Chapter 154 Proof of infinity of pairs of prime numbers with interval 6

Tian Yanzhen told Qiao Yu to get out of here. After Qiao Yu walked out of the office, he started to make plans.

The benefits of giving a report at the annual mathematics meeting are obvious. Qiao Yu thinks that the tutor and Mr. Yuan probably felt that it was humiliating that they had not been able to apply for the project before, so they fought for this opportunity.

Let him talk about this idea in front of everyone, which somewhat means letting the entire mathematical community comment on the theory.

It can be understood as a cry to the Chinese mathematics community: "Everyone, come and see if we are selfish in giving Qiao Yu the opportunity to work on a major project!"

But of course this must be under the premise that he has not submitted a paper to Ann.Math. Judging from the attitude of his tutor, Qiao Yu can tell that giving a report at a conference is still useful without publishing a paper in a top journal.

After all, the former is only speaking to the Chinese mathematics community, while the latter is speaking to the world mathematics community.

But now there are only seventeen or eight days left on the 25th. Prepare a paper that can be delivered as a one-hour report at the annual mathematics meeting, and you can’t lose the instructor’s face...

It does seem a bit difficult.

This is the disadvantage of being too good. The instructors think that he can do anything!

So on the way back to the dormitory, Qiao Yu felt what he should have felt at his age - melancholy!

Totally unprepared!

During this period of time, he was focused on completing the proof of the multi-modal space system, and was working on the proof of the extension of two dimensions to three dimensions. This work will take at least one or two months to complete.

Well, Qiao Yu has to admit that without any direction, it must be a joke to ask him to spend half a month to come up with a paper out of thin air?

This is a bit troublesome...

Soon Qiao Yu sat in front of his computer and began to think hard.

The main reason is that the timing of the conference is too much. It happens to be the beginning of November, the month Ann.Math is published every year, and it happens to be a single month every year.

This is also the reason why Qiao Yu feels that his and Senior Brother Chen's papers may not be published this year.

If you submit your manuscript in October, no matter how quickly it is reviewed, it will probably not be until November. In addition to the typesetting time, the fastest publication time will be January next year, or even March.

This is under the condition that everything goes well. If the reviewer has any questions about the paper, they can discuss it back and forth, and they may have to push it back.

This is also the reason why many university teachers who have signed 3+3 employment agreements with universities are under great pressure.

Generally, this kind of agreement has clear requirements for the number of papers to be published during the assessment period, such as three papers to be published in journals of a certain level in three years.

It may not sound difficult. But for young teachers who have just entered colleges and universities, they not only have to complete the most arduous teaching tasks, but also do research and go back and forth with reviewers.

Projects have always had more monks than meat, and it is difficult to publish articles on time without the recommendation of a big boss. There are only so many academic positions that universities can provide.

Most people are unwilling to acquiesce. After all, going to teach in a certain three or even junior college basically means severing oneself from the mainstream academic world, and they will stay like that for the rest of their lives.

Thinking of this, Qiao Yu suddenly felt that he was not in such a difficult situation. After all, it was impossible for him to encounter the above situation.

It is nothing more than writing a paper that will make Director Tian and Mr. Yuan feel that they will not lose face. Although the time is shorter, as long as there is a general direction, it should not be a big problem.

The key is direction.

Then Qiao Yu turned his attention to prime numbers...

Just as he told Zhang Yuantang, Tao Xuanzhi, Lott Dugan and other big guys, his original intention to construct the axiom system of generalized modal number theory was to solve the prime number problem.

So apart from this axiom system, he thinks about prime numbers the most in his daily life.

We even tried to use this axiom system to solve some prime number problems. And there was a lot of progress.

For example, regarding the twin prime conjecture, Qiao Yu felt that he could use the method he constructed to reduce the bounded distance between prime numbers to two digits, or even a single digit greater than 2.

Since Zhang Yuantang proved that the interval is less than 60 million, the collective efforts of the mathematical community have only pushed this value to 246.

This number has not changed since 2014, because with the method given by Zhang Yuantang, it is already a limit to prove it. It is basically recognized in the mathematics community that further mathematical ideas and tools will be needed to complete it.

For Qiao Yu, he had never thought of writing a paper on this issue before. The main reason was that he could not make this value equal to 2 for the time being.

Because we want to be equal to 2 and completely solve the twin prime conjecture, there are still some technical problems that remain unresolved.

After all, tools such as modal density and modal paths have not yet been fully proven, and once they get to that point, accuracy must be considered.

For example, can the local oscillation of the modal density function satisfy the twin prime trajectory? Only after these are proved can we start to formally explore this issue.

However, as long as it is not equal to 2, the accuracy requirements are actually not that high, and it can be proved using the existing tools of the generalized modal number theory axiom system.

And a paper like this is definitely enough to cope with the conference. Not to mention that this is a report given on the morning of the third day of the conference, not an opening report.

The most important thing is that if it is such a paper, he will not need eighteen days, but ten days at most. After all, the proof ideas are already in his mind.

The only problem is that such a paper still needs to use many concepts from the first stage of his axiom system of generalized modal number theory. But this paper has not been published yet...

Quoting the results of his unpublished paper at the conference to demonstrate a result, Qiao Yu could imagine how confused the mathematicians in the audience would be and how controversial it would be.

But with only such a small amount of time, it is almost impossible for him to choose a proposition again.

So Qiao Yu decided to leave this issue to Director Tian for decision-making.

Although he did not communicate with Director Tian and Mr. Yuan when he sent the paper to Ann.Math, it was indeed his fault.

However, Director Tian and Mr. Yuan asked him to give a report at the Chinese Mathematics Annual Conference without telling him in advance, so it can be said that both parties are responsible for this matter.

Anyway, if he has to give a report, then this is the paper. If this paper doesn't work, then quickly find someone else.

After all, starting from topic selection in half a month to write a mathematics paper that is qualified to attend the top domestic conferences, in Qiao Yu's opinion, it is as outrageous as giving him a pile of sand and asking him to pinch out a chip.

Not even a god can do it!

Of course, you can't say that after the phone call is made.

"Hey, Director Tian, ​​I have a very bold idea about the conference paper you just mentioned!"

After hearing this sentence, the other person didn't know whether it was because he was reading a paper. He was silent for more than ten seconds before the voice came through.

"You think it's very bold? Okay, let's talk about it."

"What do you think of my paper, which mainly talks about reducing the upper bound interval of prime numbers to single digits greater than 2? But I'm not sure how far it can be reduced, but I think single digits should be no problem." Qiao Yu immediately

said.

The silence lasted longer this time, and Director Tian's voice came again for more than twenty seconds, but this time it was much more serious and had a hint of inquiry.

"Single digits? Are you sure you can do it?"

Qiao Yu immediately replied affirmatively: "Of course, I can definitely do it, but I have to use my new theory. So the question is, I can use new methods to reduce this gap, but this new method is not there yet.

Published by the mathematical community, do you see it?"

Generally speaking, as a graduate student, you cannot make your supervisor feel entangled. Otherwise, your graduate career will most likely not be perfect.

But there are always a small number of people who have this power and are lawless. Obviously Qiao Yu falls into this category.

After three sentences, the instructor was silent three times, and the silence time continued to extend... If he hadn't heard a slight breathing sound, Qiao Yu would have suspected that the phone was disconnected.

Finally, when Qiao Yu felt a little nervous, the instructor's voice finally came again.

"You write it first and then talk about it after you finish it."

"Okay, Director Tian, ​​please wait a minute! I will write the paper as soon as possible."

When the busy signal came from the phone, Qiao Yu also let out a sigh of relief.

At least his problem is solved.

As long as he writes this paper, it doesn't matter whether he gives a report at this conference or not.

Anyway, Director Tian asked him to write the paper first. If it doesn’t work, you can’t blame him.

You can't give him half a month to write two papers that can give a 60-minute report at such a top domestic conference, right?

Unless someone doesn't understand mathematics at all, no one would make such an outrageous request.

After he felt relaxed, Qiao Yu opened WeChat with a smile, and then clicked on Senior Brother Chen's chat interface.

"Hey, Senior Brother Chen, you have caused me so much trouble..."

As the person in charge of the project and also an oppressed junior brother, he can completely channel the pressure. Based on the principle that whoever benefits should bear greater pressure, I can only say that senior brother Chen is blessed again.

…

United States, Princeton.

Lot Dugan has always believed that a good paper is like an excellent work of art. Therefore, the better the paper, the more it needs an excellent reviewer.

So after reading Qiao Yu's paper, he directly found five of the world's top reviewers. One of them was his friend.

Coincidentally, his friend also recommended a reviewer who could be said to be a top reviewer. Then six of them were recruited.

Some of them have no intention of taking over the paper at all.

Such as Andrew Wiles.

After he received a call from Lot Dugan, he refused directly. The excuse he used was that he was very busy recently...

Lot Dugan did not give up and suggested that the other party read the introduction of the paper carefully before making a decision.

Andrew Wiles gave Lott this face, read the introduction, and then commented: "Are you sure the author is not talking nonsense?"

Then Lot Dugan helped Pierre Delini to evaluate the paper before the Fields Medal boss had even read it.

"Of course, do you know what Pierre said about this paper? He said that this will be the greatest milestone work of this century, bar none. This is also the reason why I chose you as a reviewer... Andrew!

After all, you have also made history. There is no doubt that your work is one of the greatest mathematical milestones in the past world, so I want to believe that you must be interested in the relay work of this world."

As a journal editor-in-chief, he is as skillful as a salesman, just to help find matching reviewers for Qiao Yu's paper, which is enough to prove how responsible Lot Dugan is for his work.

It was this sentence that made Andrew Wiles no longer refuse and readily agreed to become the reviewer of this paper.

Then Andrew Wiles began sending emails to Lot Dugan.

The first email was to the effect that this article does have some interesting ideas.

Second letter, this article contains many creative ideas that should not be said to be subversive, which may be true.

In the third letter, he seemed to be right. I tried hard to find the loopholes but couldn't. Maybe I need to find the faults word for word.

The fourth letter, I have to admit that what Pierre Delini said seems to be right. This paper can start an era, because I can’t seem to find anything wrong. I can’t wait to know who the author of the paper is! So this paper passed !

Lott Dugan replied to Andrew Wiles' email and sent Qiao Yu's current results.

Then he forwarded the four emails sent to him by Andrew Wiles to Pierre Delini.

After all, these contents actually have nothing to do with privacy, and he also directly asked Andrew Wiles for his opinion in one sentence.

"Thank you very much, Professor Wiles. I will inform Professor Delini of your comments on this paper and Pierre's evaluation. He will definitely feel that you have once again become a bosom friend with mutual understanding."

Soon, Pierre Delini gave him feedback.

"My opinion is basically the same as Andrew's, so you can publicize my comment to more people."

…

Generally speaking, the higher the ranking, the longer the review cycle. Especially for mathematics papers, it is nothing new to calculate the review cycle in years.

Of course, it is not that the reviewers deliberately delayed for so long. The key issue is that generally the articles that can be published in such journals either solve major problems or contribute new ideas, and the proof process is often very cumbersome.

Especially from the perspective of the editorial office, the more important the paper, the more cautious the editor will be when selecting reviewers.

After all, it's hard to build academic credibility, but it's easy to destroy it. A few times is enough.

It’s like some journals give people the impression that they can publish them by paying the page fee, and then they become recognized as water journals in the industry. As long as you look at the name of the journal, you can know what is going on.

However, as Andrew Wiles and Pierre Delini responded almost simultaneously that the paper was approved, Lot Dugan felt that Qiao Yu and Chen Zhuoyang’s paper should be published in November.

After all, these two papers are actually not very long, only twenty-five pages in total.

The elderly Professor Wiles was able to pass the review so quickly, and other reviewers should have less problems.

Of course, he didn't want to push too hard, but in order to ensure that if the other four reviewers could complete the review this month, it would be published in November, he simply called the editor in charge of typesetting...

"Hi, John, I hope you can do a favor...that is, do two copies of the typesetting work for the next issue. I sent you an email, and the two attached papers will be proofread first.

Yes, save the front page. If these two papers can be reviewed and approved before this month passes, then these two papers will be published in the November magazine."

Well, actually this is not too exaggerated. Qiao Yu has not broken Zhang Yuantang's record back then.

His paper on the bounded interval of prime numbers was accepted in just three weeks. At that time, it set a record for the fastest acceptance of a paper in Ann.Math's 130-year history.

If Qiao Yu's paper can be published in November, it will probably rank among the top three in terms of publication speed.

Of course, doing this is not free of charge.

Journals have always been mutually successful with high-quality papers. After Qiao Yu told Lott Dugan about his ambitions, Lott Dugan naturally hoped that all papers on the axiom system of general modal number theory could be published in Ann.

Posted by .Math.

After all, there are four top journals in mathematics, not just one. A luxurious and efficient review team is also a reflection of competitiveness for top journals.

Qiao Yu is a very smart man, and Lot Dugan believes that this future star of mathematics can feel his painstaking efforts.

…

At this time, Qiao Yu had no time to think about this, and he did not communicate with Lot Dugan.

Anyway, according to the previous publication schedule of Ann.Math, even if his paper is published in November, it will be at least in the middle of the year.

The Chinese Mathematics Annual Conference will be held in early November. It will definitely be impossible to make it in time anyway, so he has not considered when the paper will be published.

All his thoughts were focused on quickly writing the paper and submitting it to Director Tian, ​​and settling the matters related to the report first.

After all, having confidence and completing the thesis are two different things. The thesis mainly includes three key points.

The first is the modal geometrization of the prime spacing. The original prime spacing problem is that in the prime pair (p, p′), there are infinite pairs of prime numbers that satisfy p′p=d, where d is a fixed value.

After conversion, in the modal space M, whether there are infinite pairs of modal points (r_p, r_p') that satisfy the modal distance d_M (r_p, r_p') = d.

First, we need to prove that this transformation is reasonable. This part can be directly borrowed from a small part of the paper he submitted to Ann.Math...

In this section, he directly quoted some theorems in the paper sent to Ann.Math.

The second part is to prove a key theorem. In the modal space M, there is a modal path Γ, so that the upper bound of the modal distance d_M(r_p, r_q) can be reduced to a single digit. At the same time, for the modal path

Perform density analysis on the point and give the verification results.

The third part is the final homomorphic transformation. Through these mapping relationships, the characteristics of the geometric model are re-transformed into the language of number theory...

It sounds simple, but it was actually very hard for Qiao Yu to get started. It took him ten full days to complete the first draft. In the end, Qiao Yu reduced the number 246 to 6.

That is to say, Qiao Yu proved that there are infinite pairs of prime numbers with an interval of 6. It is not far away from completing the proof of the twin prime conjecture.

In fact, Qiao Yu felt that the scope could be narrowed down a little, but he felt that it was not necessary. Further narrowing the scope would add more technical details. Even to 4, Qiao Yu felt that it would take more than ten pages, which would obviously

Make the proof lengthy.

It’s just a conference paper, that’s about it.

Then I spent another five days carefully checking the article step by step to see if there were any problems.

This has become Qiao Yu's obsession. After reviewing the manuscript of Senior Brother Qin from Yujiang University, Qiao Yu feels that he really does not allow the appearance of something that makes people laugh because of a little carelessness.

operation.

The final paper was twenty-one pages long, and the title was also very simple: "Proof of the Infinity of Prime Number Pairs with an Interval of 6."

After the inspection, Qiao Yu sent it to Director Tian and Mr. Yuan via email on October 25th on time. In short, there must be no mistakes made this time.

After the paper was sent out, there was no news. But Qiao Yu didn't care anymore. He had already completed the paper. As for whether he could present it in the Mathematical Society, that was a matter decided by the tutors.

As for him, he can relax for another two days.

…

October 30, Huaqing, Qiuzhai, multi-functional conference room.

If someone breaks in here today, they will find that there are many bigwigs gathered in the conference room.

A bunch of academicians sat together at the conference table.

Yuan Zhengxin, Tian Yanzhen, Pan Yuedong, Li Luhe...

Not only Yanbei and Huaqing University, but also those from the Chinese Academy of Sciences, Nanjin University in the nearby satellite city, and Beijing Normal University...

Really, the dozen or so professors sitting in the conference room can basically represent half of the Chinese mathematics community.

Not only that, there are also three internationally renowned Chinese mathematicians who participated in this conference via remote video, Zhang Yuantang, Zhang Shuwen and Tao Xuanzhi.

Each person gets a copy of Qiao Yu’s latest paper.

There was no way, the situation this time was indeed very special, so five days ago, when Qiao Yu sent the paper to Tian Yanzhen and Yuan Zhengxin, the two bosses met to discuss it.

During this period, I also made a phone call to Lot Dugan.

Then the two big guys made a list, selected Chinese and Chinese mathematicians who were qualified to review Qiao Yu's paper, and then started making phone calls one by one.

After the paper was sent out, today's meeting took place.

However, after Tian Yanzhen discussed with Yuan Zhengxin, Qiao Yu was not allowed to attend the meeting today.

Mainly because some things are hard to explain. For example, Qiao Yu posted two Ann.Math articles without telling his tutor.

As a result, the tutor wanted him to give a report at the annual mathematics conference, but he couldn't give it, so he temporarily kicked out a paper.

The whole thing was so shocking that the cause and effect could be written into their memoirs later, but both of them felt that it was not necessary for their colleagues to understand it so clearly for the time being.

Of course, even if Qiao Yu did not come, it would be difficult for many academicians present to evaluate this paper.

After all, many people in the mathematical community have not heard of a series of brand-new concepts such as the modal axiom system in the paper.

But the proof process looks like that. This feeling is very strange.

However, Tao Xuanzhi's speech solved many people's doubts.

"I have read this paper carefully in the past five days and found no mistakes. Of course... he cited some new theories that have not been released to the public..."

Having said this, Tao Xuanzhi was silent for a moment, because he also felt that this matter was a bit difficult to evaluate, and then continued: "Coincidentally, I reviewed two papers at the invitation of Ann.Math not long ago.

It’s about this modal axiom system framework.

As far as I know, the six reviewers of these two papers have all given opinions agreeing to be published. Therefore, there is a high probability that these two papers will be published in the last issue of Ann.Math this year.

So I personally think there is not much problem with the argumentation process of this paper. Including the modal space, path existence theorem, and modal density function mapping theorem he cited, as well as the related transformation process."

In the conference room, everyone had different expressions.

Tian Yanzhen and Yuan Zhengxin behaved very calmly, and they have been able to accept this accident for a long time.

As for others, some are confused and some are surprised...

After a while, Academician Pan from the Academy of Sciences asked: "Well, although it may be a bit presumptuous, Professor Tao, can I ask, do you know who else reviewed the two papers you mentioned besides you?

"

Tao Xuanzhi nodded and replied: "Besides me, there are Professor Pierre Delini, Professor Andrew Wiles, Professor Richard Taylor, Professor Andrew Granville and Peter Schur.

Professor Ci."

Sometimes reviewers are reluctant to let people know that they have reviewed certain manuscripts.

But this case is obviously not included.

In fact, when these reviewers are willing to review a paper, it usually means that they really don't mind letting the outside world know that they are reviewers.

As a result, the big guys in the conference room were speechless again.

Good guys, there are five Fields Medal winners. Although the other one did not win the Fields Medal, he did win a Fields Silver Medal, which is the only one in history.

This team of reviewers all thought that the other two papers had no problems, which made those who originally wanted to question them simply shut their mouths.

After another long moment of silence, Yuan Zhengxin coughed twice and said, "Professor Zhang Yuantang, do you think there are any flaws in Qiao Yu's paper?"

This is a very polite question.

After all, one of the earliest important issues regarding the twin prime conjecture was whether the minimum interval between prime numbers is finite.

You must know that in 2008, a group of the world's top number theory experts held a meeting at the National Institute of Mathematical Sciences to discuss this issue.

But the meeting ended in failure.

Zhang Yuantang is the first mathematician to answer this question. Even though his result is that the bounded distance between prime numbers is 70 million...

But his proof directly answered this important question. It can be said to be a progress from scratch on the milestone of number theory. It was later reduced to 246, all based on the tools provided by his paper. It is also said that he is the founder of this problem.

Not an exaggeration.

"I went to Yanbei University to give lectures in August this year and met with Qiao Yu. He told me that in order to solve a series of prime number problems, he planned to design a new axiom framework.

At the time, I thought this was a very magical idea. But what was even more amazing was that he did it in October, not only actually building a new axiom framework.

What's more, when I tried to find the unreasonable part of the proof, I failed... I couldn't believe that this was done by a sixteen-year-old boy.

But one thing I am sure of is that a new number theory track is about to begin. In modal space, we are no longer studying specific numbers, but elements that contain all possible states.

Give every number a geometric meaning... I don't even know how to evaluate this framework, but it is obvious that he is on the road to success.

So if I were to simply evaluate this paper, I think it is right. As I said just now, I tried very hard to find faults, but failed.

Of course, all this is based on the premise that the definition of modal space is logically self-consistent. As for whether the definition of modal space is reasonable, I think Professor Tao Xuanzhi has already given the answer. I am finished."

After carefully listening to Zhang Yuantang's comments, Tian Yanzhen waited for a moment to give everyone enough time to think before officially speaking.

"Ahem, um... why don't we just vote directly? Because Qiao Yu is the joint training target of Yanbei and Huaqing, Mr. Yuan and I abstained from voting.

If you think this paper is suitable for presentation at this year's annual meeting, please raise your hands."

There was no long hesitation, and soon everyone in the conference room raised their hands.

Thanks to Chinese Baby, the little squid with sore balls for the reward and encouragement
Chapter completed!