"..., this is an elliptic curve. But it is not an ellipse in ordinary conic sections, but a smooth projective curve with a genus of 1. If the characteristic is not equal to 2, then the affine equation is y^2= x^3 ax^2 bx c. You must still remember the precondition of the BSD conjecture, right? The elliptic curve on the complex city is a Riemann surface with genus 1, and the elliptic curve on the overall domain is a finitely generated commutative group. Abelian varieties are higher-dimensional generalizations of elliptic curves.
So at this time, I feel that I need to transform the elliptic curve into the Weierstrass form. This is a method I found after reading a lot of relevant theories. This kind of transformation is a very mechanical operation, and the prerequisite is that there is at least one equation. Rational number points. But obviously this step is established. We have proved it before, so we can get these two formulas..."
As Qiao Yu spoke, he wrote with a pen on the small table.
Lan Jie listened carefully, stretching his neck to see Qiao Yu's overall problem-solving process and the plane diagram he drew casually 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 plus and minus infinity. The closed elliptic curve on the left is the key to solving. Given any solution to this equation, We can use equations to restore the values ​​we require."
"The most critical part of this step is that the triplet (a:b:c) must be a projection curve, so that you can multiply it by any constant and the equation will be established. Next, you will use two-way rational equivalence. I just directly find a rational point on this elliptic curve that is the most convenient to solve, and then put it into the original equation to find the solution.
In fact, this step is easy. In elliptic curve theory, the chord tangent 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, they can be generated by addition. New rational number point. The next step is to directly construct the tangent line. 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 It's P.
...We then find the point where P intersects the curve again by making a tangent to point P, and then calculate again. If we cannot get an integer solution, we continue to connect P and 2P to find the third intersection point with the curve and then connect it with O Connect the points to find the fourth intersection point. If that doesn't work, repeat this step to find the fifth intersection point...
In short, this step is repeated until the corresponding integer solution is found. However, this step will definitely not work by hand. It can only be calculated by computer. After finding the value, iterate with the geometry program. Finally, calculate 9P to be an integer. , and then use the obtained value of 9P to do 9 iterations of the geometric program, and finally get the values ​​​​of a, b, c of the above equation. The whole problem-solving idea is like this."...
Qiao Yu spoke for an entire hour at one breath, but felt his mouth was dry. After finishing speaking, he took out the mineral water stuck on the back of the seat in front of him and drank a few sips. Then he asked: "What? So, Teacher Lan, do you think my solution is universal?" Lan Jie came back to his senses, glanced at Qiao Yu, and did not answer immediately.
After all, in order to judge whether this solution is universal, he must first fully understand it.
He asked Qiao Yu to explain because he thought that Qiao Yu would not use too complicated number theory when solving this equation. After all, Qiao Yu had always given him the impression that he was talented, but he had not studied mathematical systems.
Yes. But he was different. He also systematically studied abstract algebra and introductory number theory courses in college, so he didn't understand them.
But obviously he was wrong.
When listening to Qiao Yu's explanation, he even recalled the fear of being dominated by advanced algebraic geometry during his green years in college.
Projective geometry and modulus spaces were really confusing. He worked hard to learn them, but in the end he just barely passed the test and got the credits. Of course, there were many talented students in the class, and he could get full marks by just learning casually.
This is also the reason why he chose combinatorial mathematics during his graduate studies and returned to Star City to become a high school mathematics teacher after graduation.
It’s really not that he doesn’t want to do scientific research and continue to study for a doctorate, and then strive to become a teacher in a university. The main reason is that his ability is limited and he really can’t study anymore.
So he really didn't fully understand Qiao Ze's idea of ​​solving this equation.
As we all know, if you want to judge whether a certain mathematical solution method is universal to a class of equations, you must first fully understand the entire solution idea. This is very embarrassing.
I thought that with the mathematical knowledge he accumulated in college, he would be able to give an answer after listening to Qiao Yu's live explanation.
But now he needed to make a choice between being embarrassed and finding a way to cover it up. After pondering for about ten seconds, Lang Jie chose to be honest.
Because he's really not good at pretending.
"Qiao Yu, to be honest, I'm not good enough to judge...so you can only try this problem yourself. Find a few similar equations and use your method to solve them. If you can get the correct answer in the end
If the answer is correct, you can start writing a paper.
I can't help you with how to solve the specific problems in the paper. But I can teach you how to write the paper. After all, writing mathematics papers has specific format and writing requirements, and there are also some common general standards." Qiao Yu was surprised.
He glanced at Lang Jie.
Because this problem is really difficult, it can be said to be the most difficult equation solving problem he has ever encountered since he started studying mathematics seriously, so he was somewhat showing off when he explained it, and he really explained it very carefully. But the good guy
He actually said he didn't understand?
Wait... I said that I didn't put much effort into teaching or number theory in college. It's okay if it's just for the elementary level, which means it's just the content I learned as an undergraduate. I want in-depth micro theory... I'm the main subject at this stage
It is knot mathematics, which studies and teaches problems such as the combinatorial properties of structures, permutations, combinations, graphs, sets, and sequences...
And after you start working, you know high school mathematics... As for the content in the Olympiad, it will not be very in-depth. It will only cover some of the most basic things in advanced algebra and number theory. The main purpose is to cultivate students who use elementary mathematics.
Method, the ability to solve problems. So...
Facing Qiao Yu's probing gaze, Lan Jie stuttered in his explanation. Well, this is indeed a tribute.
After all, the subject of mathematics is divided into countless directions. Number theory is obviously the one that requires the most talent. It is actually normal to not understand it.
The key is that Qiao Yu's age and experience are too hurtful.
"Oh, that's it...I understand, this is a professional specialty, and these things happen to be not the direction you are good at." Qiao Yu said thoughtfully, even using the honorific again. Lan Jie Zhang He opened his mouth, but didn't say anything.
This child has already given enough face. If he talks more, it will look like sophistry and is meaningless. He admits that he doesn't understand, so why not be more calm.
So Lan Jie raised his hand and patted Qiao Yu on the shoulder and said, "Don't worry, believe me, one day in the future, the big guys on that forum will all be proud to be invited to your report! By then, If you still remember me, invite me to sit in the first row, and by the way, let my mentor sit next to me."
Qiao Yu didn't know why, so he asked: "What will the report be?"
Lang Jie was really patient at this time, because he knew very well that there was not much he could teach the child in front of him.
"For example, if you solve a difficult problem that no one else has solved in mathematics, it is as if you have found the numerical solution to this equation. You write a paper and it is recognized. At this time, some well-known universities or mathematics research institutions will You will be invited to share your problem-solving ideas and ideas with mathematicians in the same field."
After speaking, Langer paused and continued: "Of course, the limit of mathematics is that there is no limit. The more difficult and famous the problem you solve, the higher the level of the report will be! And it is not just about solving a certain problem. , if you can invent some new mathematical tools and gain widespread recognition, you will also be invited to give reports at some very important mathematics conferences."
After listening, Qiao Yu was thoughtful.
Lan Jie seemed to have thought of something, and took the initiative to add: "Well... don't always think about those very high-level mathematics conferences that invite you to give reports and pay 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 naturally come from you, so open up the situation a little bit."
"Oh! I understand!" Qiao Yu nodded, looking like he was being taught, but he didn't really listen, Lan Jie couldn't tell.
But he was too lazy to care.
With Qiao Yu's current mentality, when he really reaches that position, he will naturally understand it, and maybe he will understand it better than him.
"Don't worry, Teacher Lan. If one day I can really be invited to attend the lecture you mentioned, I will leave two empty seats in the first row every time. You can take whoever you want." You don’t have to bring a mentor every time." Qiao Yu promised with some excitement.
It was probably only when he was showing off like this that this kid would show off his youthful spirit. Lan Jie could only smile bitterly.
When he goes to attend a mathematics conference, who else can he bring with him if he doesn’t bring his tutor? When faced with difficult problems, he will still fall asleep.
These are all independent of human will.
"Okay, come on! I'm looking forward to that day from now on." Lan Jie said seriously.
During the next high-speed train trip, the two of them became quiet again.
Qiao Yu continued to browse his phone with confidence, but he no longer looked at the comments of the big guys on the forum. Lang Jie's words really made him lose a layer of filter towards the so-called big guys, and even more No more worries.
Those guys seemed very powerful just because they lived ten or twenty years longer than him.
The most important thing is that when Qiao Yu found out that after carefully explaining to Teacher Lan, the good old man still frankly admitted that he didn't understand, he also determined the value of this equation. No wonder those professors thought he was a graduate student and specialized in number theory. Graduated.
Ordinary mathematics undergraduates still don't fully understand the problem-solving process after listening to it. How could those professors imagine that he is just a junior high school student who has 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 a month! Qiao Yu felt that he understood the self-confidence of adults who always doubted the rationality.
When I was in elementary school, a jealous kid went to the teacher to complain, saying that he helped others write, cheated on exams, etc., but the teacher didn't even believe it.
How can a guy who can only score ten or twenty points in every exam, who performs poorly in all aspects, help other students do these things? Even if he is half-believed, as long as there is no actual evidence, nothing will be done in the end. So. As long as he doesn't take the initiative to log into the forum in front of those professors, those big guys will probably beat him to death without suspecting him.
Without this layer of worries, Qiao Yu just felt relaxed. He simply continued to watch the video on his mobile phone. As for Lan Jie beside him, he went into a daze.
I really can’t stand the phone anymore.
Before, he wanted to test the upper limit of Qiao Yu's level, but what just happened made him realize that his previous idea was a little ridiculous.
To be honest, he really couldn't figure out why a guy who had never experienced the systematic study of algebra and number theory could solve such a difficult equation with his bare hands. Really, he had already admired Qiao Yu very much before. But now he discovered that he still underestimated this child
This self-learning ability...
Lan Jie thought of the instructor's emotion during 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 others 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 exchange. Lang Jie still remembered that the Irish young man was about the same age as him, but he had already studied for a Ph.D., but he had just entered the postgraduate entrance examination... .Now it seems that he has also discovered such a treasure, even more powerful than the other party.
At this moment, Lan Jie also had a very strong desire to take Qiao Yu to show off to his mentor...
If I could be in the tutor's office, point at Qiao Yu and say: "Teacher, look, this is my student, brought here specifically for you to test him." It would be a great feeling just thinking about it.
And speaking of it, Xiaozhou is actually not that far from Linhai. If he can convince Qiao Yu to go to his alma mater, Shuangdan University, with him... When this thought flashed through his mind, Lan Jie suddenly felt that he was too far. Shameful!
The principal has entrusted him with an important task, counting on Qiao Yu to win an IMO gold medal in the first grade of Tie. Exposing the child so early will directly attract a bunch of professors to compete for him, which will most likely disturb Qiao Yu's mentality. Not to mention those Professors are mostly human beings. After a few contacts with Qiao Yu, they can probably figure out Qiao Yu's temperament. That would be really dangerous.
There is no way, this child's weakness is simply too obvious.
Lan Jie even suspected that Qiao Yu deliberately showed his endless desire for money to seduce adults who were interested in him to invest in him as much as possible. So it was better to wait a little later.
Just one year...
Chapter completed!