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Chapter 69 Sorry, I'm not good enough...

"...You see, this is an elliptic curve. However, it is not an ellipse in a normal conic curve, but a smooth projective curve with genus 1 on the field. If the characteristic is not equal to 2, then the affine equation is y^2=x^3+ax^2+bx+c.

You must remember the prerequisite of the BSD conjecture, right? The elliptic curve over the complex field is a Riemann surface with genus 1, and the elliptic curve over the global field is a finitely generated commutative group. Abelian clusters are high-dimensional generalizations of elliptic curves.

So at this time, I felt that I had to transform the elliptic curve into Weierstrass form. This is a method I found after reading a lot of related theories. This kind of transformation is a very mechanical operation, and the prerequisite is that there is at least one rational point in the equation.

But obviously this step is valid, we have proved it before, so we can get these two formulas..."

As Qiao Yu spoke, he wrote on the small table.

Langjie listened attentively, stretching his neck to watch Qiao Yu's overall problem-solving process and the plane diagram he drew using the coordinate system.

"...Obviously, we now have a classic elliptic curve with two real parts. The line on the right obviously extends continuously to positive and negative infinity. The closed elliptic curve on the left is the key to the solution. Given any solution to this equation, we can use the equation to restore the value we need."

"The key point of this step is that the triple (a:b:c) must be a projective curve, so that the equation can be established no matter what constant is multiplied. Next, I need to use bidirectional rational equivalence. I will directly find a rational number point on this elliptic curve that is most convenient to solve, and then substitute it into the original equation to find the solution.

In fact, it is very simple at this point. In the theory of elliptic curves, the chord cutting technique is a key tool for generating new rational number points. As long as two known rational number points are found on the elliptic curve: P1 and P2, new rational number points can be generated by addition.

The next step is to construct the tangent directly. At this time, an Abelian group is naturally formed. We need to introduce the zero element in the group O. According to the rules, when any point P is added to O, the result is still P.

...We then draw a tangent to point P to find the point where P intersects the curve again, and then calculate again. If we cannot get an integer solution, we continue to connect P and 2P to find the third intersection with the curve and then connect it with point O to find the fourth intersection. If that doesn't work, repeat this step to find the fifth intersection...

In short, just repeat this step until you find the corresponding integer solution. However, this step cannot be calculated by hand, you can only use a computer to calculate it. After finding the value, use a geometric program to iterate.

Finally, 9P is calculated to be an integer, and then the value of 9P is used to perform 9 geometric program iterations, and finally the values ​​of a, b, and c in the above equation can be obtained. This is the whole idea of ​​solving the problem. "

……

Qiao Yu spoke for a full hour without stopping, and felt his mouth dry. After he finished speaking, he took out the bottle of mineral water plugged into the back of the seat in front of him and drank a few gulps. Then he asked, "How about it, Teacher Lan, do you think my solution is universal?"

Lang Jie came back to his senses, glanced at Qiao Yu, and did not answer immediately.

After all, in order to determine whether this solution is universal, he must first fully understand it.

He asked Qiao Yu to explain because he thought Qiao Yu would not use too complicated number theory to solve this equation. After all, Qiao Yu always gave him the impression that he was talented, but he had not studied mathematics systematically.

But he was different. He had systematically studied abstract algebra and introductory courses on number theory in college, so he would not fail to understand.

But apparently he was wrong.

While listening to Qiao Yu's explanation, he even recalled his youthful years in college, when he was terrified by advanced algebraic geometry.

Projective geometry and moduli space are really confusing. He studied very hard but just barely passed and got credits. Of course, there were many talented students in the class who could get full marks just by studying casually.

This is also the reason why he chose combinatorial mathematics in his graduate studies and returned to Xingcheng to become a high school mathematics teacher after graduation.

It’s not that he doesn’t want to do scientific research, continue studying for a doctorate, and then strive to become a teacher in a university.

The main reason is that my ability is limited and I really can’t read anymore.

So he really didn't fully understand Qiao Yu's idea of ​​solving the equation.

As we all know, if we want to judge whether a mathematical solution method is universal for a class of equations, we must first fully understand the entire solution idea.

This is very embarrassing.

I thought that with the mathematical knowledge he had accumulated in college, he would be able to give an answer after listening to Qiao Yu's live explanation.

But now he needs to make a choice between embarrassment and trying to cover it up.

After hesitating for about ten seconds, Ranger chose to be honest.

Because he is really not good at pretending.

"Qiao Yu, to be honest, my level is not high enough to judge... so you can only try this problem yourself. Find a few similar equations and solve them using your method. If you can get the correct answer in the end, you can start writing the paper.

I can’t help you with how to solve the problem in the paper. But I can teach you how to write the paper. After all, the writing of mathematical papers has specific format and writing requirements, and there are also some common general standards. "

Qiao Yu looked at Langjie in surprise.

Because this problem was indeed difficult, it could be said to be the most difficult equation-solving problem he had ever encountered since he started studying mathematics seriously, so when he explained it, he had a bit of the intention of showing off, and he really explained it in great detail.

But the nice guy actually said he didn’t understand?

"Huh... I told you that I didn't put much effort into algebraic geometry and number theory in college. It's okay if it's just the basics, that is, the content learned in undergraduate school. More advanced number theory... I majored in combinatorial mathematics in my master's degree, which is to study the combinatorial properties of discrete structures, such as permutations, combinations, graphs, sets, and sequences...

And after you join the work, you know high school mathematics... As for the content in the Olympic Games, it will not be involved in depth, but only the most basic part of advanced algebra and number theory. The main thing is to cultivate the ability to solve problems with elementary mathematical methods. So..."

Facing Qiao Yu's inquiring gaze, Lang Jie explained with some stammering.

Well, this is indeed a way to save face.

After all, the subject of mathematics is divided into countless directions... and number theory is obviously the direction that requires the most talent.

It’s normal not to understand.

The key is that Qiao Yu’s age and experience are too hurtful.

"Oh, I see... I understand. It's a matter of specialization, and these things just happen not to be your forte." Qiao Yu said considerately, even using the honorific again.

Ranger opened his mouth but said nothing.

This kid has already given me enough face. If I say more, it will seem like I am quibbling and it is meaningless. Since I have already admitted that I don't understand, I might as well be more open about it.

So Lang Jie raised his hand and patted Qiao Yu on the shoulder, saying, "Don't worry, believe me, one day in the future, the big guys on that forum will be proud to be invited to your lecture! If you still remember me by then, invite me to sit in the first row. Oh, and let my mentor sit next to me."

Qiao Yu was confused and asked, "What is the report meeting?"

Langjie was very patient at this time, because he knew very well that he had little to teach the child in front of him. "For example, if you solve a difficult mathematical problem that no one else has solved, it's like you have found the numerical solution to this equation. If you write a paper and it is recognized, then some well-known universities or mathematical research institutions will invite you to share your ideas and thoughts on solving problems with mathematicians in the same field."

After finishing speaking, Ranger paused and continued, "Of course, the limit of mathematics is that there is no limit. The more difficult and well-known the problem you solve, the higher the level of the report conference! And it's not just about solving a difficult problem. If you can invent some new mathematical tools and gain wide recognition, you will also be invited to give reports at some very important mathematics conferences."

After listening, Qiao Yu was thoughtful.

As if thinking of something, Ranjie added, "Um... don't always think about those high-level mathematics conferences inviting you to give a report and paying you a lot of money! You will know when you reach that level. As long as you can sit in that reporting position, the money will come naturally. Open your mind."

"Oh! I see!" Qiao Yu nodded, looking as if he had learned something, but Ranjie couldn't tell whether he had really listened or not.

But he was too lazy to care.

Judging from the intelligence that Qiao Yu has shown so far, when he really reaches that position, he will naturally understand it, and maybe even understand it better than him.

"Don't worry, Teacher Lan. If one day I am really invited to attend the kind of lecture you mentioned, I will definitely leave two empty seats in the first row. You can bring anyone you want. You don't necessarily have to bring a tutor every time." Qiao Yu promised excitedly.

Perhaps only when showing off like this, this kid would show a bit of youthful spirit.

Ranjie could only smile bitterly...

Who else could he bring with him to attend a mathematics conference if not his supervisor?

To be honest, he went to listen to the lectures with his tutor purely for the sake of appearance, not because he really wanted to listen to these things. Firstly, high school math teachers don’t need to know so many fancy things; secondly, even math teachers will feel drowsy when listening to math problems that they don’t understand.

These are all beyond human control.

"Okay, come on! I'm looking forward to that day from now on." Ranger said seriously.

……

During the rest of the high-speed rail journey, the two of them fell silent again.

Qiao Yu continued to browse on his phone with full confidence, but he no longer read the comments of the big guys on the forum.

What Ranjie said really made him lose a layer of filter towards the so-called big guys, and he no longer worried about them.

Those guys seemed very powerful, but that was only because they had lived ten or twenty years longer than him.

The most important thing is that when Qiao Yu found that after he explained it carefully to Teacher Lan, the good old man still frankly admitted that he didn’t understand, he also confirmed the value of this equation.

No wonder those professors thought he was a graduate student, and a graduate student specializing in number theory.

An average undergraduate mathematics student still cannot fully understand the problem-solving process after listening to it. How could those professors possibly imagine that he is just a junior high school student who just graduated?
So now he takes off safely.

He is very experienced in this area.

When he was in elementary school, he relied on this trick to earn an average of two to three thousand yuan per month!
Qiao Yu felt that he understood very well the self-confidence of adults who always doubted rationality.

When I was in primary school, there were jealous kids who went to the teacher to complain that he helped others write, helped others cheat on exams, etc., but the teacher didn't even take the letter...

How can a guy who only scores a dozen or twenty points in every exam and performs poorly in all aspects help other students do these things? Even if you are half-believing and half-doubting, as long as there is no actual evidence, it will end up in nothing.

Therefore, as long as he didn't take the initiative to log into the forum in front of those professors, those big guys would probably never suspect him.

Without this worry, Qiao Yu felt relieved and simply continued watching the video on his phone.

As for Ranger beside him, he fell into a daze.

I really can't stand looking at my phone anymore.

He had wanted to test out the upper limit of Qiao Yu's level before, but everything that had just happened made him realize that his previous idea was a little ridiculous.

To be honest, he really couldn't understand why a guy who had never undergone systematic learning of algebra and number theory could solve such a difficult equation with his bare hands. Really, he had already thought highly of Qiao Yu before, but now he realized that he still underestimated this kid.

This self-learning ability...

Ranjie thought of a sigh from his tutor at a group meeting. The original words were: There is a kind of genius who can see through the essence of abstract problems at a glance. What they can understand in one day may take you three years, so you have to work hard!
At that time, this sentence was used to describe a young doctoral student from the University of California, Berkeley, who came to the school for an exchange. Langjie still remembered that the Irish boy was about the same age as him, but he had already studied for a doctorate, while he had just passed the postgraduate entrance examination not long ago...

Now it seems that he has also discovered such a treasure, and his ability is even better than that of the other party.

At this moment, Lang Jie also had a strong desire to take Qiao Yu to show off to his teacher...

If I could point at Qiao Yu and say to him in the tutor's office, "Teacher, look, this is my student. I brought him here especially for you to test him."

That feeling is so great just thinking about it.

Besides, Xiaozhou is not that far from Linhai. If I can convince Qiao Yu to go with me to visit my alma mater, Shuangdan University...

When this thought flashed through his mind, Ranger suddenly felt that he was too shameful!

The principal has entrusted him with an important task, counting on Qiao Yutie to win an IMO gold medal. If the child is exposed so early, it will directly attract a bunch of professors to compete for him, which will most likely affect Qiao Yu's mentality.

Not to mention that most of those professors are shrewd people. After a few contacts with Qiao Yu, they will probably be able to understand Qiao Yu's personality. That would be really dangerous...

There's no way, this child's weakness is too obvious.

Langjie even suspected that Qiao Yu deliberately showed his infinite desire for money in order to seduce those adults who were interested in him to invest as much as they could in him.

So let's do it later.

But only one year...

We’ll talk about it when Qiao Yu wins the gold medal in the world competition and gets the qualification to be admitted to Huaqing or Yanbei!
　Thanks to _Wanlaitianyuxue and Laoli Space for the rewards and encouragement!
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(End of this chapter)