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Please prove: 1+1=2
Goldbach conjecture can be roughly divided into two conjectures:
■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.
from p>1729 to 1764, Goldbach and Euler kept correspondence for 35 years.
in a letter to Euler on June 7, 1742, Goldbach put forward a proposition. He wrote:
"My question is this:
Take any odd number, such as 77, and you can write it as the sum of three prime numbers:
77 = 53+17+7;
take an odd number, such as 461,
461=449+7+5,
which is also the sum of these three prime numbers. 461 can also be written as 257+199+5, which is still the sum of the 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 have been obtained in every experiment, it is impossible to test all odd numbers. What is needed is general proof, not one other test. "
Euler wrote back: "This proposition seems to be 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, so odd number 2N+1 can be written as the sum of three prime numbers.
But the establishment of Goldbach's proposition does not guarantee the establishment of Euler's proposition. So Euler's proposition is more demanding than Goldbach's proposition.
Now these two propositions are generally referred to as Goldbach conjecture
[ Edit this paragraph] A brief history of Goldbach conjecture
In 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). Such as 6 = 3+3, 12 = 5+7 and so on. On June 7th, 1742, Goldbach wrote to Euler, a great mathematician at that time. In his reply on June 3th, Euler said that he believed this conjecture was correct, but he could not prove it. Describing such a simple problem, even a leading mathematician like Euler can't prove it, and 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, 1 = 5+5 = 3+7, 12 = 5+7, 14 = 7+7 = 3+11, 16 = 5+11, 18 = 5+13, ... Someone checked the even numbers within 33×18 and greater than 6, and Goldbach's conjecture (a) was established. But strict mathematical proof needs the efforts of mathematicians.
Since then, this famous mathematical problem has attracted the attention of thousands of mathematicians all over the world. 2 years have passed, and no one has proved it. Goldbach conjecture has thus become an unattainable "pearl" in the crown of mathematics. People's enthusiasm for Goldbach's conjecture problem has lasted for more than 2 years. Many mathematicians in the world have tried their best, but they still can't figure it out.
it was not until the 192s that people began to approach it. In 192, Brown, a Norwegian mathematician, proved by an ancient screening method and came to a conclusion that every even number n (not less than 6) larger than it 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 reduced the number of prime factors in each number from (99) until every number is a prime number, thus proving Goldbach' s conjecture.
The best result at present is proved by China mathematician Chen Jingrun in 1966, which is called Chen Theorem: "Any sufficiently large even number is the sum of a prime number and a natural number, and the latter is only the product of two prime numbers." This result is usually referred to as a big even number and can be expressed as "1+2".
Goldbach's conjecture proves progress correlation
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 "s+t" problem) is as follows:
In 192, Norwegian Brown proved "9+9".
In 1924, Latmach of Germany proved "7+7".
in 1932, Esterman of Britain proved "6+6".
in 1937, Lacey of Italy proved "5+7", "4+9", "3+15" and "2+366" successively.
in 1938, the Soviet Union's Bukhsiteb proved "5+5".
in 194, the Soviet Union's Bukhsiteb proved "4+4".
In 1948, Rini of Hungary proved "1+ c", where c is a large natural number.
in 1956, Wang Yuan of China proved "3+4".
in 1957, Wang Yuan of China proved "3+3" and "2+3" successively.
In 1962, Pan Chengdong of China and Barba of the Soviet Union proved "1+5" and Wang Yuan of China proved "1+4".
in 1965, Buchteber and vinogradov Jr. of the Soviet Union and Pompely of Italy proved "1+3".
in 1966, Chen Jingrun of China proved "1+2".
It has been 46 years since Brown proved "9+9" in 192 and Chen Jingrun captured "1+2" in 1966.
[ Edit this paragraph] The meaning of Goldbach conjecture
"Described in contemporary language, Goldbach conjecture has two contents, the first part is called odd conjecture, and the second part is called even conjecture. The conjecture of odd numbers points out that any odd number greater than or equal to 7 is the sum of three prime numbers. The conjecture of even numbers means that even numbers greater than or equal to 4 must be the sum of two prime numbers. " (Quoted from Goldbach's Conjecture and Pan Chengdong)
I don't want to say anything more about the difficulty of Goldbach's Conjecture. I want to talk about why the modern mathematics community is not interested in Goldbach's Conjecture, and why many so-called folk mathematicians in China are interested in Goldbach's Conjecture.
In fact, in 19, the great mathematician Hilbert made a report at the World Congress of Mathematicians and raised 23 challenging questions. Goldbach conjecture is a sub-problem of the eighth question, which also includes Riemann conjecture and twin prime conjecture. In modern mathematics, it is generally believed that the most valuable thing is the generalized Riemann conjecture. If Riemann conjecture is established, many problems will be answered, while Goldbach conjecture and twin prime conjecture are relatively isolated. If these two problems are simply solved, it will not be of great significance to solve other problems. So mathematicians tend to find some new theories or tools while solving other more valuable problems, and solve Goldbach's conjecture "by the way".
] For example, a very meaningful question is the formula of prime numbers. If this problem is solved, it should be said that the question about prime numbers is not a problem.
why are folk mathematicians so obsessed with Kochi's conjecture and don't care about more meaningful problems such as Riemann's conjecture?
An important reason is that Riemann conjecture is difficult for people who have never studied mathematics to understand what it means. Goldbach's conjecture can be read by elementary school students.
It is generally believed in mathematics that these two problems are equally difficult.
most folk mathematicians solve Goldbach's conjecture with elementary mathematics. It is generally believed that elementary mathematics cannot solve Goldbach's conjecture. To take a step back, even if there was an awesome person who 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 Diliu challenged the mathematics field 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 problem-variational method. Now, Jacob's method is the most meaningful and valuable.
Similarly, Hilbert once claimed that he had solved Fermat's Last Theorem, but he did not disclose his own method. When someone asked him why, 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 that this "golden chicken" can give birth to more theories. ]
[ Edit this paragraph] Reportage: Goldbach Conjecture
1.
Let px(1,2) be the number of prime numbers P suitable for the following conditions: x-p=p1 or x-p=p2p3, where P1, P2 and P3 are all prime numbers. [this is not easy to understand; When you don't understand, you can skip these lines. Use x to represent a sufficiently large even number.
p-1 1
life CX = ii-ii 1-
p \ xp-2 p <; 2 (p-1)2
p> 2
for any given even number h and sufficiently large x, xh (1,2) is used to represent the number of prime numbers p satisfying the following conditions: p≤x,p+h=p1 or h+p=p2p3, where P1, 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 reference [1], which are detailed as follows.
2.
The above is quoted from a paper on analytic number theory. This passage is quoted from its "Introduction" and puts forward this question. It is followed by "(2) several lemmas", which are full of various formulas and calculations. Finally, the "(3) result" proves a theorem. This paper is extremely difficult to understand. Even a famous mathematician may not be able to 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 proved is now called "Chen Theorem" by all countries in 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 China Academy of Sciences.
Chen Jingrun was born in Fujian in 1933. When he was born into 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 in those days, he would have prospered, but his father refused to join. Some colleagues say that he is really out of touch with the times. His mother is a kind and overworked woman, and she gave birth to twelve children. Only six survived, of which Chen Jingrun was the third. There are brothers and sisters in the world; There are younger brothers and sisters. If there are many children, they will not be loved by their parents. They are becoming more and more a burden to their parents-redundant children, redundant people. From the day he was born, he came into this world like a person who was declared persona non grata.
he didn't even enjoy much childhood happiness. Mother worked hard all day long to love him. When he can remember, a fierce war broke out. The Japanese invaded Fujian province. He is so young that he lives on tenterhooks. My father went to a post office in Sanming City, Sanyuan County as a 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, Maoyushan forest had become a miserable world. All the men were slaughtered by the Kuomintang bandit troops, and no one survived. There are not even any old men left. Only women are left. Their lives are particularly bleak. Flower gauze is too expensive again; 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. Planes don't bomb here, and the mountains are getting a little prosperous. But moved to a concentration camp. In the middle of the night, the whip often echoes painfully; From time to time, there are gunshots that kill martyrs. The next day, those who came out to work in chains looked even more gloomy.
Chen Jingrun's young mind has suffered great trauma. He is often conquered by panic and confusion. He didn't have 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 like he's alone. It's just that he is thin and weak. You can't 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 was too sensitive and felt the cannibalism of those people in the old society too early. He was made into an introverted person, 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.
In mathematics, there is another very famous "(1+1)", which is the famous Goldbach conjecture. Although it sounds amazing, its topic is not difficult to understand. You can understand its meaning as long as you have the mathematics level in the third grade of primary school. It turns out that in the 18th century, the German mathematician Goldbach accidentally discovered that every even number not less than 6 is the sum of two prime numbers. For example, 3+3 = 6; 11+13=24。 He tried to prove his discovery, but he failed repeatedly. In 1742, the helpless 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 the even numbers greater than 6 until they reached 33,,. The results show that Goldbach's conjecture is correct, but it just can't be proved. So this conjecture that every even number not less than 6 is the sum of two prime numbers [(1+1) for short] is called Goldbach conjecture, which has become an elusive "pearl" in the crown of mathematics.
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