England, home of Andrew Wiles.

The phone rang suddenly, and Andrew Wiles was woken up, and of course his wife was also woken up.

At this time, it was the third watch in the middle of the night in the UK, and it was about three o'clock, which was the rest time.

Andrew Wiles picked up his phone with a confused expression.

Who called him so late?

At first glance, it was Simon Donaldson.

这个老家伙这么晚不睡，给他打骚扰电话？

Don't know that sleep is important for older people like them?

Answering the phone, he said angrily: "Simon, if you don't give a reason that satisfies me, I will definitely remember this incident of you waking me up tonight." ”

"Hey, Andrew, don't worry, I was woken up by someone else too." Simon Donaldson said, "Guess what's going on with your students who haven't come here yet?" ”

Wilese was stunned for a moment.

What does this say...... Li Mu?

Calculating the time, it seems that it is indeed daytime on the Huaguo side.

"What's going on again?" He asked, confused?

"Just now, my friend in the United States suddenly called me to tell me that there was news from Huaguo, and Li Mu was proving the Polygnac conjecture and the Hardy-Littlewood conjecture."

Wiles: "??? ”

He had a question mark written all over his face: "Are you sure?" Really? ”

"What are you lying to?" Simon Donaldson said: "I'm watching his proof live right now, and the Polignac conjecture has been solved by him, and now it's Hardy Littlewood's turn to conjecture." ”

For a moment, Wiles was not sleepy.

Just kidding, how could he miss such a big event in the world of mathematics?

"Give me the live broadcast URL, I'll watch it now."

"I sent it to you, you can translate for me later, what Li Mu is saying, after all, you have studied Chinese for so long."

"I can't guarantee that I can fully translate it, and you know that Chinese is difficult to learn."

Wiles said as he rolled over and got out of bed, giving his wife a look and letting her go back to sleep on her own.

His wife had a helpless look on her face.

Sometimes being the wife of a mathematician seems to be a common thing in the face of such a thing.

Perhaps for most mathematicians, mathematics is a little more important than love.

Wiles came to the study, grabbed a cup of coffee, and turned on the computer.

entered the live broadcast room sent by Donaldson.

This is a Chinese website, so the text of the page is also in Chinese.

Although the web translation function can be used, Wiles still relied on his Chinese ability to carefully identify it.

Li Mu...... Prove...... The Polygnac conjecture and the Hardy-Littlewood conjecture! ”

"It's true!"

He hurriedly clicked on the live broadcast and saw Li Mu inside writing mathematical formulas on the blackboard.

"This is proof...... Hardy Littlewood conjecture! ”

With just one glance, Wiles could see what Li Mu was writing.

"This is the theory of divergence...... It actually used the absolute value of non-Archimedes...... Is it true? ”

Simon Donaldson's voice came over the phone again.

"Of course it's true, translate it for me and practice your simultaneous interpretation skills."

Wiles: "Don't worry, I haven't even understood it myself......"

西蒙·唐纳森却还是催促着：“快给我翻译！ Even, karma, this step of Li Mu is too crucial, looking at what he wrote on the blackboard, it seems that he has successfully found the superposition relationship between the twin prime pairs and the prime number theorem? Oh, what is he talking about? ”

In this way, a Fields Medal winner and a Fields Special Medal winner got up in the middle of the night to listen to the report of a young Chinese man.

And in fact, there are many more like them.

A large number of well-known mathematicians in the world have entered this live broadcast room.

It doesn't matter what time zone these mathematicians are in, whether it's a break or not.

Even mathematicians from some countries, such as Japan, Han, and Xinapo, were attracted by the report because their time zones were not much different from those of China, and they were basically still working hours.

As a result, the students of the mathematics departments of universities in these countries temporarily received the news of the teacher's leave before class, and even more, they were in class, and then the teacher opened the live broadcast room and took them to watch Li Mu's certificate.

The name is beautiful: witness history.

It's really just what their teachers want to see.

Of course, the result of this situation is that both teachers and students are confused.

Who made the content too esoteric.

Not only did they not understand Chinese, but they also did not understand what was being said.

For them, the only thing to be thankful for is that the mathematical formula written by Li Mu on the blackboard can still make them associate and figure out how to prove it.

However, even if you want to do this, it is very difficult, and only those top mathematicians, and mathematicians who have a deep understanding of the relevant knowledge, can easily understand, as for other mathematicians who are slightly less able, or who study other fields, they can only be half-understood, or they are confused from beginning to end.

I can't read it, I can't understand it at all.

……

Kyoto University Auditorium.

Li Mu on the stage did not know that his report had attracted so many internationally renowned mathematicians.

He continued to prove on stage with all his heart.

Whether it is the theory of divergence or the absolute value of non-Archimedes, it is the method that he has come up with recently.

And the source of these inspirations lies in Professor Lin Yao's report that day.

Although the topic of Lin Yao's report that day was "Non-Archimedesan Submorphic Mapping of Supersurfaces in Projective Clusters", although it was not much related to the Hardy-Littlewood conjecture, mathematics and mathematics were common, and methods were also common.

Once you find a relationship, you can use it.

“…… By theorem 2.1 and theorem 2.2, the algebraic extension of the finite field of the eigenp, whose non-Archimedean absolute group is isomorphic to ^Zp at ......"

【A(F)→ H1(GF， T (A))……】

在黑板上再度写下了一个式子。

Li Mu stood up straight, with a slight smile on his face.

At the same time, the mathematicians who understood in the audience were surprised again.

"Wait...... Did he turn the whole problem into an algebraic geometry problem? ”

Qiu Chengtong's eyes narrowed.

Through a series of transformations, Li Mu completely transformed the original Hardy-Littlewood conjecture into a problem in algebraic geometry at this step.

"Does he want to solve it by the methods of algebraic geometry?"

Qiu Chengtong's heart has already given birth to this idea.

Use the methods of algebraic geometry to solve problems in number theory!

What a crazy thing it is.

In the past, one of the mathematicians who did this was named Gerd Faltins.

It was one of the top mathematicians in the world today who used the methods of algebraic geometry to prove the Model conjecture in number theory, and finally won the Fields Medal.

而现在李牧也要用这种方法来完成证明？

Zhang Yitang next to him also had the same expression.

He had seen many geniuses, and he himself was a genius, but he didn't expect Li Mu to plan to do this.

At the same time, there were also many mathematicians in the live broadcast room who were aware of Li Mu's intentions, and they all gasped.

"Can this really be done?"

In the United Kingdom, Andrew Wiles and Simon Donaldson have been on the phone to exchange the content of Li Mu's report.

They are all top mathematicians, so they can also understand the meaning through what Li Mu wrote.

And for this question, they couldn't help but remain silent for a long time.

Until the end.

"I hope he could."

Even if Li Mu used other methods to prove this conjecture, it probably wouldn't make them so excited.

But if this problem is solved by using algebraic geometry, then the implications for this are very far-reaching in the mathematical community.

This will reinvigorate mathematicians' confidence in the realization of the Langlands program and the unification of algebraic geometry and number theory.

This report will also become a classic report in the field of mathematics.

……

On the rostrum.

Li Mu turned his head slightly and smiled: "I believe that some friends have seen my thoughts." ”

"So at this point, we will also officially enter the field of algebraic geometry—"

"And here, let me first briefly introduce a new theory to you."

"I call it k-mode theory."

"For the time being, you can think of it simply as a combination of k-theory and modulo space."

His words once again shocked the mathematicians present.

Combining K-theory with modulo space?

K theory is closely related to algebraic geometry, algebraic number theory and other fields, and modulo space is the key research object of algebraic geometry.

There have been no precedents for these two to be used together in the past, but they are rare, because there has never been a systematic approach that allows the two methods to be perfectly combined.

And now Li Mu's meaning ...... Is it to achieve this?

Li Mu didn't explain much, turned his head, and began to write on the blackboard.

Everyone in the audience held their breath, even if they couldn't understand it, they knew that Li Mu was doing something big.

With the listing of a blackboard, Qiu Chengtong showed a sudden expression.

"It turns out that each point in the modulo space is calculated according to the K0 functifier to generate a semigroup of projective modulo isomorphisms...... By the way, coupled with the incompleteness of the modulo space, he will probably use this to estimate the distribution of twin prime pairs......"

作为顶尖的数学家，邱成桐的数学直觉也当然很强。

Almost quickly, he saw Li Mu's purpose.

But although he saw it, if he was asked to do it, he could only choose to give up.

To do this, it is too technically too difficult and too difficult.

In particular, the calculation that needs to be carried out later will test the control of the whole method.

He might have been able to give it a try when he was younger, but now, he has to serve his old age.

After that, Li Mu did start a lot of calculations as he expected.

His calculations brought a tightrope feeling to others on the scene, and once one step was wrong, it would be an absolute error.

However, Li Mu is like a humanoid computer, handling the entire complex calculation process perfectly, and his intuition in mathematics is even more vivid.

And such calculations also require enough blackboards.

So people saw the staff next to them dragging up a small blackboard from time to time, until all 20 small blackboards were dragged up—

[To sum up, π2(N) is approximately equal to ∫dt/(lnt)^2≈2Ct(N/(ln)^2N]

[where Ct is the twin prime constant.] 】

[Certification.] 】

Li Mu wrote the last three lines on the last blackboard and the last blank space.

"At this point, I think the Hardy-Littlewood conjecture is officially history."

"End of all the contents of this report."

"Let me conclude with the honor and pride to introduce you, the Polygnac Lee theorem, and the Hardy-Littlewood Lee theorem."

Li Mu smiled slightly, and then bowed to the audience.

Thunderous applause.

他真的做到了，用代数几何的方法，解决了一个数论问题！

(End of chapter)