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Chapter 154 Proof of the Infinity of Prime Pairs with an Interval of 6

Tian Yanzhen told Qiao Yu to get out immediately. After Qiao Yu walked out of the office obediently, he started to think about something.

The benefits of giving a report at a mathematics conference are obvious. Qiao Yu thought that it was probably because his supervisor and Professor Yuan felt embarrassed that he had not been able to apply for the project before, so they fought for this opportunity.

Letting him present this idea in front of everyone would at least allow the entire mathematics community to make their own judgment.

This can be understood as a message to the Chinese mathematics community: "Everyone, come and see whether we have any selfish motives in securing an opportunity for Qiao Yu to get a major project!"

But of course, this is based on the premise that he has not submitted his paper to Ann.Math. From his supervisor's attitude, Qiao Yu could tell that making a report at a conference is not as useful as 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 17 or 18 days left until the 25th. I have to prepare a paper that can be presented for an hour at the mathematics conference, and I can't lose face for my supervisor...

It does seem a bit difficult.

This is the disadvantage of performing too well, the tutor thinks he is omnipotent!
So much so that on the way back to the dormitory, Qiao Yu felt the feeling that he should have experienced at his age - melancholy!
Totally unprepared!

During this period, he has been concentrating on completing the proof of the multimodal space system, and is currently working on the proof of extending from two-dimensional to three-dimensional. This work alone will take at least one or two months to complete.

Well, Qiao Yu had to admit that without any direction, asking him to produce a paper out of thin air in half a month was probably a joke.

This is a bit troublesome...

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

The main problem is that the meeting was held at an unusual time, which happened to be the beginning of November, the month when Ann.Math is published each year, and it also happened to be an odd month each year.

This is also the reason why Qiao Yu feels that the paper he and Senior Brother Chen wrote may not be published this year.

If you submit your manuscript in October, the review process will probably take until November, even if it’s done quickly. Add in the typesetting time, and the earliest time for publication will be January next year, or possibly even March.

This is the case when everything goes well. If the reviewers have any questions about the paper, they may have to discuss it back and forth, and the paper may have to be postponed.

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

Generally, this type of agreement has clear requirements on how many papers should be published during the assessment period, such as publishing three papers in journals of a certain level in three years.

It sounds easy, but for young teachers who have just entered colleges and universities, they have to complete the most arduous teaching tasks, do research, and negotiate with reviewers.

There are always too many people for too little work in the project, and it is difficult to publish an article on time without a recommendation from a big shot. Universities can only offer so many academic positions.

Most people are unwilling to compromise. After all, if they really go to teach at a third-tier university or even a junior college, it basically means they are cutting themselves off from the mainstream academic community, and their lives will be like that for the rest of their lives.

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

It was nothing more than writing a paper that would make Director Tian and Professor Yuan feel that they would not lose face. Although the time was a little shorter, as long as there was a general direction, it shouldn't 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 bigwigs, his original intention of constructing a generalized modal number theory axiom system was to solve the prime number problem.

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

We have even tried to use this axiom system to solve some prime number problems, and have made 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 6000 million, the collective efforts of the mathematical community have only pushed this value to 246.

This number has not changed since 2014, because the method proposed by Zhang Yuantang has already reached its limit. It is generally recognized in the mathematical community that new mathematical ideas and tools are needed to complete the task.

Qiao Yu had never thought about writing a paper on this issue before, mainly because he was unable to make the value equal to 2 for the time being.

Because in order to make it equal to 2, there are still some technical problems that have not been solved to completely solve the twin prime conjecture.

After all, tools such as modal density and modal path have not been fully proven, and when we really get to that point, we have to consider accuracy.

For example, can the local oscillation of the modal density function satisfy the twin prime trajectory? These have to be proven before we can start to formally explore this issue.

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

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

The most important thing is that if it is a paper like this, he doesn’t need eighteen days, but only ten days at most. After all, the proof ideas are already in his mind.

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

At the conference, Qiao Yu cited the results of his unpublished paper to prove a result. He could imagine how confused the mathematicians in the audience would be and how much controversy it would cause.

But with so little time, it was almost impossible for him to choose a new proposition.

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

Although he did not discuss the paper with Professor Tian and Professor Yuan before publishing it to Ann.Math, it was indeed his fault.

However, Director Tian and Professor Yuan did not tell him in advance that they wanted him to give a report at the China Mathematical Conference, so it can be said that both parties are responsible for this matter.

If he has to do a report, then it's this paper. If this paper is not good, then find someone else immediately.

After all, writing a mathematical paper qualified for presentation at a top domestic conference in just half a month from topic selection is, in Qiao Yu's opinion, as outrageous as giving him a pile of sand and asking him to make a chip out of it.

Even if a god comes, it won’t work!
Of course, you can't say that after the call is connected.

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

After hearing this, the person on the other end was silent for more than ten seconds, perhaps because he was reading a paper, before a voice came over.

"You think that's bold? Come on, tell me."

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

This time the silence lasted even longer, for more than twenty seconds before Director Tian's voice was heard again, but this time it was much more serious, and there was a hint of inquiry in it.

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

Qiao Yu immediately replied affirmatively: "Of course, I can definitely do it, but I must use my new theory. So the problem is, I can use a new method to narrow this gap, but this new method has not been published in the mathematical community, what do you think?"

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

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

These three sentences made the tutor silent three times, and the silence kept getting longer...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 tutor's voice finally came again.

"Just write it first, and we'll talk about it after you finish it."

"Okay, Director Tian, ​​take your time! I'll finish the paper as soon as possible."

When the busy tone came from the phone, Qiao Yu also breathed a sigh of relief.

At least his problem was solved.

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

Anyway, Professor Tian asked him to write the paper first, so if it is not usable, it cannot be blamed on him.

It's impossible to ask him to write two papers that can be presented in a 60-minute report at such a top domestic conference in just half a month, right?
No one would make such an outrageous request unless they know nothing about mathematics.

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

"Hey, Brother Chen, you've caused me so much trouble..."

As the project leader and also an oppressed junior, he can completely pass on the pressure. Based on the principle that whoever benefits should bear greater pressure, we can only say that Senior Brother Chen is blessed again.

……

Princeton, USA.

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

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

Coincidentally, his friend also recommended a top reviewer, and then we had six reviewers in total.

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

For example, Andrew Wiles.

When he received the call from Lott Dugan, he rejected it directly. He used the excuse that he was very busy recently...

Lott 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 the benefit of the doubt, read the introduction, and then commented: “Are you sure the author isn’t talking nonsense?”

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

"Of course. Do you know what Pierre said about this paper? He said it would be the greatest milestone of this century, bar none. That's why I chose you as the reviewer... Andrew!

After all, you have made history. Your work is undoubtedly one of the greatest mathematical milestones of the last century, so I think you must be interested in the world's relay work."

As a journal editor, Lott Dugan worked like a salesman to find matching reviewers for Qiao Yu's paper, which is enough to prove how responsible he is at work.

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

Andrew Wiles then began to email Lott Dugan repeatedly.

The first email said something like: This article does have some interesting ideas.

Second, this article has many creative and rather subversive ideas, which may be correct.

In the third letter, he seemed to be right. I tried hard to find the loophole but failed. Maybe I need to look for the problem word by word.

Fourth, I have to admit that Pierre Delini seems to be right. This paper can open an era, because I can't seem to find fault with it. 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 him all of Qiao Yu's current grades.

He then forwarded the four emails Andrew Wiles had sent him to Pierre Delini.

After all, there was nothing private about the content, and he directly asked for Andrew Wiles' opinion in one sentence.

"Thank you very much, Professor Wiles. I will inform Professor Delini of your comments on this paper and on Pierre's evaluation. He will definitely feel that you have once again become bosom friends who understand each other."

Soon, Pierre Delini gave him back.

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

... Generally speaking, the higher the ranking of a journal, the longer the review period. Especially for mathematics papers, it is not uncommon for the review period to be calculated in years.

Of course, it is not that the reviewers deliberately drag out the time so long. The key issue is that the articles that can be published in such journals generally either solve major problems or contribute new ideas, and their proof process is often 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, academic credibility is hard to build but easy to destroy. A few times is enough.

Just like some journals give the impression that you can publish as long as you pay the page fee, they are recognized as water journals in the industry. As long as you look at the name of the journal, you can know what it is.

However, as Andrew Wiles and Pierre Delini gave their responses that their papers were approved almost at the same time, Lot Dugan felt that Qiao Yu and Chen Zhuoyang's papers should be published in November.

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

If Professor Wiles, who is already very old, was able to pass the review so quickly, other reviewers should have no problem at all.

Of course, he couldn't rush too much, but in order to ensure that if the other four reviewers could complete the review this month, the manuscript would be published in November, he simply called the editor in charge of typesetting...

“Hi John, I was hoping you could do me a favor… just do two copies of the next issue. I sent you an email and asked you to proofread the two papers in the attachment.

Yes, save the front page. If these two papers can be approved before the end of this month, they will be published in the November issue of the magazine. "

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

His paper on bounded intervals between prime numbers was accepted in just three weeks, setting a record for the fastest acceptance of papers in Ann.Math’s 130-year history.

If Qiao Yu's paper could be published in November, it would probably be ranked among the top three in terms of publication speed.

Of course, this is not without purpose.

Journals and high-quality papers have always been mutually beneficial. When Qiao Yu told Lott Dugan about his ambition, Lott Dugan naturally hoped that all papers on the axiom system of generalized modal number theory could be published in Ann.Math.

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

Qiao Yu is a very smart person, and Lott Dugan believes that this future math star can feel his painstaking efforts.

……

Qiao Yu didn't have time to think about these things at this time, nor did he communicate with Lotte Dugan.

Anyway, according to the previous publication time of Ann.Math, even if his paper can be published in November, it will at least be mid-November.

The Huaxia Mathematics Annual Conference will be held in early November. He would definitely not be able to make it in time, so he did not even consider when the paper could be published.

His mind was completely focused on finishing the paper as quickly as possible and submitting it to Director Tian, ​​and dealing with the report issues first.

After all, confidence and thesis completion are two different things. Thesis mainly includes three key points.

The first is the modal geometry of prime spacing. The original prime spacing problem is that in the prime number pair (p, p'), there are infinitely many pairs of prime numbers that satisfy p'p = d, where d is a fixed value.

After the transformation, in the modal space M, are there infinite pairs of modal points (r_p, r_p') satisfying 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 from the paper sent to Ann.Math.

The second part is to prove a key theorem: in the modal space M, there exists a modal path Γ, which reduces the upper bound of the modal distance d_M(r_p, r_q) to a single digit. At the same time, the density analysis of the points on the modal path is performed to give the verification results.

The third part is the final homomorphic transformation. Through these mapping relationships, the characteristics of the geometric model are 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 finish the first draft. In the end, Qiao Yu reduced 246 to 6.

In other words, Qiao Yu proved that there are infinite pairs of prime numbers with an interval of 6. He is not far from completing the proof of the twin prime conjecture.

In fact, Qiao Yu felt that the scope could be narrowed down a little bit, but he felt that it was unnecessary. 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 enough.

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. Ever since reviewing the manuscript of Qin, a senior student at Yujiang University, Qiao Yu felt that he really would not allow a little carelessness to cause a ridiculous operation in the paper.

The final draft of the paper is 6 pages long, and the title is also very simple: "Proof of the Infinity of Prime Pairs with an Interval of ".

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

After the paper was sent out, there was no news. But Qiao Yu didn't care. He had already completed the paper. As for whether he could give a talk at the mathematics society, that was up to the instructors.

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

……

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

If someone breaks in here today, they will find that the conference room is full of bigwigs.

A group of academicians gathered around the conference table.

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

Not just Yanbei and Huaqing University, but also Huazhong University of Science and Technology, Nanjing University in the satellite city next door, Beijing Normal University...

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

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

Everyone had a copy of Qiao Yu's latest paper.

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

During this time, he also called Lott Dugan.

The two big guys then made a list, picked out all the Chinese and Chinese-American mathematicians who were qualified to review Qiao Yu's paper, and then started calling them one by one.

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

However, after Tian Yanzhen discussed with Yuan Zhengxin, he did not allow Qiao Yu to attend the meeting today.

The main reason is that some things are hard to explain. For example, Qiao Yu published two papers in Ann.Math without telling his supervisor.

As a result, he was unable to give the report that his supervisor wanted him to give at the annual mathematics conference, so he had to rush out a paper.

The whole thing was too shocking, and the cause and effect could be written into memoirs later, but both of them felt that there was no need to let their colleagues know so clearly for the time being.

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

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

But the proof process seems to make sense. This feeling is very strange.

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

"I have carefully read this paper in the past five days and did not find any mistakes. Of course... he cited some new theories that have not yet been announced to the public..."

At this point, Tao Xuanzhi was silent for a moment, because he also felt that this matter was a little difficult to comment on, and then he continued: "Coincidentally, I was invited by Ann.Math not long ago to review two papers on this modal axiom system framework.

As far as I know, all six reviewers of these two papers have given their approval for publication. So 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 that there is no big problem with the argumentation process of this paper. Including the modal space, path existence theorem, and modal density function mapping theorem he cited. And the related transformation process. "

In the conference room, everyone had different expressions.

Tian Yanzhen and Yuan Zhengxin behaved very calmly and have been able to accept this accident after such a long time.

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

After a while, Academician Pan of 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 answered, "Besides me, there are Professor Pierre Delini, Professor Andrew Wiles, Professor Richard Taylor, Professor Andrew Granville and Professor Peter Schultz."

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

But this case is obviously not included.

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

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

Wow, there are five Fields Medal winners. Although another one did not win the Fields Medal, he won a Fields Silver Medal, which is the only one in history.

If this team of reviewers all thought that there was no problem with the other two papers, then those who wanted to question would simply shut up.

After another long 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 questions about the twin prime conjecture was whether the minimum interval between prime numbers is finite.

You should know that in 2008 a group of world-leading number theory experts held a meeting at the National Institute of Mathematical Sciences of the United States to discuss this issue.

But the meeting ended in failure.

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

But his proof directly answered this important question. It can be said that it was an improvement from scratch in the milestone of number theory. The subsequent reduction to 246 was based on the tools provided by his paper. It is not an exaggeration to say that he is the founder of this problem.

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

I thought it was a magical idea at the time. But what was even more magical was that he did it in October, and not only did he actually build a new axiomatic framework.

More importantly, when I tried to find the irrational part of the proof, I failed... I couldn't believe that a sixteen-year-old boy could do this.

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

Giving each number a geometric meaning… I don’t even know how to evaluate this framework, but it’s clear that he is on the road to success.

So if I were to simply evaluate the paper, I think it is correct. As I said, I tried very hard to find faults, but failed.

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

After listening carefully to Zhang Yuantang's comments, Tian Yanzhen waited for a moment, giving everyone enough time to think before he formally spoke.

"Ahem, well... why don't we just vote directly? Since Qiao Yu is a joint training target of Yanbei and Huaqing, Elder Yuan and I will abstain from voting.

Please raise your hand if you think this paper is suitable for presentation at this year’s annual meeting.”

Without much hesitation, everyone in the conference room raised their hands.

　Thanks to Chinese to Baby and Little Squid for their encouragement.
　　
　
(End of this chapter)