Xueba's life simulator

Chapter 146 Goldbach Conjecture
The three major modern mathematics problems mentioned by Li Zhizhi before refer to the four-color conjecture, Fermat's last theorem and Goldbach's conjecture.

The four-color conjecture is also known as the four-color theorem and the four-color problem. It was first proposed by a university student named Gudry Guthrie in Yingguo in 1852. If you select some adjacent limited areas, you can use four colors to dye these areas, so that every two adjacent areas are dyed with different colors."

It has been 164 years since the four-color problem was raised, and it has not been solved yet.

Fermat's last theorem was proposed around 1637 by Fermat, a French scholar, when he read the Latin translation of Diophatus' "Arithmetic". He once wrote next to Proposition 11 in Volume 8 of the book: It is impossible to divide a cubic number into the sum of two cubic numbers, or a fourth power into the sum of two fourth powers, or in general divide a power higher than two into the sum of two equal powers . Regarding this, I am convinced that I have discovered a wonderful proof, but unfortunately the blank space here is too small to write it down."

The last sentence of this passage written by Fermat in the book has been quoted by countless posterity.

Fermat's last theorem was solved in 1994. During the past 300 years, countless scholars who are interested in this problem have been continuously studying the problem. It took 357 years from the time it was proposed to the complete solution.

The Goldbach conjecture was proposed by Goldbach in a letter to Euler in 1742. It has been 274 years since the original content of the conjecture proposed by Goldbach is "Any integer greater than 2 can be written as three prime numbers Sum."

The reason why Goldbach mentioned this conjecture in his letter to Euler was that he could not prove the conjecture himself, so he wanted to write a letter to ask Euler, an academic genius at the time, to help prove it. Euler obtained a bachelor's degree at the age of 15. He obtained a master's degree at the age of 16, and succeeded Daniel Bernoulli as a professor of physics at the age of 24. It is absolutely no exaggeration to say that he is a genius.

This Daniel Bernoulli is the proponent of Bernoulli's principle. The expression of Bernoulli's principle is called Bernoulli's equation, but anyone who involves fluid mechanics will learn this equation.

Goldbach's full name is Christian Goldbach, a Prussian who once served as the teacher of Russian Tsar Peter II. Because of his frequent visits to Europe, he met Leibniz, Euler and Bernoulli, etc. Keep in touch with them for a long time, and then they will write a letter to Ouola to help prove this matter.

But even a genius like Euler failed to prove the conjecture proposed by Goldbach until his death.

When Goldbach put forward this conjecture, the reason why he said that "any integer greater than 2 can be written as the sum of three prime numbers" was because in the mathematical circles at that time, people still believed that 1 was also a prime number.

But the current mathematical community no longer believes that 1 is also a prime number, so Goldbach's original conjecture has now become "Any integer greater than 5 can be written as the sum of three prime numbers."

After Euler saw the conjecture mentioned in the letter Goldbach sent him, although he didn't know how to prove the conjecture, he improved the conjecture.

他在回信中写道“我虽然现在还不知道该如何证明这个猜想，但我觉得这个猜想可以改成：任一大于2的偶数可以写成两个素数之和，比如4=2+2，6=3+3，8=3+5……”。

This improvement of Euler can be said to be an epic enhancement.

Euler's enhanced version of Goldbach's conjecture is the most common version now.

However, for more than 200 years, mathematicians around the world have proved Goldbach's conjecture and Fermat's last theorem through relay proofs, just like a suspension bridge with no planks and only two suspension ropes.

The proof of the relay is that people keep laying planks forward on the suspension bridge. After the first person lays the first plank, the second person can stand on the first plank and lay the second plank. By laying wooden planks backwards on top of the human body, one day we will be able to cross this suspension bridge and completely prove Goldbach's conjecture.

If the road is found to be inaccessible after going to the back, then the plank needs to be laid from scratch again.

On the way to prove Goldbach's conjecture, Hardy and Littlewood of Yingguo first invented the "circle method", and in 1923 they proved that under the premise that the generalized Riemann conjecture is established, each sufficiently large Any odd number can be written as the sum of three prime numbers.

In 1919, the Norwegian mathematician Brown improved the sieve of Eratosthenes, and proved that all even numbers of sufficient size can represent the sum of two numbers, and the number of prime factors of these two numbers is not More than 9.

The number of prime factors is how many prime factors can be divided into, and the prime factor decomposition is the content of the fifth grade of elementary school, so I won’t talk about it here.

In layman's terms, any sufficiently large even number can be written as the product of no more than 9 prime numbers plus the product of no more than 9 prime numbers.

Brown's conclusion was later called "9+9".

If 9 can be reduced to 1, it is equivalent to proving that a sufficiently large even number can be expressed as a prime number + a prime number, which is why people often hear people say that proving Goldbach's conjecture is proving "1+1".

In fact, regarding this point, Zhou Ming heard his teacher say that Chen Jingrun proved "1+1=2" when he was a child in school. At that time, he really thought it was a proof of 1+1=2, and he believed it for many years.

直到到后来看了相关的科目文章，周明才明白这里说的“1+1”并不是证明1+1=2，而陈景润证明的也并不是1+1，而是“1+2”。

自布朗证明了“9+9”之后，这条路便开始有人走了，先后由德国的拉特马赫于1924年证明了“7+7”，瑛国的埃斯特曼于1932年证明了“6+6”……

By 1966, Chen Jingrun followed this path and proved that "1+2" ​​is true, that is, "any sufficiently large even number can be written as a prime number plus the sum of the products of two prime numbers."

It can be proved that after "1+2", no one can take a step forward on this road.

Chen Jingrun and the others have come to this road, which is called almost prime number.

In addition, there are three ways to prove Goldbach's conjecture, namely: exception set, three prime number theorem for small variables and almost Goldbach's problem.

The proof process of Goldbach’s conjecture that Zhou Ming is going to write now uses the Hardy-Littlewood circle method and Brown’s improved sieve method, but it is fundamentally different from the method used by Chen Jingrun and the others. After all, it is a method that has been continuously improved by many mathematicians after decades of hard work.

For people in the future, the improvement is carried out step by step, and when Goldbach's conjecture is finally proved, people will not feel that the method used is too subversive compared to the previous method.

But for people now, Zhou Ming's proof of Goldbach's conjecture is a subversive improvement on the existing method on the approach of almost prime numbers.

Just like using Zhang Yitang’s method to reduce the distance between twin primes to 256 is close to the limit, almost primes follow the sieve method. Chen Jingrun proved it to the establishment of "1+2", which is already in a sense. The power of the sieve method has been brought into full play.

Because if the weighted sieve method wants to prove the "1+1" of Goldbach's conjecture, then it is necessary to take x=2 in the weighted sieve, and this will make it difficult to estimate the main term and the remaining term, and this is why The reason why no one has been able to go further on this road since 1966.

To thoroughly prove Goldbach's conjecture requires new ideas or new mathematical tools, or subversive improvements in existing methods.

……

After Li Mingzhi left, Zhou Ming's office became extremely quiet. You could hear footsteps outside the office and rustling outside the window, and even the sound of Zhou Ming's pen writing mathematical formulas on the draft paper. Like someone telegraphing Morse code.

"Ding!"

The message reminder from the mobile phone attracted Zhou Ming's attention, causing Zhou Ming to stop writing.

Zhou Ming took a look at the phone and unlocked it, and found that it was not important news, so he turned off his phone directly to prevent it from disturbing him again.

In this way, the time passed by every minute and every second. When Zhou Ming felt a little hungry, he looked at the time and found that it was almost two o'clock in the afternoon.

People are iron rice and steel, and if they don't eat a meal, they will be very hungry.

In order to fill his stomach, Zhou Ming had no choice but to put down the pen in his hand, and went to the cafeteria to have a quick lunch. After wolfing down his lunch, Zhou Ming returned to the office soon to continue writing his proof process.

"Hi, Professor Zhou."

"Hi, Professor Zhou."

On his way to the canteen and back to the office from the canteen, Zhou Ming met several students who knew him and greeted him.

Since Zhou Ming appeared on several Internet searches and was reported by some official media in 2015, many students on the campus of HKUST knew Zhou Ming. Every time Zhou Ming met students who greeted him at school, Zhou Ming also said hello to him every time. He responded to them with a smile.

But this time Zhou Ming just nodded and left in a hurry, which made those students who had greeted Zhou Ming several times before feel a little curious.

Does Professor Zhou have anything to do today?Why does it look like you are in a hurry?

After Zhou Ming returned to the office, he buried him in the draft paper that proved Goldbach's conjecture.

I don't know how long it had been since the sun went down at night, so Zhou Ming checked the time and found that it was past nine o'clock.

"This time has passed too fast. Why do you have the feeling of 'I don’t know how time has changed in the mountains, but Chunzhou has listened to the barbarians for six times'?" Zhou Ming said to himself with a smile, "It's better to go back first, and by the way, Find a small shop on the way to settle the dinner."

Zhou Ming tidied up his desk briefly. He first organized the papers that had been filled with the certification process into a stack and put them away. Then he picked up a stack of unused printing paper and put it together with the used paper. Take its place.

"Hello, Dean Li. I want to ask for a few days off."

Last night Zhou Ming randomly found a small shop downstairs in the community, had dinner, and then went home to take a hot bath, and washed the changed clothes before he went.

After finishing all this, it was almost eleven o'clock, so Zhou Ming went straight to bed.

Early the next morning, Zhou Ming called Li Mingzhi early, planning to ask him for leave.

"Ask for leave? What's the matter? Did something happen? Why do you want to ask for leave?" Li Mingzhi asked with concern when he heard that Zhou Ming was going to ask for leave and thought he was sick.

"No, I'm going to concentrate on writing the proof process of Goldbach's conjecture at home in the past few days, and I don't want to waste time by running around the school dormitory." Zhou Ming explained.

"Oh, so that's the case, that's okay. You are working to write the proof of Goldbach's conjecture at home. How can it be considered a leave of absence? You don't count as a leave of absence. You can do your things at home without worrying about the school. This way."

When Li Mingzhi heard that Zhou Ming said that he was taking leave to stay at his residence and study Goldbach's conjecture, he immediately told him so.

After talking about this, Li Mingzhi was still not at ease, and he continued: "By the way, your research on soybeans should not be over yet, right? Tell me about the experimental field where you grow soybeans, and I will help you go there every day. Look, it saves you having to go out for this matter every day."

"Ah?" Zhou Ming didn't expect Li Zhizhi to be so considerate, he was very moved, and said without any excuses, "My experimental field is..."

In addition to talking about the experimental field, Zhou Ming also talked about other things, and Li Mingzhi agreed.

"Okay, okay, no problem, no problem, don't worry, you can leave these things to me, and you can just prove Goldbach's conjecture with peace of mind." Li Mingzhi was not unhappy about these things Zhou Ming said, on the contrary, he was happy Said to Zhou Ming.

The method and proof process used by Zhou Ming to prove the twin prime number conjecture made Li Mingzhi realize how powerful Zhou Ming is, so he also had high expectations for Zhou Ming to prove Goldbach's conjecture, and naturally he didn't want him to be delayed by other things.

In Li Mingzhi's view, helping Zhou Ming is not just helping Zhou Ming, but doing things for mathematics and bringing glory to the mathematics community in Huaguo.

Faced with various requests from Zhou Ming, Li Mingzhi agreed to them one by one.

When Zhou Ming was writing the process of proving Goldbach’s conjecture in his room at night, the other side of the world was located at Princeton University in Princeton, New Jersey on the east coast of the Ugly country, and the faculty and staff had just started working.

A staff member of Princeton University who specializes in sending and receiving emails and other related matters was preparing to deal with today's emails, and quickly noticed Zhou Ming's previous emails.

This staff member knew Zhou Ming's name, after all, his superiors had said before that he should pay attention to whether there was an email from this person.

Now that he noticed Zhou Ming's name, the staff member quickly clicked on the email from Le Zhou Ming, and quickly read the content of the email.

Although Zhou Ming did not accept the invitation and did not agree to come to Princeton University to give an academic lecture, Zhou Ming invited professors from his school’s Institute of Mathematics to go to China’s University of Science and Technology to listen to his academic report on proving Goldbach’s conjecture. It was indeed bigger than him accepting the invitation.

Although this staff member did not have a deep understanding of mathematics, Goldbach guessed that he had heard of his name. After being surprised, he quickly sent this email to the Institute of Mathematics .

Soon, Zhou Ming's e-mail was sent to the mailboxes of every mathematics professor in the Institute of Mathematics of Princeton University.

(End of this chapter)