One
Px(1,2) is the number of primes p that fit the following conditions:
x-p=p=p1或-p=p2p3
where p1, p2 and p3 are all primes. (This is not easy to understand; if you can't read it, you can skip these lines.) ）
Use x to table a sufficiently large even number.
命Cx=∏(p-1)/(p-2)∏(1-1/(p-1)2)
p|xp>2
P>2
For any given even number h and sufficiently large x, xh(1,2) is used to denote the number of primes p satisfying the following conditions:
p≤x, p+h=p1 or h+p=p2p3,
where p1, p2, and p3 are all primes.
The purpose of this paper is to demonstrate and improve all the results mentioned by the authors in Ref. [10], which are detailed below.
Two
The above is quoted from a paper on analytic number theory. This paragraph, quoted from its "(a) Introduction", raises this question. It is followed by "(b) several lemmas", full of various formulas and calculations. Finally, there is "(3) Result", which proves a theorem. This paper is extremely difficult to understand. Even famous mathematicians, if they are not specialized in this branch of mathematics, may not be able to read it. However, this paper has been recognized by the international mathematical community and is well-known all over the world. The theorem it proved is now unanimously named "Chen's theorem" by countries around the world, because its author's surname is Chen and his name is Jingrun. He is now a researcher at the Institute of Mathematics of the Chinese Academy of Sciences.
Chen Jingrun is a native of Fujian, born in 1933. When he was born into this real world, his family and social life did not show him the bright colors of a rose. His father was a post office clerk and was always running around. If he had joined the Kuomintang that year, he could have soared, but his father refused to participate. Some colleagues said that he really didn't know the times. His mother was a good, overworked woman, and had twelve children. Only six lived, of which Chen Jingrun ranked third. There are older brothers and sisters, and younger brothers and sisters. If a child has given birth to too much, it will not be a child loved by both parents. They are increasingly becoming a burden to parents - superfluous children, superfluous people. From the day he was born, he came to this world as if he had been declared persona non grata.
He didn't even enjoy much of his childhood. His mother toiled all day long, and she couldn't care about loving him. As far as he can remember, a fierce war broke out. The Japanese devils entered Fujian Province. He was still so young, so he lived his life in fear. Father went to Sanming City, Sanyuan County, the head of a post office. A small post office in an ancient temple in the mountains. This place was once a revolutionary base. But by then, the lush forests had become a world of misery. All the men were slaughtered by the Kuomintang bandits, and there were no survivors. Not even a single elderly man remains. Only women remained.
Their lives are particularly bleak. The price of flower gauze is too expensive, and the big girls are still naked. After Fuzhou was occupied by the enemy, more people fled into the mountains. There are no planes to bomb here, and the mountains are gradually flourishing. But another concentration camp was moved. In the middle of the night, the sound of whipping echoes in pain, and from time to time there are gunshots of martyrs. The next day, those who came out in shackles to work looked even more gloomy.
Chen Jingrun's young mind was greatly traumatized. He was often overcome by panic and confusion. He didn't have fun at home, and he was always bullied in elementary school. He felt like an ugly duckling. No, it's human, he still feels like he's a human being. It's just that he's thin and weak. The mere appearance of this wreck is not flattering. Accustomed to being beaten, never begging for mercy. This made the other party beat him harder, and he was more tenacious and enduring. He was overly sensitive, and he sensed the cannibalism of those people in the old society too early. He was created as an introvert with an introverted personality. He fell in love with mathematics alone. It's not because he's pressed, it's just because he loves math, and calculus problems take up most of his time.
When he entered junior high school, Jiangsu College moved from the distant occupied area to this mountainous area. The professors and lecturers in the college also came to the local junior high school to teach part-time, which could also improve their lives in exile to some extent. These teachers are very knowledgeable. There is a Chinese teacher with the highest level. Everyone adores him. But Chen Jingrun doesn't like language. He likes two out-of-town math and science teachers. Teachers from other places like him too. These teachers often boast about science to save the country. He didn't believe that science could save the country. But we cannot save our country without science, especially mathematics. And math is all about it. People discriminated against him, punched and kicked, which only made him fall in love with mathematics even more. The boring algebraic equations filled him with happiness and became the only pleasure.
At the age of thirteen, his mother died. He died of tuberculosis, and from then on, the son wanted to kiss his mother in his dreams, and his father got married again, and his stepmother was even worse to him.
The Anti-Japanese War was won, and they returned to Fuzhou. Chen Jingrun entered Trinity Middle School. After graduation, he went to Ying Wa College to study in high school. There was a math teacher there who used to be the head of the aviation department at National Tsing Hua University.
Three
The teacher is very knowledgeable and tireless. In his math class, he gave his classmates a lot of interesting math knowledge. Students who don't love mathematics can be attracted to him, let alone those who love mathematics.
Mathematics is divided into two main parts: pure mathematics and applied mathematics. Pure mathematics deals with the relationship of numbers to spatial forms. In the part dealing with the relations of numbers, an important branch of the nature of integers is discussed, called "number theory". The great French mathematician Fermat of the seventeenth century was the founder of Western number theory. However, ancient Chinese people have already made special contributions to number theory. Zhou Ji is the oldest classical mathematical work. There is also an earlier "Sun Tzu Sutra". One of the remainder theorems is the first of its kind in China. Later, it was transmitted 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 terms of number theory. Zu Chongzhi in the fifth century calculated pi more than a thousand years earlier than that of Otto in Germany. Scientists led by Joseph (referring to Stalin) named a valley on the moon "Zu Chongzhi". The second half of the thirteenth century was the culmination of ancient Chinese mathematics. Qin Jiushao, a great mathematician of the Southern Song Dynasty, wrote "Nine Chapters of the Book of Numbers". His solution to the simultaneous equation was more than 500 years earlier than that of the great Italian mathematician Euler.
Zhu Shijie, a great mathematician of the Yuan Dynasty, is the author of "Four Yuan Jade Jian". His solution of multivariate higher-order equations was more than 400 years earlier than that of the great French mathematician Bizhu. After the Ming and Qing dynasties, China lagged behind. However, Chinese seem to be particularly gifted with mathematics. China should produce great mathematicians. China is a good hotbed of mathematics.
Once, the teacher gave these high school students a lecture on a well-known problem in number theory. He said that at the beginning, Peter the Great of Russia built Petersburg and hired a large number of great European scientists. Among them were the great Swiss mathematician Euler (who wrote more than 800 books) and a German secondary school teacher named Goldbach, who was also a mathematician.
In 1742, Goldbach discovered that every large even number could be written as the sum of two prime numbers. He examined many even numbers, and they all showed that this was true. But this needs to be proven. Because it has not yet been proven, it can only be called a conjecture. He could not prove it himself, so he wrote to the famous mathematician Euler and asked him to help prove it. Until death, Euler could not prove it. Since then, this has become a difficult problem that has attracted the attention of thousands of mathematicians. For more than 200 years, many mathematicians have tried to prove this conjecture, but without success.
At this point, the classroom became a boiling pot of water. The young students, who were like the first flowers, chattered.
The teacher also said that the queen of natural science is mathematics. The crown jewel of mathematics is number theory.
The Goldbach conjecture is the crown jewel.
The students' eyes widened in surprise.
The teacher said, you all know even and odd numbers. We also know prime numbers and composite numbers.
We taught this in the third grade of elementary school. Isn't that the easiest? No, it's the hardest one. This question is very difficult, very difficult. If anyone can do it, it's amazing, it's amazing!
The young people quarreled again. What's the big deal about this. Let's do it. We do it. They boasted about Haikou.
The teacher laughed too. He said, "Really, I had a dream last night."
I dreamed that one of you was a classmate, and he was terrible, and he proved the Goldbach conjecture. ”
The high school students burst out laughing.
But Chen Jingrun didn't laugh. He was also shaken by the teacher's words, but he couldn't laugh. If he laughs, some of his classmates will stare at him with white eyes. Since entering high school, he has become more and more lonely. His classmates thought he was weird, dirty, and sickly, and ignored him. They looked at him with contempt and mockery.
He became a deviant who walked alone, alone, talking to himself, and being alone. In the sky, a lonely goose.
The next day, classes were held again. A few hard-working students happily presented the teacher with a few answer papers. They said that they had already made it and would be able to prove the German conjecture. It can be proven in many ways. Nothing remarkable. Hah!
"Forget it!" said the teacher with a smile, "Forget it!"
"We're done, we're done. We've figured it out!"
"Forget it, okay, okay, I mean, forget it, what are you doing with all this effort? I can't read your papers, I don't need to read them.
Is it that easy? You want to ride your bike to the moon. ”
The classroom erupted in laughter. Those students who didn't hand in the papers laughed at the few who handed in the papers. They themselves laughed, and they all stamped their feet with laughter, and their stomachs burst with laughter. Only Chen Jingrun didn't laugh. He frowned. He was excluded from all this joy.
The following year, the teacher went back to Tsinghua University. He is now Shen Yuan, vice president of Beijing Institute of Aeronautics and Astronautics, and chairman of the National Society of Aeronautics and Astronautics. He should have forgotten these two math lessons a long time ago. How could he know how deeply he was engraved in the memory of his student Chen Jingrun.
Teachers are easy to forget because they have many classmates, but students often remember their teachers when they were younger.
Four
He was in his third year of high school that year. Because he could not pay the tuition, he did not go to school in the first half of 1950 and studied at home for a semester. He did not graduate from high school, but he applied for the examination with the same qualifications, and he was admitted to Xiamen University. At that time, there was only a Department of Mathematics and Physics at the university. When I was in my second year of college, I had a math group, but there were only four students. By the time I was in the third year, there was a mathematics department, and there were still these four people in the department. Because of their particularly outstanding achievements and the country's urgent need to train talents, the four of them completed their studies ahead of schedule; moreover, they were immediately assigned jobs and received preferential treatment, which was envied by others. In the autumn of 1953, Chen Jingrun was assigned to Beijing to work as a mathematics teacher at No. X Middle School. What a blessing this must be!
However, when he was at Xiamen University, his life was good. There are only four college students in the same group and department, but there are four professors and one teaching assistant to guide the study.
How hungry and greedy he was to drink from the flowers to make the fragrant and rich mathematical honey! How freely he navigates the realm of abstraction, and we all have a common mathematical language like DX and DY. Heart-to-heart, interconnected. During the three years, no one discriminated against him, and he was not scolded or beaten. He had little contact with people, lived a golden life, and immersed himself in the ocean of mathematics. I really didn't expect him to graduate so soon. He couldn't help but shudder at the thought of him going to be a teacher, standing on the podium, stared at by dozens of pairs of sharp, clever, and sometimes mischievous eyes!
His conjecture was immediately proven. He was completely unfit to be a teacher. He was so thin and sickly, but his students were tall and strong. He is the least good at talking, and his throat hurts when he says a few more words. How he envied those good teachers who were kind and seductive. When he returned to his room after class, he called himself stupid. Insulting yourself is much worse than someone else's. He never took care of himself and didn't pay attention to nutrition. He became ill and had a fever of 38 degrees Celsius. Taken to the hospital, he was found to have tuberculosis and peritoneal tuberculosis. During the year, he was hospitalized six times and had three surgeries. Of course, he wasn't able to teach well. But he didn't give up on his profession. Not long ago, the Chinese Academy of Sciences published Hua Luogeng's famous book "Theory of Stacked Prime Numbers". As soon as it was placed on the shelves of the bookstore, Chen Jingrun bought it. He plunged headlong into it. Very profound writing, very difficult! But he delved into it. Admitted to the hospital, he also secretly avoided the eyes and ears of doctors and nurses and studied it. At that time, he also thought that if this continued, the school would have no reason to welcome him.
He thought he might be unemployed, and what could he do? Fortunately, he cut back on food and clothing, and didn't buy a toothbrush. He never spends a penny casually, and he has saved almost all of his income. He relented, went home when he was unemployed, and continued to work on his mathematical research. Saving this money is a guarantee for him to do mathematics. This guaranteed that he would still be able to study mathematics even if he lost his job, and that was his life: his life was mathematics.
As for what happens when the savings are gone? He doesn't know, what will happen then?
This is a difficult question, and a conjecture that has not yet been answered. And this conjecture turned out to be correct. His illness did not heal, and he could not be rehired in secondary school.
The president of Xiamen University came to Beijing for a meeting at the Ministry of Education. A leader of the middle school met him, talked about it, was very dissatisfied, and put forward a lot of opinions: How did you cultivate such a high-achieving student?
Wang Yanan, president of Xiamen University and translator of Marx's Capital, was very surprised when he heard the opinion. He had always considered Chen Jingrun to be the best student in their school. He disagreed with what he had heard. He believes that this is the assignment of the student's work, and the assignment is not appropriate. He agreed to let Chen Jingrun return to Xiamen University.
I heard that he could go back to the Department of Mathematics of Xiamen University, and it was strange to say that Chen Jingrun's illness had improved. However, Wang Yanan arranged for him to be a caretaker in the library of Xiamen University. He was not allowed to manage the books, but only to concentrate on the study of mathematics. Wang Yanan is worthy of being a critic of political economy, he understands the theory of value and understands the value of people. Chen Jingrun also did not live up to the training of the old principal. He really studied Hua Luogeng's "Theory of Stacked Prime Numbers" and Da Houben's "Introduction to Number Theory". Chen Jingrun ate them thoroughly. His experience is not unprecedented.
At the beginning, Xiong Qinglai, a great mathematician and educator of the older generation in mainland China, was the introducer of modern mathematics in mainland China and taught at Tsinghua University in Beijing. At the beginning of the thirties, a young man who dropped out of school after graduating from junior high school and was completely self-taught after dropping out of school, sent an article on the solution of algebraic equations to Xiong Qinglai. Xiong Qinglai took a look at it and saw the heroic posture and brilliance in this article. He immediately invited its author, surnamed Luo Geng, into Tsinghua Garden. He arranged for Hua Luogeng to be a clerk in the Department of Mathematics of Tsinghua University, so that he could study on his own while listening to a large number of lectures. After that, Hua Luogeng was sent abroad to study in Cambridge, England. After returning to China, Xiong Qinglai, who had already served as the president of Yunnan University in Kunming, introduced him as a professor at the United Nations University. Hua Luogeng later went abroad again to teach at universities in Princeton and Illinois. After the founding of the People's Republic of China, Hua Luogeng immediately returned to China, and he presided over the work of the Institute of Mathematics of the Chinese Academy of Sciences.
Chen Jingrun also quickly wrote a special article on number theory in the library of Xiamen University, and the article was sent to the Institute of Mathematics of the Chinese Academy of Sciences. As soon as Hua Luogeng read the article, he saw the heroic and brilliant brilliance in the article, and also put forward a suggestion to transfer Chen Jingrun to the Institute of Mathematics as an intern researcher. Exactly: Xiong Qinglai recognizes Luo Geng with his eyes, and Hua Luo Geng knows Jingrun.
At the end of 1956, Chen Jingrun once again came to the capital Beijing from the southern coast.
In the summer of 1957, mathematician Xiong Qinglai also returned to the capital of the motherland from abroad.
At this time, the young and long salty gathered, and the sages were all over. At that time, there were many famous mathematicians such as Xiong Qinglai, Hua Luogeng, Zhang Zongsui, Min Sihe, Wu Wenjun, and many other celebrities; there was also a new generation of Junyan, such as Lu Qikeng, Wan Zhexian, Wang Yuan, Yue Minyi, Wu Fang, and so on; and there were also rising stars, such as Lu Ruchao, Yang Le, Zhang Guanghou, and so on, who had entered Peking University to study. In the disciplines of analytic number theory, algebraic number theory, implicit number theory, generalization analysis, geometric topology, etc., there are already many talents, and another Chen Jingrun has been added. Everyone holds the pearl of the spirit snake, and every family holds the jade of Jingshan. It is all the rage, and the lineup is neat. The conditions were met, and Hua Luogeng made a deployment. Focus on applied mathematics, but also on the crown jewel, the Goldbach conjecture!
Five
To understand what Goldbach's conjecture is, just review the math that I learned earlier in the third grade. Those 1, 2, 3, 4, 5, ten hundred thousand numbers, 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 kind of number, such as 2, 3, 5, 7, 11, 13, etc., which can only be divided by 1 and its own number, and cannot be divisible by other integers, which is called a prime number. In addition to 1 and its own number, it can also be divisible by other integers, such as 4, 6, 8, 9, 10, 12, etc., which are called composite numbers. An integer that is divisible by a prime number is called the prime factor of the integer. For example, 6 has two prime factors, 2 and 3. For example, 30 has three prime factors: 2, 3 and 5. Well, that's enough for now.
In 1742, Goldbach wrote to Euler that every even number not less than 6 is the sum of two prime numbers. For example, 6 = 3+3. For example, 24 = 11 + 13 and so on. Someone has done this on even numbers one by one, all the way up to 330 million, which shows that this is correct. But what about bigger numbers, bigger numbers? Guess right. Conjectures should be proven. It's hard to prove it. No one in the entire XVIII century has been able to prove it.
The whole nineteenth century also failed to prove it.
It was only in the twenties of the twentieth century that some progress began to take place.
It has long been wanted to prove that each large even number is the sum of two numbers that "do not have too many prime factors". They wanted to set up the encirclement in this way, and they wanted to prove that Goldbach's proposition of a prime number plus a prime number (1+1) was correct step by step.
In 1920, the Norwegian mathematician Brown used an ancient sieve method (which is a method of studying number theory) to prove that each large even number is the sum of two numbers with "no more than nine prime factors". Brown proved that the product of the nine prime factors plus the product of the nine prime factors, (9+9), is correct. This is the result of the sieving process. However, such an encirclement is still very large, and it must be gradually reduced. Sure enough, the encirclement gradually narrowed.
In 1924, the mathematician Rademahal proved (7+7), in 1932 the mathematician Essslemann proved (6+6), in 1938 the mathematician Buchstab proved (5+5), and in 1940 he proved (4+4). In 1956, the mathematician Vinogradov proved (3+3). In 1958, mainland mathematician Wang Yuan proved (2+3). The encirclement is getting smaller and smaller, getting closer to (1+1). However, all of the above proofs have a weakness, which is that none of the two numbers can be guaranteed to be prime.
As early as 1948, the Hungarian mathematician Lannyi set up another encirclement. Another battlefield was opened up, and I wanted to prove that every large even number is the sum of a prime number and a number with "no more than six prime factors". And he proved it (1+6). However, there was no progress for another decade.
In 1962, Pan Chengdong, a mathematician from mainland China and a lecturer at Shandong University, proved (1+5), which was a step forward, and in the same year, Wang Yuan and Pan Chengdong proved (1+4) again. In 1965, Buchstab, Vinogradov and the mathematician Pompi Alley all proved (1+3).
In May 1966, a bright signal was raised into the sky of mathematics, and Chen Jingrun announced in the seventeenth issue of Science Bulletin, a journal of the Chinese Academy of Sciences, that he had proved (1+2).
Since Chen Jingrun was selected to be transferred to the Institute of Mathematics, the buds of his intellect have bloomed one by one. He improved the results of Chinese and foreign mathematicians on the whole point problem in a circle, the whole point problem in a sphere, the Hualin problem, the three-dimensional divisor problem, and so on. With these achievements alone, his contribution is already huge. But when he had a good basis, he advanced against Goldbach's conjecture with astonishing tenacity. He forgot to sleep and eat, day and night, devoted himself to thinking, probing the essence, and performing a lot of calculations. Devoted to math made him in a daze. Once, when he hit a tree, he asked who had hit him, and he devoted all his mind and reason to the solution of this problem, and he paid a high price for it. His eyes were sunken. His cheeks were flushed with tuberculosis. The laryngitis was severe, and he coughed incessantly. Bloating, abdominal pain, unbearable. Sometimes they don't know anyone anymore, but they still remember numbers and symbols. He trudged through the rugged mountain roads of mathematics, struggling to move his steps. On the plateau of abstract thinking, he ascended to the steep rock, descended and ascended! A well-intentioned misunderstanding flew into his eyes. Ignorant taunts drilled into his ear canal. He was dismissive; He didn't have time to argue; Eat frost and drink snow, and one step is one step when you go up!
He was panting and sweating. I often feel that he can't support it anymore. But he climbed anyway. With the limbs, with the fingers and claws. How many times have I gone up and fallen down? Even the iron shoes should have been broken a long time ago. People ridiculed him for wearing shoes that were torn: a pair of shoes that were airy and breathable so that he would not get beriberi. I don't know how many times there have been terrible slips! He couldn't count how many times he had failed. He was undeterred. He summed up the lessons of failure, picked it up, welded it on, and used it as a nylon rope and metal ladder for mountaineering. Eat a trench, grow a wisdom. Fail once, take a step forward. Failure is the mother of success, and merit is built up of failure. He crossed the snow line and reached the snowy peaks and modern glaciers, and felt even more severely deprived of oxygen. How many times the mountain has been frozen in ice, how many times has the avalanche buried him! He is like those heroic mountaineers who conquered Mount Everest, climbing, climbing, climbing! And vicious slander, malicious slander is like a dark cloud that changes the sky and a nine-level wind. However, the enthusiastic support cleared the clouds for him, and the sunshine of love warmed him again. He was relentless towards his goal, he kept going, he kept climbing. Overcome the unattainable steepness of the first staircase, and appear before the second staircase. He only knows how to climb, above the abyss, he only climbs, between the infinite scenery. Sheet after piece of paper flew like snow, covering the earth. Numbers, symbols, lemmas, formulas, logic, and reasoning are accumulated on the floor slab, which is three feet deep. Suddenly turned into a group of mountains under the knees, and there were thousands of snow lotuses. He finally made it to the top and climbed the (1+2) steps.
He proved this proposition and wrote a long paper of more than 200 pages.
Mr. Min Sihe carefully read the original manuscript of the paper for him. Checked and checked, checked and checked. Yes, his proof is correct and reliable. He told Chen Jingrun that last year, people proved that (1+3) used large, high-speed electronic computers. And your proof (1+2) is entirely up to you. No wonder the essay is written for a long time. It's too long, and he was advised to simplify it.
The last sentence of the first paragraph of this article refers to the "literature [10]" that he announced in the form of a briefing in the "Science Bulletin", but only mentioned the results, and his proof has not yet been published. He was revising his long paper. It was at this moment that Chen Jingrun was suddenly swept into the turmoil of the political revolution. The rolling waves have struck the ideologies of all exploiting classes. The unprecedented Great Proletarian Cultural Revolution exploded one after another on the land of China, just like the successful tests of the spiritual atomic bomb and the hydrogen bomb.