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 three prime numbers, and 46 1 can also be written as 257+ 199+5, or the sum of three prime numbers. So I found that any odd number greater than 5 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 general proof, not individual inspection. "

Euler wrote back that 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 can't prove this proposition either.

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 proposition is more demanding than Goldbach proposition.

Now these two propositions are collectively called Goldbach conjecture.

A brief history of Goldbach conjecture

1742, Goldbach found in 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, etc. 1742 On June 7th, Goldbach wrote a letter to the great mathematician at that time. But he can't prove it. 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 overcome it, but they have not succeeded. Of course, some people have done some specific verification work, such as 6 = 3+3 and 8 = 3+5. 10 = 5+5 = 3+7,12 = 5+7,14 = 7+7 = 3+1,16 = 5+/. Someone looked up even numbers greater than 6 within 33× 108, and Goldbach conjecture (a) was established. However, 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 have tried their best to solve it, but they still can't figure it out.

It was not until the 1920s that people began to approach it. 1920s, proved by Norwegian mathematician Brown with an ancient screening method, and concluded that every even number n (not less than 6) larger than it can be expressed as (99). This method of narrowing the encirclement is very effective, so scientists began to gradually reduce the prime number contained in each number from (99).

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 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 proved "6+6".

1937, Lacey in Italy successively proved "5+7", "4+9", "3+ 15" and "2+366".

1938, Bukxitib of the Soviet Union proved "5+5".

1940, Bukhsiteb 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 of the Soviet Union, George W. vinogradov and Pemberley of Italy proved "1+3".

1966, China Chen Jingrun proved "1+2".

It has been 46 years since Brown proved "9+9" in 1920 and Chen Jingrun captured "1+2" in 1966. For more than 40 years since the birth of "Chen Theorem", people's further research on Goldbach's conjecture has been in vain.

■ Brownian sieve correlation

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+(2n-3) = 2i and 2i. 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, such as remembering that one pair is p 1 and p2, then both p 1 and p2 are prime numbers, that is, n=p 1+p2, then Goldbach's conjecture is proved. The first part is a natural idea. The key is to prove that' at least one pair of natural numbers has not been filtered out.

But because the big even number n (not less than 6) is equal to its corresponding odd number series (starting with 3 and ending with n-3), it is the sum of odd numbers added from beginning to end. Therefore, according to the sum of odd numbers, it has all possible correlations with prime number+prime number (1+kloc-0/) or prime number+composite number (1+2) (including composite number+prime number 2+ 1 or composite number+composite number 2), namely/. And the cross occurrence "category combination" of 1+0 can be derived as 1+ 1+1and 1+2,1+and. 1+2 and so on. Because 1+2 and 1+2 do not contain 1+ 1, 1 cannot cover all possibilities ". And 1+2 is excluded, then 1+ 1 is proved, otherwise 1+ 1 is not proved. However, the facts are: 1+2 and 2+2, and 1+2. Or the sum of the products of a prime number and two prime numbers), the basic basis for the existence of some laws (for example, 1+2 exists, 1+ 1 does not exist). So 1+2 and 2+2, and 1+2 (or at least one) "

Because the distribution of prime numbers itself changes in disorder, the change of prime number pairs is not simply proportional to the increase of even numbers. When the even number increases, the value of the prime number pair rises and falls. Can the change of prime pairs be related to the change of even numbers through mathematical relations? Can't! There is no quantitative law to follow in the relationship between even values and their prime pair values. For more than 200 years, people's efforts have proved this point, and finally they choose to give up and find another way. So some people use other methods to prove Goldbach's conjecture. Their efforts have only made progress in some fields of mathematics, and have no effect on Goldbach's conjecture.

Goldbach conjecture is essentially the relationship between an even number and its prime pair. There is no mathematical expression to express the relationship between an even number and its prime pair. It can be proved in practice, but it can not solve the contradiction between individual even numbers and all even numbers logically. How can an individual be equal to the average person? Individuals and the general are the same in nature, but opposite in quantity. There are always contradictions. Goldbach conjecture is a mathematical conclusion that can never be proved theoretically and logically.

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. It is generally believed that the most valuable thing in modern mathematics is the generalized Riemann conjecture. If Riemann conjecture holds, many questions will be answered. 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. Therefore, mathematicians tend to find some new theories or tools to solve Goldbach's conjecture 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 of 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 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.

Folk mathematicians solve Goldbach's conjecture mostly by elementary mathematics. Generally speaking, elementary mathematics cannot solve Goldbach's conjecture. To say the least, even if a great man 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 an exercise in math class.

At that time, Bai Ge tried to challenge the mathematical world and put forward the problem of the steepest descent line. Newton solved the steepest descent line equation with extraordinary calculus skills, and John White tried to skillfully solve the steepest descent line equation with optical methods. Jacob Bai tried to solve the 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.

Similarly, Hilbert once claimed to have solved Fermat's Last Theorem, but he did not disclose his 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.

Proof of Goldbach conjecture

Goldbach conjecture has puzzled people for more than 200 years, but it has never been proved. The simpler it looks, 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 conjecture.

Prime numbers are the basis of integers, that is, numbers that are not divisible by other numbers except 1 are prime numbers, and numbers 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, which was first proposed by Goldbach in 1742, but it has not been proved after more than 200 years. In fact, Goldbach conjecture is simpler than people think. One is that even numbers can be decomposed into the sum of two prime numbers, which is not unique. An even number can be decomposed into various sums of two prime numbers, and with the increase of even number, 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, it has been proved that prime numbers are infinite. 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. Because we usually think that 1 is not a prime number, there may be a combination of N- 1. We need to find out that N+K and N-K are both combinations of prime numbers, which can be done for relatively small numbers. For infinite numbers, we need to prove that the possibility that N+K and N-K are both 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.

Find the euler theorem of the number of prime numbers, and we can get the approximate number of prime numbers from this theorem. The number of prime numbers less than 2N is greater than the formula 1, 2n×1/2× (1-kloc-0//3 )× (1-kloc-0//5 )×. This formula contains prime numbers. It is not a good algorithm to find prime numbers within 2N by using known prime numbers, but the way to prove Goldbach's conjecture is similar to this formula.

For N+K and N-K, a * * * has N- 1 combinations, in which both numbers are prime numbers. The number A is similar to the above formula, and its minimum value can be calculated by the following formula 2, and A must be greater than the value of formula 2, formula 2, (n- 1) × {1/. Where p < √ 2N < P+M (the middle number is the square root of 2N), the next prime number greater than p is recorded as P+M, and the second prime number greater than p is recorded as p+L. The number of braces in the above formula is represented by f, and for p+m < √ 2h < P+L, the probability that even numbers are decomposed into two prime numbers in this interval is (h-66).

An even number between P2 (the square of p) and (P+M)2, where P2+ 1 can be divided into two prime numbers. The minimum value a is the smallest, but this value a is greater than 1, so at least one group of numbers are prime numbers, even numbers between (P+M)2 and (P+L)2. Substituting P2+ 1 and (P+M)2+ 1 into Equation 2, we can know that this probability is increased because the minimum m is 2, for example, we go to P = 1 1, P+M = 13.

120 is twice as much as 60, and 12 1 is less than the square of 1 1, and it is substituted into Equation 2; 59×1/2×1/3× 3/5× 5/7 ≈ 4.2, but 60 is divisible by 3 and 5, and the above formula is actually 59× 1/2× 2/3× 4/5/7 ≈ 65438. Then the probability that N+K and N-K are divisible by J at the same time is reduced to (J- 1)/J instead of (j-2)/j. In addition, when N-K is small, N-K may become prime numbers, which also increases the probability that these two numbers become prime numbers. Equation 2 is the minimum value, and it is not the sum of two prime numbers. In fact, 122 can be decomposed into the sum of four groups of prime numbers, which is close to the calculation result of the formula. This is because 122 divided by 2 equals 6 1, and 6 1 is a prime number, so there is no need to adjust the formula, but for n is a sum number, the result of adjustment can only be increased, so for any even number 2N, the minimum value decomposed into two prime numbers is increased.

The distribution of prime numbers is a definite sequence, but it is not a simple sequence. The probability of random distribution does not consider this deterministic distribution, so Goldbach conjecture can not be proved by random distribution theory, nor can the deterministic distribution of prime numbers be obtained. This is the difficulty of Goldbach's conjecture. To prove Goldbach's conjecture, we need to use prime number distribution and symmetry to eliminate it. This paper skillfully uses this point and proves it from the possibility that 2N can be decomposed into two prime numbers.

Note: p+m) 2 represents the square of p, because computers are not convenient to write. The following (P+M)2 represents the square.

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. You can skip these lines when you don't understand. X represents a sufficiently large even number.