Kanter said: "Mr. Pang, I will definitely ask Ms. Say to consider this plan you put forward, but whether it can be implemented will have to be discussed by the United Nations Security Council." ”

These few of them are all a small number of human elites who know that the Three-Body Civilization is about to invade the earth, and they have also read the information of the Three-Body Civilization, and when the basic science of the earth has been locked by Zhizi, they have almost no hope that the human civilization can survive the war with the Three-Body People in four hundred years.

But the plan proposed by Pang Xuelin is the first counterattack plan that makes them feel a glimmer of hope so far, and at the same time, there is also a trace of awe for Pang Xuelin, the proposer of this plan.

Pang Xuelin smiled: "Then there is Mr. Laucante." ”

Next, the group chatted for a while.

Pang Xuelin got the reason why he, Fred, and Grant were found so quickly from Shi Qiang.

It turned out that the "Judgment Day" had already been targeted by the US military's Virginia-class nuclear submarine before entering the Windward Strait.

They came out of the Judgment Day in a lifeboat that night and were also under the surveillance of the nuclear submarine.

It was only later that they landed in Cuba and then went into hiding in Santiago that they lost their tracks.

Although the "Guzheng Operation" was successful a few days later, no information about the Three-Body Civilization was found in the "Judgment Day", and the Adventists who disembarked from Port-au-Prince were almost wiped out, and no information about the Three-Body Civilization was found.

The whereabouts of the three of them were only then taken seriously, and with the active cooperation of the Cuban government, they were soon located at the homestay where the three were hiding.

Subsequently, the US military dispatched the "Delta" troops deployed in Guantanamo, preparing to quietly wipe out the three people, but they did not expect that the three of them had internal strife, and only Pang Xuelin survived.

Moreover, in the attic of the B&B, the "Delta" troops also found a hard disk that had been burned to store the materials of the Three-Body Civilization.

Originally, the human side had already decided that the operation against the "Judgment Day" had failed, and the Adventists had completely destroyed the relevant materials of the Three-Body Civilization.

No one expected that the next day, things took a turn for the worse, and in the public mailboxes of the United Nations and the five permanent members of the Security Council, they received an email containing all the information of the three-body civilization, and the sender was Pang Xuelin, who was injured by Fred with a gun.

Therefore, Pang Xuelin also received the attention of the United Nations, and when he was still in a coma, he came to the New York University Medical Center by special plane.

The information that Shi Qiang said was not much different from Pang Xuelin's guess, and they chatted for about half an hour before the three of them said goodbye and left.

Pang Xuelin also breathed a sigh of relief, although the injury was not light this time, he still achieved the result he wanted.

With the protection of the United Nations, I can do research in the three-body world with peace of mind.

It is now 2007 in the three-body world, and there are still two years to go before the implementation of the wall plan, which is enough to wave on its own.

He closed his eyes, called up the system, and began to study the full text of the proof of the BSD conjecture given by the system.

……

The BSD conjecture is the full name of the Behe and Swinathon-Dale conjecture.

Since the fifties of the last century, mathematicians have discovered that elliptic curves are closely related to number theory, geometry, cryptography, etc.

For example, Wiles proved Fermat's final theorem, and one of the key steps was to use the relationship between elliptic curves and modular forms (Taniyama-Shimura conjecture).

The BSD conjecture is related to elliptic curves.

In the sixties of the last century, Behe and Swinatone-Dale of the University of Cambridge in England used computers to calculate the rational solutions of some polynomial equations, and found that such equations usually have an infinite number of solutions.

But how do you give an infinite number of solutions?

The solution is to classify first, and the typical mathematical method is to isonominate and thus obtain the congruence class, that is, the remainder after being divided by a number.

But it is not possible for every finitely many numbers to be needed, so mathematicians chose prime numbers, so in part, the problem is also related to the Zeta function of the Riemann conjecture.

After a long period of computation and data collection, Behe and Swinathorn-Dale observed some laws and patterns, and thus proposed the BSD conjecture: let E be an elliptic curve defined on the algebraic number field K, and E(K) is the set of rational points on E, and it is known that E(K) is a finite generation commutative group. Note L(s,E) is the Hasse-Weil L function of E. Then the rank of E(K) is exactly equal to the order of L(E,s) at the zero point at s=1, and the first non-zero coefficient of the Taylor expansion of the latter can be accurately expressed by the algebraic properties of the curve.

The first half is often referred to as the weak BSD conjecture, and the second half is a generalization of the BSD conjecture's class number formula.

At present, mathematicians have only proved that the weak BSD conjecture of rank=0 and 1 holds, and there is still nothing that can be done about the strong BSD conjecture of the Rank≥2 part.

Previously, Pang Xuelin also followed the route taken by Gross and Coates, trying to launch the BSD conjecture of rank≥2 on the basis of rank=0 and 1, but found that he gradually walked into a dead end.

In the last six months, he has not made any progress.

Therefore, he is very curious about what kind of thinking is used in the proof process given by the system.

Pang Xuelin opened the BSD conjecture proof paper and read it.

The proof of the BSD conjecture is more than 60 pages long, which is a bit too streamlined for a conjecture of the millennial puzzle.

However, this does not matter, when Perelman proved Poincaré's conjecture, he only used more than 30 pages, because the process was too brief and many people could not understand it, and at the strong request of the mathematical community, Perelman reluctantly added two more articles, and then refused to give more.

But that doesn't stop Perelman's greatness.

Therefore, the length of the paper does not matter, the key depends on the quality of the paper.

Pang Xuelin did not start to read it carefully from the beginning, but first skimmed it roughly.

A cursory glance helps him to understand the proof idea of the BSD conjecture as a whole.

But soon, Pang Xuelin's brows furrowed.

At the beginning of the paper, a completely different idea from the current mathematical community is given.

The first part of the paper is written about the proof of the congruent problem, i.e., that there is an infinite number of prime factors that are the congruents of any specified positive integer.

Then, it is deduced that the BSD E_D holds that D is the product of an 8k+5 prime and a number of 8k+1 primes, as long as the 4-fold mapping of \Bbb Q(\sqrt{-D}) is single.

That's interesting.

Although there are already attempts in mathematics to prove the BSD conjecture by means of the same remainder problem.

However, this road is too difficult and is still in the germination state, and there are not many achievements in the international mathematical community at present.

The appearance of this paper shows that the current popular BSD conjecture proof methods will eventually lead to a dead end.

Proving the BSD conjecture by the same remainder problem is the correct way of thinking.

Pang Xuelin held his breath and continued to watch.

……

Given the prime number p, (1)p \equiv 3(\mod 8) :p is not the same remainder but 2 p is the same remainder; (2) p \equiv 5(\mod 8):p is the same remainder; (3) p \equiv 7 (\mod 8) :p and 2 p are both congruent remainders.

(Weak BSD conjecture) The BSD conjecture holds true for E_D. In particular, r_D>0 if and only if L(1,E_D)=0.

Assuming that the weak BSD conjecture is true, then (1) theoretically we can determine whether D is the same remainder; (2) The Tunnell theorem gives an algorithm for determining whether D is the same remainder in a finite step; (3) It can be proved that D \equiv 5,6,7 (\mod 8) is r_D odd, so such D is the same remainder.

……

According to the height theory of the Heegner point, the famous Gross-Zagier formula, it can be linked to L'(1,E).

Based on the work of Eichler and Shimura on modular elliptic curves and the newly proven Taniyama–Shimura conjecture (modular theorem), the L(s,E) analysis can be extended to the entire complex plane and the corresponding Riemann conjecture holds.

……

When you look at it, you don't know how time has passed.

I don't know how long it took, Pang Xuelin finally read the whole paper roughly, and breathed a long sigh of relief.

Although there are still many details and many problems to be solved for this paper, Pang Xuelin feels that there is no problem in the overall proof idea.

And for the proof of the entire BSD conjecture, Pang Xuelin also has a feeling of sudden enlightenment.

With the right ideas, he would have been able to fully derive the proof process of the BSD conjecture even without this paper.

Pang Xuelin opened his eyes at this time, and when he turned his head, he found that it was already dark before he knew it, and the blonde and blue-eyed little nurse he had seen before was busy beside him.

Seeing Pang Xuelin open her eyes, she couldn't help but smile and said, "Oh my God, Pang, you're finally awake!" ”

Pang Xuelin was slightly stunned, his eyes swept over the nurse MM's identity card, and he said suspiciously: "Olivia, I'...... How long have I been asleep? ”

Olivia said: "You have slept for three days and three nights, and the doctor is still worried that something is wrong with you, and in the past two days, you have been given a CT of the brain, and various blood tests, and the results show that your body is healthy, but you are asleep, and no one can understand why you have slept for so long." ”

Pang Xuelin couldn't help but be taken aback, although he had done this kind of liver explosion research in the real world, most of them were interrupted because of the need for sleep and supplementary food.

Unexpectedly, lying on the hospital bed this time, he studied for three days and three nights, and after waking up, he did not have the feeling of exhaustion that burst his liver, but a kind of indescribable refreshment.

Could it be that after closing your eyes and entering the system, even if you are doing research inside, it is only equivalent to entering a deep sleep?

If this is the case, then the efficiency of your own research may be improved with the help of the system.

Pang Xuelin's eyes couldn't help but light up.

For a long time, Pang Xuelin did not consider himself a genius, and his achievements in the academic world were insignificant compared to those famous figures in history.

But Pang Xuelin also has his own pursuits.

He hopes that one day, he will be able to solve millennial problems in his own right, and that one day, his name will be on a par with the glittering mathematicians of history.

Therefore, he needs to continuously improve his learning and research efficiency.

Maybe in the eyes of others, Pang Xuelin is already a genius, but Pang Xuelin himself doesn't think so.

The reason why the so-called genius scholars in the world can reach the height of being a god is not that he is naturally smarter than others, but because he has good study habits and efficient learning efficiency.

Not to mention anything else, the reason why Pang Xuelin himself has been able to achieve today's achievements is because he has been studying more than ten hours a day for ten years.

Even so, he is only a young mathematician who has just emerged in the international mathematical community, and there is still a long way to go before those top bulls.

Genius is one percent inspiration and 99 percent perspiration, but without 99 percent perspiration, where can that one percent inspiration come from!