12345 and 1 100 million are called positive integers. Those numbers that are divisible by 2 are called even numbers. The remaining numbers are called odd numbers. There is also a number, such as 2, 3, 5, 7, 1 1 3, which can only be divisible by 1 and its original number, and cannot be divisible by other integers, so it is called a prime number, and can be divisible by other integers other than 1 and its original number. If an integer is divisible by a prime number, the prime number is called the prime factor of the integer. For example, 6 has two prime factors, 2 and 3; 30 has three prime factors: 2, 3 and 5. Okay, that's enough for now. /kloc-at the beginning of the 0/8th century, Peter the Great of Russia wanted to build a large-scale project and build Petersburg. To this end, a large number of European scientists have been hired to invest in design and construction. One of them is a German mathematician named Goldbach. 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 this is indeed correct. Therefore, he guessed that all even numbers must be written as the sum of two prime numbers. However, this needs to be proved. Because it has not been strictly proved, it can only be said to be a guess. Therefore, he wrote to the famous Italian mathematician Leo Hof Euler. In the letter, he proposed that every even number not less than 6 is the sum of two prime numbers. For example: 6 = 3+3; 24= 1 1+ 13 and so on. Specifically: (a) Every even number not less than 6 is the sum of two odd prime numbers. (b) Every odd number not less than 9 is the sum of three odd prime numbers. This is the famous Goldbach conjecture. Later generations called conjecture (1) "Goldbach conjecture on odd numbers". Since 2n+ 1=2(n- 1)+3, it is also correct to infer conjecture (b) immediately from the correctness of conjecture (a). Euler, a famous mathematician, studied Goldbach's problems very seriously. Perhaps, at first, he thought the problem was easy to prove, because it was the simplest and most basic. However, it is often the simplest and most basic questions that are the most important. To Euler's surprise, it turned out that the work was not smooth. The mathematician who has made outstanding contributions to number theory has tried his best, but it has proved that he has made no progress. He didn't even find the right evidence. Euler verified these two conjectures over and over again. Although he didn't prove them, he was convinced of their correctness through the mathematician's intuition. 1742 On June 30th, he wrote to Goldbach: I think this is a definite theorem, although I can't prove it yet. As a natural scientist, Euler is excellent. He didn't cover up his guess because of his failure. He published Goldbach's letter to the whole world. Like Euler, the great mathematicians in the18th century opened their eyes in surprise and said with one voice: Goldbach conjecture, the jewel in the crown!

Dust-covered

A knowledgeable math teacher in our country once introduced Goldbach's conjecture to high school students with great interest. He told his classmates that every big even number can be written as the sum of two prime numbers. This is Goldbach's conjecture, this is the jewel in the crown! The students all opened their eyes in surprise. The teacher said that you all know even numbers and odd numbers, and you all know prime numbers and composite numbers. We have already learned this. Isn't this the simplest? No, this question is the most difficult, very difficult. If someone can work it out, that would be great! Students quarreled: 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. 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, and it's not a big deal. Ha ha! Haha! " "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, and those students who didn't hand in their papers laughed at those who did. They laughed themselves, stamped their feet and burst into laughter. Is this problem really that difficult? Is this pearl really that hard to pick? It's really hard! /kloc-from the 0/8th century to the 20th century, many major breakthroughs have been made in natural science, the basic theories of many disciplines have been updated, and epoch-making major inventions have emerged. Not only that, mankind is relying on the living earth to uncover the secret of self-reproduction; The study of nuclear physics has gone deep into the nuclear energy level of quarks. However, this simplest problem, the most basic problem, every big even number can be written as the sum of two prime numbers. Goldbach guessed that the jewel in the crown was still hanging there quietly, showing her pride and beauty to mankind. Every big even number can be written as the sum of two prime numbers, which can be expressed in a concise and inaccurate way, namely (1+ 1). Goldbach's conjecture is (1+ 1). For this (1+ 1), Euler, a great mathematician, spent all his energy, but by the time he finished the mileage of his life, he had not seen the dawn of (1+ 1). /kloc-in the 8th century, after Euler announced Goldbach's conjecture, many famous mathematicians devoted themselves to research. What's more, a mathematician devoted his life to research. However, in the whole18th century, mathematicians did not produce any results in the face of (1+ 1). /kloc-in the 0/9th century, the industrial revolution began in the west. In the whole19th century, science and technology developed rapidly. It is worth mentioning that almost all the basic disciplines of modern science were laid in this century. For example, in physics, Newton's law of universal gravitation has been successfully applied to mechanical devices to calculate the mass of the earth; Charles of France discovered the relationship between gas volume and temperature and revealed the physical properties of gas. The essence of light has also been discovered, and the talented physicist Falk of France successfully measured the speed of light in the laboratory; German doctor Meyer and Englishman Joule both discovered the law of conservation of energy; Molecules and atoms have also been discovered. In chemistry, quite a few elements have been found. 1872, Mendeleev of Russia discovered the periodic law of elements and listed 63 elements in the periodic table. In biology, cells and cell division have been found; Know that biological production is the combination of male germ cells and female germ cells; Moreover, the theory of heredity began to be established. Darwin in Britain also made investigations all over the world and discovered the evolution of living things. In addition, bacteria, viruses, vaccinia, etc. It has also been recognized. Pasteur in France also developed immunity. People also discovered electricity, magnetism and so on. /kloc-almost all disciplines have made new progress in the 0/9th century, and developed science urgently needs mathematics. /kloc-what is mathematics like in the 0/9th century? This oldest discipline appeared 4000 years ago. 19th century, the revolution of electrical technology led to the rapid development of electric power application and electrical communication technology, thus, the branch of applied mathematics based on calculus developed rapidly. In algebra, the study of algebra is promoted by solving quintic equations, and abstract algebras such as group theory, field theory, ring theory and beam theory are produced. In geometry, the talented Russian mathematician Lobachevsky founded non-Euclidean geometry. Pure mathematics, which uses axioms and theorems for theoretical research, has also developed rapidly in the19th century. Everything is thriving. The treasures in the science hall are dazzling. However, Goldbach guessed that this beautiful crown pearl was still covered with dust and no one could pick it. Don't forget, mathematicians' IQ and sensitivity have always been first-class. They are too familiar with the proposition (1+ 1), this great conjecture. Nothing is more attractive than proving a difficult problem, but no mathematician has ever succeeded! After Euler, many persistent and tenacious mathematicians began to explore hard again. Gauss, Dirichlet, Riemann and Hadamard fought bravely one after another, but they all failed. So some people say that Goldbach's conjecture cannot be proved. 1892, the 5th International Mathematical Society was held in Cambridge, England. German mathematician Goldbach's compatriots announced at the conference very pessimistically that it is impossible to prove Goldbach's conjecture, even if the proposition weaker than Goldbach's conjecture-[(e)] has a positive integer k, so that every positive integer ≥ 2 is not greater than the sum of k prime numbers, which is beyond the power of contemporary mathematicians. In a speech by the Mathematical Society in Copenhagen, British mathematicians thought that Goldbach conjecture might be the most difficult mathematical problem that has not been solved so far. From Goldbach's conjecture to the end of 19 century, there was no substantial achievement in the study of this magical proposition, and even no effective method was put forward. At the beginning of the 20th century, developed mathematicians and evolutionary mathematicians were still powerless in the face of Goldbach conjecture (1+ 1). Goldbach guessed that you, a beautiful pearl, really don't want the world to explore?

Difficult exploration

Just when some famous mathematicians made pessimistic predictions and felt overwhelmed, they didn't think, or didn't realize, that the research on Goldbach's conjecture was about to start again. The March was an attack from several directions. It should be affirmed that although Euler, Gauss and others did not prove Goldbach's conjecture, they made brilliant achievements in number theory and function theory, which provided powerful tools for mathematicians in the 20th century to study conjecture and laid an indispensable solid foundation. Mathematicians in the 20th century rallied and prepared to continue to challenge Goldbach's conjecture. First of all, in 1920, British mathematicians Harding and Littlewood initiated and developed a brand-new method in the theory of heap prime numbers, which is called Hardy Littlewood Raman Nuzhan circle method. If the circle method is successful, it will be very powerful. Because it not only proves the correctness of the conjecture, but also obtains the asymptotic formula of the table number expressed by the sum of odd prime numbers, which is impossible by other methods so far. Although Harding and Littlewood have not proved any unconditional results, their "circle method" and its preliminary exploration are very important contributions to the study of Goldbach's conjecture and analytic number theory, and point out a promising research direction for people. In 1937, Hysmans proved that every sufficiently large odd number can be expressed as the sum of the product of two odd prime numbers and the product of no more than two prime numbers. In 1937, Buchstaber unconditionally proved that every odd number large enough is the sum of three odd prime numbers by using Hardy Littlewood Ramanujan circle method and his original triangle sum estimation method. This basically solved conjecture (b) and is a very significant contribution. In 1938, China Renhua proved the general result: for any given integer r, every sufficiently large odd number can be expressed as the sum of two odd prime numbers plus the r product of another odd prime number. That is, P 1+P2+PK3, where P 1, P2 and P3 are odd numbers. The "circle method" is very successful in the study of conjecture (b), but it has little effect on the study of conjecture (a) and can't get any important results. Secondly, let's look at the "screening method". At the same time of putting forward the "circle method", in order to study conjecture (a), a powerful elementary method in number theory-"screening method" has also begun to develop. It is too difficult to solve conjecture (1). Therefore, people imagine whether it is possible to prove that every large enough even number is the sum of the products of two small prime factors, and thus seek a method to solve conjecture (a) by gradually reducing the number of prime factors. For the convenience of description, we use the proposition (a+b) to express the following proposition: Every sufficiently large even number is the sum of the product of no more than one prime number and the product of no more than b prime numbers. In this way, if the proposition (1+ 1) is proved, the conjecture (a) is basically proved. "Screening method" is an ancient method, which was created by Greek scholars to find prime numbers more than 2000 years ago. Because this primitive "screening method" has no theoretical value, it has not been developed for a long time. It was not until around 1920 that the mathematician Brown improved the "sieve method" for the first time, and since then, it has opened up a very extensive and fruitful new way to study many number theory problems such as conjecture (a) by using the "sieve method". Brown made a great contribution to number theory, and later people called his method Brown method. Brown's "screening method" has strong characteristics of combinatorial mathematics, which is complicated to apply and difficult to use, but Brown's thought is very enlightening. 194 1 year, another visionary mathematician Kuhn first proposed a better "weighted screening method", and later many mathematicians made in-depth research on various forms of "weighted screening method", thus constantly improving the role of "screening method". 1950, Selberg made another major improvement on the ancient "screening method" by solving the quadratic extreme value, which was called "Selberg screening method". It is not only convenient for application, but also achieves better results than the "brown screen method". Modern mathematicians began to March to Goldbach's conjecture from the two battlefields of "circle method" and "screening method". After hard struggle, mathematicians have made great achievements in both directions. 1920, Brown proved the proposition (9+9); In 1924, Rad mahar proved the proposition (7+7); 1932, isler proved the proposition (6+6); In 1937, Ricks proved the propositions (5+7), (4+9), (3+ 15) and (3+336); In 1938, Buchstaber proved the proposition (5+5). From 1939 to 1940, he proved the proposition (4+4). The above results are all obtained by Brown's "screening method". 1950, Selberg announced that the proposition (2+3) could be proved by his method, but he did not publish his proof for a long time. Later, people used his "screening method" to get the result: 1956, and Wang Yuan proved the proposition (3+4); 1957, vinogradov proved the proposition (3+3); 1958, Wang Yuan proved proposition (2+3) and proposition (A+B), where a+b ≤ 5; However, all the above results have the same weakness, that is, we are not sure that at least one of the two numbers is a prime number. In order to get this result, we need to prove the proposition (1+b). As early as 1948, Hungarian mathematician Lan Ian found another way and opened up another battlefield, trying to prove that every big even number is the sum of a prime number and a number with no more than six prime factors. He proved (1+6). 1962, Pan Chengdong, a mathematician and lecturer of Shandong University, proved (1+4). 1965, Buchstaber, vinogradov and the mathematician Pompeii Alley all proved (1+A). At this point, it is not far from Goldbach's conjecture. However, in this near-final journey, the brilliance of this pearl has not been seen. People entered the silent waiting again.

A footstep away

Like the middle school students mentioned earlier in this article, we may ask (1+ 1). Is it so difficult? Especially in modern times, the computing speed of computers has reached billions of times. Can't you solve the math problem (1+ 1)? Put this question aside for a while and don't answer it. Let's see how mathematicians work hard for the jewels in the crown. We may not know much about how ancient and western mathematicians worked. Let's look at the situation of modern mathematicians in China. Among the mathematicians who study Goldbach's conjecture in China, the most representative is Chen Jingrun of Institute of Mathematics, Chinese Academy of Sciences. Chen Jingrun was born in 1933 in Fujian. When he was born into this real world, his family and social life did not show him the gorgeous colors like roses. His father is a post office clerk, always running around. His mother is a kind and overworked woman. She gave birth to 12 children, but only six survived, of which Chen Jingrun was the third. There are brothers and sisters in the world and brothers and sisters in the world. Chen Jingrun was very partial to mathematics in middle school. 1950 was admitted to Xiamen University. Because of excellent grades, I graduated without graduation. Later, after many twists and turns, he was transferred to the Institute of Mathematics of the Chinese Academy of Sciences. Having said that, he made Goldbach's conjecture and another miracle appeared. At the beginning, Xiong Qinglai, a great mathematician and educator of the older generation in China and the importer of modern mathematics in China, taught in Tsinghua University. In the early 1930s, a young mathematician who neglected his studies after graduating from junior high school and was completely self-taught sent Xiong Qinglai an article on solving algebraic equations. As soon as Xiong Qinglai saw it, he saw the heroism and splendor in this article. He immediately invited the young Geng Hualai, the author of the book, to Tsinghua campus. He arranged for Hua to work in Tsinghua Library, teach himself and listen to lectures. Later, Hua was sent to Cambridge University in England to study. After returning home, Xiong Qinglai, president of Yunnan University in Kunming, introduced him as a professor at the General Assembly. Hua later went abroad and taught at universities in Princeton and Illinois. After the founding of People's Republic of China (PRC), Hua immediately returned to China and presided over the work of the Institute of Mathematics of China Academy of Sciences. Chen Jingrun also wrote a special article on number theory in Xiamen University Library and sent it to Institute of Mathematics, Chinese Academy of Sciences. After reading the article, Hua also saw the heroism and splendor in the article, and also put forward suggestions to transfer Chen Jingrun to the Institute of Mathematics as an internship researcher. Exactly: Xiong Qinglai has a good eye for Jing Run, and Hua has a good eye for Jing Run. 1at the end of 956, Chen Jingrun came to Beijing from the south coast. 1957 In the summer, Xiong Qinglai, a mathematician, also returned to Tsinghua from abroad. At this time, there is a long salty collection and a group of talents are complete. At that time, famous mathematicians included Xiong Qinglai, Hua, Zhang Zongsui, Min Sihe, Wu Wenjun and many other talented stars. There are a new generation of Toshihiko, Lu Ruqian, Wang Yuan, Yue Minyi and Wu Fang. , such as the sunrise; There are rising stars like Yang Le and Zhang Guanghou who study in Peking University. In the fields of analytic number theory, algebraic number theory, function theory, universal number analysis, geometric topology and so on, there are already many talents, and a Chen Jingrun has been added. Everyone holds a snake bead, and every family holds Jingshan jade. It's all the rage and the lineup is neat. When the conditions are met, Hua made a strategic deployment, focusing on the development of applied mathematics, and at the same time advancing towards the jewel in the crown-Goldbach conjecture! Since Chen Jingrun was transferred to the Institute of Mathematics, his intellectual bud has blossomed. He perfected the achievements of Chinese and foreign mathematicians in the aspects of the whole point problem in the garden, the whole point problem in the ball, the Waring problem and the three-dimensional divisor problem. These achievements alone, his contribution has been great. When he had sufficient basis, he advanced to Goldbach conjecture with amazing perseverance. He forgot to eat and sleep, stayed up all night, concentrated on thinking, explored the essence, made a lot of calculations, and devoted himself to mathematics, which made him dumbfounded. Once I bumped into a tree and asked who hit him. He devoted all his mind and reason to solving this difficult problem and paid a high price for it. His eyes were deep, his cheeks were red with tuberculosis, his laryngitis was severe, and he kept coughing, abdominal pain and abdominal distension, which was unbearable ... Finally, in 1966, Chen Jingrun announced that he had proved the proposition (1+2). At that time, he did not give a detailed proof, but simply outlined his method. 1973, he published all the proofs of the proposition (1+2). It should be pointed out that during the whole seven years from his announcement of the result to the publication of all the proofs, no other mathematician has given the proof of the proposition (1+2). It seems that the international mathematical community still thinks that the proposition (1+3) is the best result. Therefore, when Chen Jingrun published all the proofs of his creative proof proposition (1+2) in 1973, it immediately aroused strong repercussions in the international mathematics field and was recognized as a very outstanding achievement, a great contribution to the study of Goldbach's conjecture and the most outstanding application of the "screening method" theory. This achievement was unanimously called Chen Theorem. Chen Jingrun's contribution, in methodology, lies in his proposal and realization of a new "addendum screening method". Because of the importance of these studies, several other simplified proofs (1+2) were published in a short time at home and abroad. Goldbach, you made a magical and solemn conjecture more than 200 years ago, which attracted many human geniuses to struggle and explore! Now, it is only one step away from this pearl.

Who took the pearls?

It has been 30 years since China and Chen Jingrun announced the proposition of proof (1+2) in 1966. During this period, international mathematicians have been exploring and updating their methods on the basis of previous studies. Some mathematicians use large computers. However, there is still no significant substantive progress. This is a daily - happened thing. For studying a major problem, it is equally difficult to take a pioneering first step and a final step to completely solve the problem. Although on the surface, the difference between the proposition (1+2) and the proposition (1+1), the solution of Goldbach's conjecture, is only "1",but the difficulty to be overcome in completing this last step is not necessarily easier than the road already taken. So far, mathematicians are not sure whether Goldbach conjecture can be finally solved along the existing methods. So far, no one can give a hypothetical conjecture proof (A). Goldbach guessed that you, the beautiful crown jewel, are still far away from the world, high above and dazzling. Only God knows when and with whom.