-Goldbach conjecture

The queen of natural science is mathematics, and the crown of mathematics is number theory. Goldbach conjecture is the crown.

That dazzling pearl. Since Goldbach put forward this conjecture in the middle of the eighteenth century, countless numbers have been put forward.

Scholars are attracted by the dazzling brilliance of this pearl and have joined the ranks of picking. however

But no one ever succeeded.

The eighteenth century has passed, and no one can prove it.

The nineteenth century has passed, and still no one can prove it.

Since the 20th century, natural science has developed rapidly, and countless science fortresses have been chased by scientists.

A conquest. In the 1920s, Goldbach's conjecture began to make progress. Mathematicians from all over the world

Round and round, gradually narrowing the encirclement. In this world-wide century competition, there is a person who is familiar to everyone.

Chen Jingrun, a China native, defeated mathematicians from all over the world and won the leading honor. Although Godba

Hector's conjecture is just a conjecture, but there is still no other scientific peak since it was put forward.

Can hide its light. History has reached the turn of the century and is about to turn a new page, but mankind is still there.

We can only enter the 2 1 century with this regret. Goldbach guessed, what kind of question is this?

Find the largest prime number

1, 2, 3, 4, 5, ... These numbers are called positive integers. In a positive integer, a number divisible by 2,

For example, 2, 4, 6, 8, ... are called even numbers. Not divisible by 2, such as 1, 3, 5, 7, ..., yes.

It's called odd number. There is also a number, such as 2, 3, 5, 7, 1 1 and so on, which can only be added to 1 and itself, but not to it.

It can be divisible by a positive integer and is called a prime number. Besides 1 and itself, it can also be divisible by other positive integers, such as 4,

6, 8 and 9 are called composite numbers. If an integer is divisible by a prime number, the prime number is called.

The prime factor of this integer. If 6, there are two prime factors, 2 and 3; And 2 10, with four elements: 2, 3, 5 and 7.

Factors.

Prime number is a very important concept in mathematics. An important reason for prime numbers, the Greek mathematician Ogilvy

Germany (Euclid, about 350 ~ 300 BC) was known as early as 2000 years ago.

Yes Euclid collected all the mathematical knowledge available at that time and wrote a mathematical book with a volume of 13.

Make the original. There is such a theorem in the book, which is now called fundamental theorem of arithmetic: each is greater than.

The natural number of 1 is either a prime number or can be expressed as the product of several prime numbers, which is not included in the prime number line.

The order of columns is unique.

For example, 630 is the product of seven prime factors (one of which appears twice):

630=2×3×3×5×7

The right part of the equal sign in the above formula is called the prime factorization of the number 630.

Fundamental theorem of arithmetic told us that prime numbers are the basic building materials for constructing natural numbers and all natural numbers.

They built it. Prime numbers are much like elements of chemists or elementary particles of physicists. Master the post

By factorizing a number, mathematicians get almost all the information about it. So primitive

Qualitative research has become one of the oldest and most basic topics in number theory. As early as Euclid's time,

It is proved that there are infinitely many prime numbers. However, for everyone, prime numbers do not seem to have any special status.

Fang. 2, 3, 5, 7, 1 1 ... Everyone can name a string at will. But what about the future? let us

Let's have a look.

We first choose a natural number and record it as n; For the number of prime numbers less than n, we call it π(n

)。 Comparing the changes of π (n)/n with different values of n, we will find that in the order of natural numbers.

Columns, prime numbers are getting less and less.

Table 1: Distribution of Prime Numbers

π(n) π(n)/n

10 4 0.400

100 25 0.250

1000 168 0. 168

10000 1229 0. 123

100000 9592 0.096

1000000 78498 0.078

/kloc-Mei Sen, a French mathematician in the 7th century, put forward a method for finding prime numbers.

Mei Sen's Random Thoughts on Physical Mathematics was published in 1644.

C) It is stated in the preface that for n = 2, 3, 5, 7, 13, 17, 19, 3 1 67,127,257, the number Mn is.

= 2n- 1 is a prime number, and Mn is the sum of all other numbers less than 257. How did he get this?

What about the conclusion? Nobody knows. But he is surprisingly close to the truth. It was not until 1947 that there was a desktop computer.

People can check his conclusion. He only made five mistakes: M67 and M257 are not prime numbers, and M6 1,

M89 and M 107 are prime numbers.

Mason number provides a good method for finding very large prime numbers. The function 2n increases rapidly with the increase of n.

Long, this ensures that Mason number Mn will soon become extremely large, and people will think of looking for people who make Mn a prime number.

Noun (short for Noun) is a prime number called mersenne prime. Elementary algebra knowledge tells us that unless n itself is a prime number, there is no

Then Mn will not be a prime number, and we only need to pay attention to the n of the prime number. But most prime numbers n also lead to

Mason number Mn is a composite number. It seems that it is not easy to find the right n-although the first few numbers make you feel that it is not difficult.

. 1998 February 12 California state university19-year-old Roland Clarkson found a new suitable n.

He found the largest known prime number by computer. This prime number is twice that of 302 1377 minus 1. this

It's a 909526-bit number. If you write this number continuously in a normal font size, its length can reach 3000.

Dom. Clarkson used his spare time to calculate for 46 days, and finally proved that this is a prime number on 65438+1October 27th. this

How big is a prime number? Let's compare it with another big prime number!

In an ordinary chessboard with 8×8 squares, we put 2 mm thick chips in the squares according to the following rules.

Code (such as British 10p coins). Number the squares first, that is 1 ~ 64. Put two chips in the first box.

Code, the second box put 4 chips, the third box put 8 chips. And so on, in the next box.

The number of chips is exactly twice that of the previous box. Therefore, there are 2n chips in the nth grid and the last grid.

There are 264 chips. Can you imagine how high this pile of chips is? 1 m? 100 meter? 10000m? Absolutely wrong.

! Believe it or not. This pile of chips will soar into the sky and surpass the moon (it is only 400 thousand kilometers away)

), more than the sun (1500,000 kilometers away), almost directly to the nearest star (except the sun) Alpha Centauri.

A star, about 4 light years away from the earth. In decimal numbers, 264 is:1844674073709551616.

264 is so impressive that in order to get 2302 1377- 1 which appears in the largest prime number at present, you need to

Play the above games on a board larger than 1738× 1738!

Finding large prime numbers has practical application value. It promotes the development of distributed computing technology. in this way

Method, it is possible to use a large number of personal computers to complete projects that would have been completed by supercomputers. In addition,

In the process of finding large prime numbers, people must multiply large integers repeatedly. Now some researchers have found that.

The method of speeding up the operation can be used in other scientific research. You can also use large prime numbers.

Encryption and decryption. The method of finding mersenne prime can also be used to check whether the computer hardware operation is correct.

Compared with infinite prime numbers, what we have found so far is extremely limited. In the meantime, we can

It is proved that there are few propositions related to prime numbers. Goldbach conjecture is just a proposition about prime numbers, A.

We humans have spent more than 250 years studying unproven propositions.

Goldbach's Conjecture

Seemingly simple numbers contain a lot of interesting and profound knowledge. In the study of number theory

In the research, we often put forward a "guess" carefully according to some perceptual knowledge, and then push it through strict mathematics.

Theory to prove it. As we said above, any composite number can be decomposed into the product of prime numbers, so the composite number

Decomposition into the sum of prime numbers? Is there any rule in this?

1742, German middle school teacher Goldbach found "any one.

Even numbers can be written as the sum of two prime numbers. For example: 6 = 3+3, 9 = 2+7 and so on. He is interested in many couples.

The figures have been verified and all show that they are correct. But it needs to be proved. Because this unproven mathematical proposition

Can only be called speculation. He couldn't prove this proposition himself, so he asked the famous Swiss numbers for verification at that time.

Scientist Euler turned to him for help. Euler was one of the most famous mathematicians at that time, although

He believed in Goldbach's conjecture, but was stumped by this seemingly simple proposition. until

After his death, Euler failed to prove Goldbach's conjecture.

Goldbach's letter puts forward two conjectures:

Any even number greater than 2 is the sum of two prime numbers.

Any odd number greater than 5 is the sum of three prime numbers.

It is easy to prove that conjecture (2) is the inference of conjecture (1), so the problem boils down to proving conjecture (1).

In fact, for this conjecture, some people check the even numbers one by one. Until hundreds of millions.

Giant, everything shows that this conjecture is correct. But what about the bigger and bigger numbers? Guess should also be right. guess

Should be proved. However, it is very difficult to prove this. 1900, the German mathematician Hilbert in the international number

In the speech of the Institute, Goldbach conjecture is regarded as one of the most important mathematical problems left over from history. He will.

"Goldbach conjecture" is included in his "23 challenges of contemporary mathematicians" And in 19 12,

Landau, a German mathematician, said in his speech at the International Mathematical Society that even if the weak proposition "(3) is proved to exist.

Positive integer a, so that every integer greater than 1 can be expressed as the sum of prime numbers not exceeding a ",is also modern.

Mathematicians are beyond their power. It should be noted that if (1) holds, a = 3 is enough. 192 1 year,

Thomas Hardy, a British mathematician, said at a mathematical conference in Copenhagen that it is as difficult to guess (1).

Compared with any unsolved mathematical problem.

However, human intelligence always breaks through the limits set by itself one after another.

Just after that, 1, that is, 1922, British mathematicians Hardy and Te Li Wood put forward the study of Goldbach.

The method of conjecture, the so-called "garden method". 1937, the Soviet mathematician Ivy Noguera Madoff applied the circle.

Method, combined with his trigonometric sum estimation method, it is proved that every odd number large enough is the sum of three prime numbers.

. Thus basically proved the conjecture (2) put forward in Goldbach's letter.

Just as some mathematicians tried to attack Goldbach's conjecture (2), other mathematicians also questioned it.

Guess (1) blew the horn of the charge. A long time ago, people wanted to prove that every big even number was 2 "

The sum of integers without too many prime factors. They want to set up the encirclement like this, and they want to do it step by step.

Prove Goldbach's conjecture that a prime plus a prime (1+ 1) is correct. So, people

Although it is very slow, scientists are approaching to prove Goldbach's conjecture step by step.

1920, the Norwegian mathematician Brown improved Ehardo's Rennie's "screening method", which has a history of more than 2,000 years.

It is proved that every large enough even number is the sum of two positive integers with the number of prime factors not exceeding 9. Relative to the final

Proposition (1+ 1), we will record Brown's result as (9+9). 1924, German mathematician Radmahal.

Er proved (7+7); 1930, the Soviet mathematician Shnirman combined his integer "secret rate".

The value of a can be estimated by proving proposition (3) with Brownian sieve method. 1932, British mathematician Esterman.

It is proved that (6+6); 1938, the Soviet mathematician Buchstaber proved (5+5); 194

In the book, he proved (4+4). 1956, mathematician vinogradov proved (3+3).

.

Chinese mathematician Hua began to study this problem as early as 1930s, and achieved good results.

Obviously, the conjecture (1) is correct for "almost all" even numbers. Shortly after liberation, he initiated and pointed out that

He directed some of his students to study this problem, and achieved many achievements, which were highly praised at home and abroad. 1965

Wang Yuan (3+4) proved the talent of China mathematicians. In the same year, the Soviet mathematician A. Vino

Gradoff proved (3+3) again. In 1957, Wang Yuan proved (2+3). The encirclement is getting smaller and smaller,

Getting closer (1+ 1). But all the above proofs have a weakness, that is, two numbers.

None of them can be sure to be prime numbers.

In fact, mathematicians have long noticed this. So they set up another kind of encirclement,

That is, trying to prove that "any big even number can be written as an integer with a prime number and another prime number factor not too much."

The sum of numbers. "1948, Hungarian mathematician Raney reopened another battlefield, and the other was interrupted.

Clear: Every big even number is the sum of a prime number and a number with no more than six prime factors. 1962

In 1998, Chinese mathematician Pan Chengdong, a lecturer at Shandong University and Soviet mathematician Barba Encai proved their independence.

(1+5), a step forward; In the same year, Wang Yuan, Pan Chengdong and Barba proved this point again (1+4).

. 1965, Buchstaber, vinogradov and the mathematician Pompeii Alley all proved it (1+3).

.

People's continuous progress in proving Goldbach's conjecture seems to make people see this.

Fully proved its hope. There are only two steps left from (1+3) to (1+ 1). Who can be the most

And take off this jewel in the crown?

1966, Chen Jingrun, a young mathematician in China, proved it (1+2) and got the conjecture so far in the world.

(1) Best score. He proved that any large enough even number can be expressed as the sum of two numbers.

Moreover, one of them is a prime number and the other is a prime number; Or the product of two prime numbers. Although "Goldbach"

Theorem "has not yet been produced, but this closest conclusion has been nicknamed China by all countries in the world.

Man's Name-"Chen Theorem"

Pick the jewel in the crown.

1933, Chen Jingrun, from Fuzhou, Fujian. His father is a clerk in the post office and his mother.

Pro is a kind but overworked woman who gave birth to twelve children and raised six. Although not.

Which parents don't want to love their children, but Chen Jingrun has brothers and sisters, ranking third.

With younger brothers and sisters, you can't be your parents' favorite child. It seems that he is redundant, and Chen Jingrun is not.

How many childhood joys have you enjoyed?

When Xiao Jingrun began to remember, the Japanese invaded Fujian Province. Young, he can only mention

Living in fear, the mind has been greatly hurt. He is not happy at home, as he always was in primary school.

It is being bullied that makes him develop an introverted personality. Chen Jingrun began to like math because of math problems.

Calculus can help him kill most of his time.

After graduating from primary school, Chen Jingrun was still a discriminated child in junior high school. War of Resistance against Japanese Aggression is over, Chen.

Jingrun was admitted to Huaying College. At that time, one of the schools was the director of the Aviation Department of the National Tsinghua University.

Math teacher. The teacher is knowledgeable and tireless in teaching, which inspires many students' love for mathematics.

Once, the teacher introduced a famous problem in number theory to the students in class, which was Goldbach.

Hector conjecture. For other students, maybe the three-minute fever will soon pass, because it is a puzzle.

Two centuries of human problems! Don't say solving, that is, for a great mathematician, want to get one.

It takes a lot of effort to make progress. However, I was fascinated by this question and was deeply impressed.

Mind, until you have paid a lifetime of hard work!

After graduating from high school, Chen Jingrun entered the Mathematics Department of Xiamen University. He was promoted because of his excellent grades.

After graduation, I stood on the platform and became a teacher. But his long-term introverted personality makes him unable to resemble it.

The high school teacher taught his students all his rich knowledge. After many twists and turns, his math day

Fu was discovered by Hua who worked in the Institute of Mathematics of China Academy of Sciences at that time, and Chen Jingrun was transferred here on 1956.

China, a temple of mathematical research, became an assistant researcher.

Since then, his mathematical talent has been fully demonstrated. In just a few years, he asked the hour in the circle.

The achievements of mathematicians at home and abroad in problems, integral problems in the ball and Waring problems have been improved. That's all.

As far as he is concerned, he has achieved great success. But he will never forget what he left in his heart in high school.

That deep imprint-Goldbach conjecture. With sufficient conditions, he set foot on this pearl.

Yes!

Unremitting efforts have achieved fruitful results. Chen Jingrun finally took another step on the road to winning the pearl.

As an important step. After making important new improvements to the screening method, he initially solved the problems in 1965 (1+2).

), wrote more than 200 pages of proof. 1966 In May, Chen Jingrun published Science in China Academy of Sciences.

17 Bulletin announced that he had proved it (1+2).

Just a year ago, foreign mathematicians proved it with high-speed computers (1+3). And Chen Jingrun can only rely on it

The conclusion drawn by hand-written mental arithmetic is better. However, the proof is too complicated and needs to be further simplified.

So, Chen Jingrun plunged into the manuscript paper again and continued his climbing road. Everything that has nothing to do with research,

Don't disturb his thoughts. In his 6-square-meter hut, between several sacks of calculus manuscript paper, Chen

Jingrun endured the unbearable hardships of ordinary people and pursued that dream tirelessly.

1973 just after the spring festival, Chen Jingrun completed the revised draft of his paper "even numbers are prime numbers and".

Do not exceed the sum of the products of two prime numbers, that is, (1+2), and publish it. Chen Jingrun proved in the paper.

：

Every big even number can be expressed as the sum of the products of a prime number and no more than two prime numbers;

Let d (n) be the number of tables in which n is the sum of two prime numbers, and prove that there is D(N).

n)/(LnN)2；

These two conclusions greatly advance the proof of Goldbach's conjecture, and are called "Chen" internationally.

Theorem ".

This achievement has aroused strong repercussions in the field of mathematics and won China a great international reputation. western

Fang reporter soon learned about it, and the news quickly spread all over the world. British mathematician Huberstein and German Numbers

When the scientist Liszt learned about it, the book Sieve Method was being printed. However, they immediately withdrew the manuscript and re-edited it.

Write and supplement Chapter 11: "Chen Theorem" and give a high evaluation: "Select methods from any aspect.

On the whole, it is the culmination of glory. "At the same time, in some foreign mathematical magazines, such as" Outstanding Achievements "

There are countless similar praises, such as "Nine" and "Brilliant Theorem". An English mathematician even wrote letters.

Tell him, "You moved all the mountains!"

Unfortunately, long-term hard research has brought Chen Jingrun a lot of pain. Although he

Thanks to the cordial care of the party and the country, I still can't cross out the proof of Goldbach's conjecture because of mental tension.

A classic mathematical problem that mathematicians all over the world have struggled for more than 250 years has the last step left.

The biggest regret in the history of mathematics in this century. Nevertheless, in more than 30 worldwide number theory problems, Chen

Jingrun conquered six or seven ways by himself, especially in proving Goldbach's conjecture.

No one can match it today.

1March 1996 19, a day that makes the whole mathematical community feel sorry.

Professor Chen Jingrun, an academician of the Chinese Academy of Sciences and a first-class researcher at the Institute of Mathematics, was invalid due to long illness, and

At the age of 63.

Century expectation

Many people don't understand the significance of studying Goldbach conjecture as a "pure number game".

And then what? You know, scientific achievements can be roughly divided into two categories. One is obvious economic value, which can be directly used in things.

In terms of qualitative wealth, it is a "valuable treasure"; However, another achievement is in the macro world, micro

The world, cosmic celestial bodies, elementary particles and other fields, their economic value can not be estimated, far away

Beyond people's imagination, it is called "priceless treasure". Chen Jingrun's "Chen Theorem" belongs to the latter.

Goldbach conjecture is very important to mathematics, in fact, it is a foundation of prime number mathematics.

This particle "is one of the most important conjectures. Solving it will promote the whole human understanding of natural science. "

Stride. Therefore, many mathematicians are committed to simplifying the proof of Chen Theorem. There are several * * * in the world at present.

A simplified proof, the simplest of which was obtained by China mathematicians Ding and Pan Chengdong.

Many methods invented and applied in the process of studying Goldbach's conjecture are not only for number theory.

It is widely used and can be applied to many branches of mathematics, which promotes the development of these branches of mathematics.

It provides endless power for the progress of the whole society. For example, prime numbers provide a good way for human beings to encode and decode passwords.

The method has played a great role in people's communication security. As the cornerstone of the natural science building, mathematics, its every

This kind of progress, even the smallest progress, may make us build the whole building more magnificent.

Decades have passed, and the attempt to prove Goldbach's conjecture was put forward that day.

It never stopped from the beginning, but the whole world once again fell into a long-term chaos. at present

Once again, mankind stands at the historical moment at the turn of the century. The rapid development of science and technology has given scientists a chance to climb.

The peak of knowledge provides far more convenient conditions than before. Especially the use of high-speed computers, make some

Mathematical problems such as "four-color theorem" are solved easily. But for Goldbach's conjecture, this crown

Pearl, in the next century, can human intelligence fully display its dazzling aura?

No one knows the answer, and the expectation of the century is calling to mankind. (Pan Zhi)