Scientists, or intellectuals, are a very special group. They may not have a lot of wealth, but in a sense, a large part of the growth of wealth in this society is based on the efforts of these people.

If we must make an analogy, they are like the engine of social wealth growth. They do the most tiring work in areas that people cannot see.

Just like after a normal family buys a car, except for maintenance or repairs, no one would deliberately open the car hood every day to see what the engine looks like.

Yes, everyone knows that the engine is important, but no one bothers to take a look at it when nothing happens.

Because most ordinary people don't know how the engine works or its principles. They just need to know that the thing can still be used and is not broken.

Mathematicians are like the engines of academia. They do the hardest and most tiring work. In fact, most of the time, everyone's attention to the mathematical community is limited to whether there are any good tools that can be used.

Even for the same scientific research funding application, the research funds for mathematics are less than those for other sciences.

Such as physics, chemistry... After all, the latter requires various experiments. Laboratory investment and various instruments require money...

So in a sense, mathematicians don't have many bright moments.

After all, only a few people can create the ultimate romance with a pen and a piece of paper.

Most people can actually become famous if they can make some improvements based on other people's research results.

Yes, in fact, there is no need for Chen Zhuoyang to feel ashamed.

Because the nature of mathematical research determines that major breakthroughs are uncommon.

Ninety percent of mathematicians' jobs are to refine existing theories or solve local problems. Of course, this does not mean that these seemingly inconspicuous results are unimportant.

Because these accumulations may give those geniuses a sudden flash of inspiration.

Various reasons also determine that for most mathematicians, being able to give a 60-minute report at an important meeting is already the highlight of life.

After all, this is not a branch venue, and the time given is very long. And there are really many people here today, and it even seems to be more people than at the last World Congress on Algebraic Geometry.

Fortunately, Qiao Yu has become accustomed to being an opinion leader among the crowd.

"The distribution of prime numbers is one of the core issues in number theory research, and its spacing problem has always been an unsolved mathematical problem. In actual research, mathematicians have proven that the upper bound of the prime number spacing can be restricted within a specific range.

Professor Zhang Yuantang's pioneering work reduced the upper bound of this distance to 70 million, and through the efforts of mathematicians around the world, it was reduced to 246. What I want to report today is based on the axiom system of generalized modal number theory, a

New geometrization methods to solve related problems.

By mapping the prime distribution into modal space, and using geometric tools such as modal density functions, modal paths, and modal convolution, we demonstrate that the bounded distance between primes can be further reduced to an upper bound of 6..."

It starts with a simple introduction.

First of all, let everyone know how this work is carried out. Without the cooperation of Lot Dugan, this aspect will be very troublesome.

Because the abstract "Based on the axiom system of generalized modal number theory, a new geometric method is derived" can make countless people in the audience confused.

But let’s not talk about all of them now, at least more than 80% of the participants will not be confused.

Because after the publication at noon yesterday, the organizer, who had been prepared for it, had already started to move.

More than two thousand papers that had been printed out in advance were distributed through the hosts of each session before dinner.

At least every registered member of the Mathematical Society has a copy. If there is no interest in this proposition, the paper will be given directly to those who are interested.

Some people directly borrowed the paper and printed a copy.

Hotels that hold such academic conferences will thoughtfully arrange printing services. Of course, if the internal staff is too busy, there will also be dedicated personnel who will send them outside for printing.

Although one night may not be enough for all participants to fully understand the paper, at least most people already know the general concept and have a preliminary understanding.

At the same time, sixty minutes is already the longest time for an academic report at a top conference, but it is actually not enough for Qiao Yu to introduce the generalized modal space framework to everyone. So after briefly talking about the abstract,

Qiao Yu immediately entered the state.

"...The modal path Γ is a continuous curve in the modal space, which is used to describe the distribution trajectory of prime numbers in the geometric space. In order to reduce the modal point spacing, the following optimization construction of the path is required:

As you can see on the big screen, this formula, where T is the path period, is used to ensure that the path has a periodically repeating structure in the modal space."

"It can be seen from the above that the curvature and distribution of the path Γk are driven by the local high-value area of ​​the modal density function ρM(r), and Γk passes through the local extreme point of the modal density function, ensuring path coverage in high-density areas.

The modal path has geometric symmetry. Assuming symmetric mapping: M→M, it satisfies:

Through the above optimization structure, the high density, periodicity and symmetry of the modal path Γ can be guaranteed..."

…

"Do you understand?" Zhang Yuanling, who was deep in thought, was interrupted by a heavy line beside him.

First he nodded slightly, then shook his head.

Well, it doesn't actually interrupt his train of thought, it can only be said to interrupt his idea of ​​further exploration.

To be honest, I didn’t fully understand it!

But this can't be blamed on him. In fact, Zhang Yuanling believed that at least 90% of the people at the meeting could not understand what Qiao Yu was talking about at this time.

It’s not that everyone’s level is not good enough, or their understanding is not good enough. The main reason is that what I’m talking about today is too special. It’s a brand new theoretical framework, and it’s only been one night that everyone has been exposed to this framework.

Before you fully understand the framework built by Qiao Yu, you can imagine how difficult it is to understand the content of this paper.

So while listening to the report, Zhang Yuanling kept recalling some conclusions and theorems in another paper.

For example, knowledge points such as the definition of modal space, modal density function, construction of modal paths, modal convolution formulas, etc...

Then use these unfamiliar knowledge points to deduce what Qiao Yu is talking about. These have to go through a very complex transformation process in the human brain.

It was easy to talk when you were young, but as you get older, your brain is not as flexible as it was when you were young.

"You don't understand either, so I feel relieved. After all, I have reviewed his ideas before." Zhongxing said in a joking tone.

It's a joke indeed.

If Zhang Yuanling knew at that time that Qiao Yu could complete this axiom system to this extent in more than two months, he would not have questioned anything at all and directly approved it.

But now it can only be said that it was a miscalculation!

"Tian Yan is so lucky that I don't know how to describe it." Zhang Yuanling said nonchalantly.

He knew that Chongzhongxing had no ill intentions. The two of them had a good relationship, and they had discussed it together in private yesterday. If Chongzhongxing came to review Qiao Yu's project document, there would definitely be doubts.

It can only be said that Zhang Yuanling was quite unlucky this time.

"Hey...it's not just good luck. After all, he is at Yanbei University." Zhongxing sighed.

Zhang Yuanling was silent.

Yes, probably only Yuan Zhengxin has the qualifications to say that Tian Yan is really lucky, and even Yuan Zhengxin is a little bit worse.

The reason is naturally the siphon effect of prestigious schools.

When more than 80% of the competition students and candidates in the society, as well as the parents, regard Yanbei and Huaqing as their first choices, other prestigious schools want to compete with these two schools for the best seedlings. Indeed, It's too difficult.

Especially a youngster like Qiao Yu who is already in his prime at the age of sixteen.

Although Shuangdan University is also a famous school and its School of Mathematics is ranked among the best in the country, it is still a little behind compared to Yanbei and Huaqing.

Seeing that Zhang Yuanling did not speak, Zhongxing sighed and said: "Hey... those were two papers yesterday. In the next at least thirty years, I think we can look forward to China having a say in the world of mathematics."

Zhang Yuanling smiled, but the smile was not so cheerful.

Just listening to the second half, it is obvious that this is a compliment, but since he is an old colleague, of course he can understand the meaning of it:

At least in the next thirty years, there is no need to consider challenging the status of Yanbei and Huaqing in the Chinese mathematics community...

Whatever ambitions you have, you can put them aside for now. These are all conceivable.

With this realization, Zhang Yuanling suddenly thought of a possibility and couldn't help but said: "Old Shen, what do you think about next year's Fields Medal..."

Zhong Xing was stunned for a moment, then shook his head slightly and said with emotion: "This... shouldn't be possible, right? After all, he is still too young, he will only be seventeen next year.

But we can look forward to that time in 2030. He will only be 21 years old at that time. But one thing I am sure of is that he will definitely be considered for many awards next year."

Zhang Yuanling nodded, indeed... seventeen years old, this age is too young.

This is definitely one of the situations that the International Mathematical Union jury needs to take into account.

Seventeen years old really allowed Qiao Yu to win the prize. How many years will he rule the world of mathematics? Not to mention that every year there are many candidates who are stuck at 40 years old. If they exceed 40 years old, they will never have a chance.

Moreover, the Fields Medal Jury Committee should consider leaving some expectations for the future... If Qiao Yu really wins the award next year, the topic of the youngest Fields Medal winner will probably be completely ended.

This also made Zhang Yuanling somewhat emotional. Who could have imagined that one day, the reason that prevents a mathematician from winning the Fields Medal is that he is too young?!

After all, this thing is too outrageous.

"So, Professor Zhang, after this meeting, I think you can choose a time to chat with Qiao Yu... Besides, he withdrew the project on his own last time, so it actually has little to do with you."

Zhongxing looked at the young man on the stage who was still talking with a complicated expression and said casually.

Zhang Yuanling thought for a while, then nodded inconspicuously, and said: "Let's finish the paper first. It's always a bad idea to rush into it. There must be common topics."

Zhong Xing also nodded. He could understand Zhang Yuanling's concerns. They were all decent people, so they couldn't completely lose face, right?

…

In the front row, Yuan Zhengxin didn't care much about what Qiao Yu said. After all, he had read the paper countless times and had communicated with Qiao Yu.

The narration at the report meeting also followed the paper, and Qiao Yu's thinking was clear at every step.

If you have this time, why not mention more of his students.

"Yesterday you said that Qiao Yu had already told you about the idea of ​​this paper?"

"Yeah." Qiao Xi responded softly.

"Can you understand?" Yuan Zhengxin asked again.

"I don't understand it just by reading the paper, but I probably understood it after Qiao Yu explained it." Qiao Xi replied.

Yuan Zhengxin nodded happily. Mathematics is like this. If you can understand it, you are already superior to 90% of ordinary people.

This has been the case even since junior high school, but it was not obvious at that time. In high school, math scores will gradually begin to show a polarization trend.

And if you want to know whether a high school student with good math scores has a talent for math, it's actually very simple. Just ask him if he is tired from studying math.

If students in high school have to work hard to get high scores by solving questions, there is a high probability that they have no talent. It is best to stay away from mathematics when applying for majors.

After all, this is a subject where after a certain level of study, hard work alone is of no use.

Yuan Zhengxin asked again: "Then have you discussed with Qiao Yu what work needs to be done to further lower the upper bound or even completely solve this proposition?"

Qiao Xi shook her head and said: "I haven't discussed it yet, but I think if there is to be a breakthrough, the structure of the modal path needs to be further optimized.

We also need to consider introducing some number theory methods, such as the sieve method, and verify the infinity of the prime numbers on the path again."

Yuan Zheng thought for a while and then said: "He didn't ask you to help with this work?"

Qiao Xi shook her head again and said softly: "He asked me to try to prove the modal existence of twin prime numbers first."

"Oh? Do you have any ideas?"

"I mentioned it briefly."

With that said, Qiao Xi took out a pen and paper and wrote down some concepts: "Qiao Yu said that if you want to introduce twin characteristics, you need to use a new mapping concept. For example, a high-order nonlinear operator, used in

Find core symmetries in modal space.

Since this type of modal mapping is nonlinear and irreversible, he designed an operator matrix structure called the supermodal operator matrix, or HOM for short. Then for any mode (α, β), there are specific elements that satisfy the following characteristics.

"

After writing this casually, Qiao Xi shook her head slightly and explained: "To be honest, I think this is very difficult. Qiao Yu often has some thoughts, well... I think the angles are very weird, and I can't keep up with him.

Ideas.”

Yuan Zhengxin smiled and said, "It's normal to not be able to keep up with his train of thought. Look at the more than 3,000 people present today, at least 3,000 of them can't keep up with his train of thought.

But you should be able to feel that Qiao Yu hopes that you can help him. So even if it is difficult, you have to think quickly and believe in yourself. Your work is very important in helping him completely solve the twin prime conjecture.

After solving this problem, he can still challenge to higher heights. So just do me a favor and don't let this kid slack off, okay? At least don't let him slack off when he is most creative!"

"Ah?" Qiao Xi found it hard to imagine that an old man who kept his word would actually say these words to her with a hint of pleading, and she was stunned for a moment.

But Qiao Xi soon came to his senses, nodded quickly and replied: "Don't worry, Teacher Yuan, I will definitely urge Qiao Yu to continue working hard."

"But it's not just that he has to work hard. Have you ever thought about the possibility that he is actually waiting for you?" Mr. Yuan asked.

"Well... I will work hard too." Qiao Xi, who had never felt that studying was a pressure, suddenly felt a mountain-like pressure falling on her slender shoulders...

Looking at Qiao Yu who was swaying freely on the stage, I couldn't help but feel a little depressed.

…

In fact, by this time the report was going on, many people in the back row had already begun to sneak away quietly. Most of them were students and mathematics enthusiasts, and of course there were a few professors.

This is actually something that can't be helped.

Listening to a lecture, if you don't understand it at all... This experience can actually be understood by everyone. It is similar to the feeling of not being able to understand it in a math class.

In short, once you don't understand it, those complicated formulas are almost like a heavenly book, not like the language of the human world. Then you will become bored, and even feel that your days are like years.

If you add in the fact that most of the students who can attend this meeting are graduate students majoring in mathematics, they may feel deeply affected, which may lead to a series of negative emotions.

For example, the mathematics I study and research seems to be different from other people's mathematics. Even if they are both studying number theory, without a certain understanding of the prerequisite framework, it is really difficult to understand the structure of the geometricalization of number theory problems...

…

All in all, this is a very painful thing. If you consider that the mathematician who gave the report is sixteen years old, this pain can easily be doubled.

Fortunately, Qiao Yu, who was sitting on the podium, didn't actually notice this. Even if he did, he didn't really care.

If he was still excited when he gave the report for the first time, then this time, Qiao Yu was completing the task, mainly because he started too high.

"...According to Lemma 7, modal convolution describes the distribution pattern of modal points on the path Γ. The pattern in the local area is as shown in the figure:"

"By controlling the maximum value of the wide basis, the distance between modal points can be limited to no more than 6. At the same time, because the global structure of the modal path Γ is periodic, its local high-density characteristics are repeated globally. Therefore, any

The modal distance of modal points r_p, r_q∈Γ satisfies: d_M(r_p,r_q)≤6

...Finally, according to Theorem 2, Theorem 3, Theorem 4, and Theorem 5, it can be seen that each modal point r_p∈M corresponds to a prime number p, and the modal path Γ describes the distribution trajectory of the prime number.

Modal distance d_M(r_pr_q) is the geometric distance between modal points, and its properties directly reflect the prime spacing ∣pq∣ in the sense of number theory...

To sum up, d_M(r_p,r_q)≤6 is equivalent to ∣pq∣≤6, and there are infinitely many pairs of prime numbers with a spacing between ∣pq∣≤6 in the sense of number theory. From now on, the proof is complete."

Qiao Yu's time control ability is very strong, and it took fifty-five minutes out of sixty minutes.

In fact, if he spoke a little faster, he could finish it in fifty minutes. This is also a more appropriate time.

Because generally invited reports require about ten minutes of Q&A time at the end.

Of course, in such a short time, you can probably answer three or five questions, so the quality of the questions is very important.

This is also the reason why various report conferences arrange a moderator. During the question and answer time, the moderator will select the questioner and ask questions about what the speaker said.

However, today's lecture will be quite special. Many people do not have that much time to fully digest the content of the generalized modal axiom system. Therefore, regarding his paper, I only listened to the fifty-minute lecture and could not find out anything valuable.

question.

So I just left it for five minutes and let the host take care of it. If anyone asked, just answer casually. If no one asked, the host would say a few words and everyone could go to eat early.

Of course, if there are any problems, you can wait until everyone can communicate with you later.

The fact was just as Qiao Yu imagined, it was obvious that the people in the audience were not very enthusiastic about asking questions.

At a casual glance, you can see that no one in the front row raised their hands... It is a brand new axiomatic framework, and it still takes some time to digest it.

Those who can be chosen to be the presenters of the report are all smart and flexible people. Especially after seeing the looks on the faces of some of the big guys in the front row, they naturally understand how to handle this situation.

"Thank you very much for Qiao Yu's wonderful report. Regarding the upper bound of 6 between prime numbers, this report has given a detailed explanation.

If you still have any questions about the paper, I believe that Qiao Yu will definitely take some time to communicate with you in more detail after this meeting. Let us thank Qiao Yu again for his wonderful discussion."

Soon there was extremely warm applause from the audience.

Although no one asked questions, except for the person who gave the report on the stage, everyone else at the meeting understood the significance of this article in the world of number theory.

At that time, all the efforts in the world only lowered the upper limit of the interval between prime numbers to 246. More than ten years later, someone finally lowered this number again, and lowered it to 6 all at once.

To be honest, 99% of the people in the audience looked at Qiao Yu with envy and a little bit of jealousy...

Even more than that.

With the publication of his and Chen Zhuoyang's papers on the Internet, anyone who has carefully read the two papers may have a rather complicated mood at the moment.

A work in one of the four top magazines, an honor so many people dream of, was given away by that ridiculously young guy on stage...

Yes, in the eyes of many people, the second paper is really just a gift.

Forget about the framework of the generalized modal axiom system, this whole set of proof logic is really not something that ordinary people can think of.

But when it comes to the follow-up verification work, it really gives many people the feeling that I can probably do it.

Even more than that...

There are always some invisible rules in academia. For example, it may sound unbelievable for tutors to use students' achievements, but this situation has always existed, and it can even be said that it does not distinguish between countries.

The students' research is directly incorporated into their own framework, and the final results are presented with the tutor as the protagonist. The data collected and analyzed by the students are directly used to publish independent papers.

Even more shamelessly, there are cases where the tutor directly names the first author on the paper that the student completed independently.

After all, not everyone in the academic world is a highly respected professor, and there are also those who are ignorant and unskilled...

The young man on the stage abruptly divided what could be done in one paper into two papers, just to give the collaborators one work at a time.

Yes, there is no need for Chen Zhuoyang to say anything. Everyone is in the academic circle. As long as they have read the paper, they probably know what is going on.

This makes many people's emotions so complicated that it is difficult to describe.

Of course, these feelings had nothing to do with Qiao Yu. He just stood up and bowed slightly to the audience. It still felt perfect.

Then he walked off the stage in a hurry. He was the second one to give a report today. Then it was time for dinner, and then he was free in the afternoon...

Qiao Yu planned to find Senior Brother Chen to accompany him around the venue and chat.

It was not because he wanted to listen to the report, but mainly because he wanted to listen to Senior Brother Chen’s rainbow farts...

Everything is fine with Qiao Xi, but it is still too difficult for Qiao Xi to praise him to the sky.

Senior Brother Chen did a good job in this regard. Unfortunately, Qiao Yu soon discovered that he had oversimplified things.

Although the morning meeting was over, Qiao Yu found that he could not walk out of the banquet hall.

As soon as I stepped off the podium, I was surrounded by a group of people.

It was literally surrounded. Of course, the situation was not very chaotic. Everyone was engaged in mathematics, and many were even doctoral-level professors.

Everyone is just doing academic discussions. But it has nothing to do with the content of today’s report. Everyone is asking questions about the generalized modal axiom system...

Although they were not very humble, as long as someone crowded around Qiao Yu and started asking questions, the others would choose to listen quietly to Qiao Yu's explanation first.

"Qiao Yu, in your paper, Lemma 3: The distribution of modal points on the modal path depends on the weighted modal density function, in which the selection of the weight function w (α, β, γ) affects the geometry of the modal distance.

characteristic.

But I don’t quite understand the basis for selecting this weight function. Is it logarithmic density? Or is it other known number theory results or geometric properties?”

"This... well, it has been proven... It is analogous to the logarithmic density 1/logp in the prime number distribution. But it is not a simple logarithmic density. There is an extension here to control the geometric properties.

Moreover, symmetry and smoothness must be ensured. For example, its defining parameters are symmetrical... Well, you see, if a periodic term sin(β) is introduced, the repeatability and global structure of the modal path will be weakened...

…”

…

Tian Yanzhen, Yuan Zhengxin and Qiao Xi stood watching from a distance. There was no way, they couldn't even squeeze in now.

Qiao Xi also probably understood why Tian Yanzhen asked her to watch Qiao Yu and not let her go anywhere yesterday.

Once the enthusiasm of mathematicians bursts out, it makes Qiao Xi feel quite scary.

So he simply suggested: "Well... why don't we just ignore him first?"
Chapter completed!