From p>1729 to 1764, Goldbach kept correspondence with Euler 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 one at will.

take another odd number, such as 461,

461=449+7+5,

which is also the sum of three prime numbers. 461 can also be written as 257+199+5, which is still the sum of three prime numbers. In this way, I find that any odd number greater than 5 is the sum of three prime numbers.

Although the above results have been obtained in every experiment, it is impossible to test all odd numbers, which requires general proof, not individual test. "

Euler wrote back that this proposition seems correct, but he could not give 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.

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+. Goldbach's conjecture holds.

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

Now these two propositions are generally referred to as Goldbach's conjecture

Goldbach's conjecture brief history

In p>1742, Goldbach found in his teaching that every even number not less than 6 is two prime numbers (numbers that can only be divisible by 1 and itself). 12 = 5+7, etc. On June 7, 1742, Goldbach wrote to Euler, the great mathematician at that time. Euler replied to him on June 3, saying that he believed this conjecture was correct, but he could not prove it. Describing such a simple problem, even a leading mathematician like Euler could not prove it, and this conjecture attracted the attention of many mathematicians. Since Goldbach put forward this conjecture, many mathematicians have been trying to overcome it. But none of them 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 and 18 = 5+. ..... and so on. Some people check even numbers within 33×18 and greater than 6, and Goldbach's conjecture (a) holds. However, strict mathematical proof needs the efforts of mathematicians.

Since then, this famous mathematical problem has attracted the attention of thousands of mathematicians in the world. 2 years have passed. No one has proved it. Goldbach's conjecture has thus become an unattainable "pearl" in the crown of mathematics. People's enthusiasm for Goldbach's conjecture 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 192s, Norwegian mathematician Brown used an ancient screening method. A conclusion is drawn: every even number n (not less than 6) larger than even number can be expressed as (99). This method of narrowing the encirclement is very useful, so scientists gradually reduce the number of prime factors in each number from (99) until every number is a prime number, which proves Goldbach' s conjecture.

The best result at present is that China mathematician Chen Jingrun. Known as Chen's 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 large even number and can be expressed as "1+2".

■ Goldbach conjecture proves that progress is related

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" for short) is as follows:

In p>192, Brown of Norway proved "9+9".

In 1924, Latmach of Germany proved "7+7".

In 1932, Britain. Italy's Lacey proved "5+7", "4+9", "3+15" and "2+366" successively.

In 1938, the Soviet Union's Buchwitz proved "5+5".

In 194, the Soviet Union's Buchwitz proved "4+4". C is a big natural number.

In p>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". In 1966, Chen Jingrun of China proved "1+2".

It took 46 years from 192 when Brown proved "9+9" to 1966 when Chen Jingrun captured "1+2". It's all for nothing.

■ correlation of Brownian screening method

The idea of Brownian screening method is as follows: any even number (natural number) can be written as 2n, where n is a natural number, and 2n can be expressed as the sum of a pair of natural numbers in n different forms: 2n = 1+(2n-1) = 2+(2n-2) = 3+(. 2i and (2n-2i),i=1,2,…; 3j and (2n-3j), j = 2, 3, ...; And so on), if it can be proved that at least one pair of natural numbers has not been filtered out, for example, remember that one pair is p1 and p2, then p1 and p2 are both prime numbers, that is, n=p1+p2, so Goldbach's conjecture is proved. The previous part of the narrative is a natural idea. The key is to prove that' at least one pair of natural numbers has not been filtered out'. At present, no one in the world can prove this part. To prove it, This conjecture is solved.

However, because the even number n (not less than 6) is equal to its corresponding odd number sequence (beginning with 3, The tail is n-3), which is the sum of odd numbers added from beginning to end. Therefore, according to the sum of odd numbers, the related type of prime number+prime number (1+1) or prime number+composite number (1+2) (including composite number+prime number 2+1 or composite number+composite number 2+2) belongs to prime number+composite number type (note: 1+2 or 2+1). The cross appearance of 1+1 and 1+2 (the appearance of incomplete agreement), the "complete agreement" of 2+1 or 2+2, and the "incomplete agreement" of 2+1 and 2+2, etc., can lead to the "category combination" of 1+1,1+1 and 1+2 and 2+2. 1+2 two "category combinations" do not contain 1+1. Therefore, 1+1 does not cover all possible "category combinations", that is, its existence is alternating. So far, if the existence of 1+2 and 2+2 and 1+2 can be excluded, 1+1 will be proved, otherwise, 1+1 will not be proved. However, the fact is that And 1+2 (or at least one) is the basis for the existence of some laws (such as the existence of 1+2 and the absence of 1+1) revealed in Chen's theorem (any sufficiently large even number can be expressed as the sum of two prime numbers, or the sum of the products of one prime number and two prime numbers), so 1+2 and 2+2, and 1+2 (or at least one) That is to say, it is inevitable. Therefore, it is impossible for 1+1 to be established. This thoroughly demonstrates that Brownian screening method can not prove "1+1".

Because the distribution of prime numbers themselves presents disorderly changes, there is no simple direct proportional relationship between the changes of prime numbers and the growth of even numbers, and the values of prime numbers fluctuate when even numbers increase. Can the changes of prime numbers be related to the changes of even numbers through mathematical relations? Can't! There is no quantitative law to follow in the relationship between even numbers and their prime pairs. For more than 2 years, people's efforts have proved this point, and finally they choose to give up and find another way. So people who use other methods to prove Goldbach's conjecture have made progress in some fields of mathematics, but have no effect on Goldbach's conjecture.

Goldbach's conjecture is essentially the relationship between an even number and its prime pairs. The mathematical expression that expresses the relationship between an even number and its prime number pair does not exist. It can be proved in practice, but it can't solve the contradiction between individual even numbers and all even numbers logically. How can the individual be equal to the general? The individual and the general are identical in quality, but opposite in quantity. Contradictions always exist. Goldbach conjecture is a mathematical conclusion that can never be proved theoretically and logically.

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 odd conjecture points out that any odd number greater than or equal to 7 is the sum of three prime numbers. Even number. 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 modern mathematicians are 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, Hilbert, a great mathematician, gave a report at the World Congress of Mathematicians, and raised 23 challenging questions. Goldbach conjecture is a sub-question of the eighth question, which also includes Riemann conjecture and twin prime conjecture. In modern mathematics, it is generally believed that the generalized Riemann conjecture is the most valuable. If Riemann conjecture is established, many questions will be answered, but Goldbach conjecture and twin prime conjecture are relatively isolated. If these two problems are simply solved, it is not of great significance to solve other problems. Therefore, mathematicians tend to find some new theories or new tools to solve Goldbach's conjecture "by the way" while solving other more valuable problems.

For example, a very meaningful problem 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?

An important reason is that Riemann conjecture is difficult for people who have never studied mathematics to understand what it means, while Goldbach conjecture can be understood by primary school students.

It is generally believed in mathematics that these two problems are equally difficult.

Most folk mathematicians use elementary mathematics to solve problems. It is generally believed that elementary mathematics cannot solve Goldbach conjecture. I'm afraid it's almost as meaningful as doing an exercise in a math class.

At that time, Brother Bai tried to challenge the math world and put forward the problem of the steepest descent line. Newton solved the steepest descent line equation with extraordinary calculus skills, John Bai tried to skillfully solve the steepest descent line equation with optical methods, and Jacob Bai tried to solve this problem in a more troublesome way. Although Jacob's method was the most complicated, But 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 that he had solved Fermat's last theorem, but he didn't 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 mathematical circles are working hard to study new tools and methods, expecting Goldbach's conjecture, the "golden chicken", to give birth to more theories.

Proof of Goldbach's conjecture

Goldbach's conjecture has puzzled people for more than 2 years, but it has always been. The simpler it seems, the more difficult it is to prove. There are many similar conjectures in mathematics, which are simple on the surface, but difficult to prove clearly. This is a * * * property of mathematical conjectures.

Prime numbers are the basis of integers, that is, numbers that are not divisible by other numbers except 1 and itself are prime numbers, and those multiplied by prime numbers are composite numbers. Every even number greater than or equal to 6 can be decomposed into the sum of two prime numbers. This is But more than 2 years have passed, and it has not been proved yet. In fact, Goldbach conjecture is simpler than people think. First, even numbers can be decomposed into the sum of two prime numbers, which is not unique, and an even number can be decomposed into a variety of sums of two prime numbers, and with the increase of even numbers, there can be more solutions. Of course, the process of proof is not based on ordinary screening or random probability. The process of proof is based on a new simple formula. Similar to mathematical induction.

First of all, the prime number is infinite, which has been proved by people. Here we just mention it. Even numbers are represented by 2N, and the sum of N+K and N-K is equal to 2N, where k < n, k is an arbitrary positive integer, which can be represented as the sum of two numbers. Since we usually think that 1 is not a prime number, there may be N-1 such combinations. We need to find out that both N+K and N-K are combinations of prime numbers, which can be done for relatively small numbers. For infinite numbers, we need to prove that the possibility that both N+K and N-K are prime numbers increases with the increase of N, so we can prove that any even number can be decomposed into the sum of two prime numbers. < P > From this theorem, we can find the euler theorem of the number of prime numbers.