Chapter 69 I'm sorry, I'm not good enough...
"...You see, this is an elliptic curve. But it is not an ellipse in a general conic section, but a smooth projection with a genus of 1 on the field. curve. If the characteristic is not equal to 2 If so, 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 in the complex field is deficit. Riemann surface with grid 1, elliptic curvature on the overall domain Lines are finitely generated commutative groups. Abelian varieties are high-dimensional generalizations of elliptic curves. So at this time, I feel that I have to convert the elliptic curve into the Weierstrass form. This is something I have read a lot about. A method that was only discovered after theory. This kind of deformation 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 true, and we have proved it before, so we will do it. Can we 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, and 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 numerical value we require."
"The most critical part of this step is that the triplet (a:b:c) must be a projection curve. Only in this way can we multiply by any constant and the square can be The equation is established. Next, we will use two-way rational equivalence. I will directly find the most convenient rational point to solve on this elliptic curve, and then insert it into the original equation to find the solution.
Actually, it’s easy once you get to this step 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, new rational number points can be generated through addition.
The next step is to directly construct the tangent. 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 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 will continue to connect P and 2P to find the third point with the curve. The intersection point is then connected to point O to find the fourth intersection point. If that doesn't work, repeat this step to find the fifth intersection point...
In short, just repeat this step until the corresponding integer solution is found. However, this step cannot be calculated by hand. It can only be calculated by computer. After finding the value, it can be iterated using a geometry program.
Finally, calculate 9P to be an integer, and then use the obtained value of 9P to do 9 geometric program iterations, and finally you can get the values of a, b, and 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 he finished speaking, he took out the mineral water stuck on the back of the seat in front of him and took a few sips. Caikai asked: "How about it, 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.
Let Qiao Yu explain because he originally 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 never studied mathematics systematically.
But he was different. He also systematically studied abstract algebra and introductory number theory courses in college, so he would not be unable to understand them.
But apparently he was wrong.
While 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 are really confusing. He studied hard but in the end he just managed to pass and got the credits. Of course, there are many talented students in the class, and you can get full marks by just learning casually.
This is also the reason why he chose combinatorics as a graduate student and returned to Star City to become a high school mathematics teacher after graduation.
It’s not that he doesn’t want to do scientific research, he will continue to study 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 this equation.
As we all know, if you want to judge whether a certain mathematical solution method is universal for a type 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 needs 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 is really not good at pretending.
"Qiao Yu, to be honest, I am 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 If you get the correct answer in the end, you can start writing a paper.
I can't help you with how to solve the problem, but I can teach you how to write the paper. After all, there are specific formats and writing requirements for writing mathematics papers, and there are some common ones. Universal Standard.”
Qiao Yu glanced at Lan Jie in surprise.
Because this question is really difficult, it can be said to be the most difficult equation solving problem he has ever encountered since he seriously studied mathematics, so he was somewhat showing off when he talked about it. Very carefully.
But the good old man said he didn’t understand?
"Huh... As I said, I didn't put much effort into algebraic geometry and number theory in college. It's okay if it's just elementary, that is, only what I learned as an undergraduate. Deeper number theory …My main focus during my master’s degree is combinatorial mathematics, which is the study of the combinatorial properties of discrete structures, such as permutations, combinations, graphs, sets, and sequences. Question...
And after joining the work, you know high school mathematics... As for the content of this aspect in the Olympiad, it will not be very in-depth, it will only cover the most basic parts of advanced algebra and number theory. . The main thing is to cultivate the ability to solve problems using elementary mathematical methods. So..."
Facing Qiao Yu's probing gaze, Lan Jie stuttered in his explanation.
Okay, this is indeed a tribute.
After all, the subject of mathematics is divided into countless directions... and number theory is obviously the one 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, 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 opened his mouth, but didn't say anything.
This kid has given enough face. If he says more, it will look like sophistry and is meaningless. Even if you admit that you don’t understand, it’s better to 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, all the big guys on that forum will be invited to participate in your report. I'll be proud! If you still remember me, invite me to sit in the front row. By the way, let my mentor sit next to me."
Qiao Yu didn't know why and 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 that, Langerton. After a pause, he continued: "Of course, the limit of mathematics is that there is no limit. The more difficult the problem you solve and the more famous it is, the higher the level of the report will be! And it's not just about solving a certain problem, if you can invent some The new mathematical tools have been widely recognized and 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 give you a lot of money! Wait until you arrive. You know that level. As long as you can sit in that reporting position, the money will come naturally. Open up the situation.”
"Oh! I understand!" Qiao Yu nodded, looking like he was being taught, but he didn't really listen, and Lan Jie couldn't tell.
But he was too lazy to care.
With the mentality that Qiao Yu is currently showing, 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 kind of report meeting you mentioned, I must leave two empty seats in the first row every time. Anyone can go. You don’t have to bring a mentor every time,” Qiao Yu promised with some excitement.
Probably only when showing off like this will this kid show a trace of youthful spirit.
Lan Jie could only smile bitterly...
Going to attend a mathematics conference, who else could he bring with him if not his tutor?
To be honest, going to listen with the instructor is purely for the sake of face. Not that he really wanted to hear these things. Firstly, high school mathematics teachers do not need to know so many fancy things; secondly, even mathematics teachers will fall asleep when listening to mathematical problems that they do not understand.
This is not based on human will.
"Okay, come on! I'm looking forward to that day from now on." Lan Jie said seriously.
...
On the next high-speed rail trip, the two became quiet again.
Qiao Yu continued to browse his phone with confidence, but he no longer looked at the reviews of the big guys on the forum.
Lang Jie's words really made him lose a layer of filter and worry about the so-called big boss.
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 that after explaining to Teacher Lan carefully, 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.
The average mathematics undergraduate still doesn't fully understand the problem-solving process. 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 consciously understands adults' self-confidence that always doubts 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 take the letter...
One test every time A guy who can only get ten or twenty points in the exam, a guy who performs poorly in all aspects, can still help other students do these things? Even if you are half-convinced, as long as there is no actual evidence, it will be settled in the end.
So as long as he doesn't take the initiative to log in to 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. I simply continued watching the video on my 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 systematic study of algebra and number theory could solve such a difficult equation with his bare hands. Really, he had already looked up to Qiao Yu very highly before, but now he found that he still looked down on the child.
This self-study 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. He can understand it in a day. It may take you three years to get something, 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 was just Not long after he took the postgraduate entrance examination...
Now it seems that he has also discovered such a treasure, even more capable than the other party.
At this moment, Lan Jie also had a 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."
That feeling, just thinking about it They all felt great.
And speaking of it, Xiaozhou is actually not that far from Linhai. If he can convince Qiao Yu to go with him to his alma mater, Shuangdan University...
When it flashed through my mind With this thought, Lang Jie suddenly felt that he was too shameful!
The principal has entrusted him with a heavy responsibility and is counting on Qiao Yu to win an IMO gold medal in Tie's first grade. Exposing the child so early will directly attract a bunch of professors to compete for him, which will most likely interfere with Qiao Yu. mentality.
Not to mention that most of the professors are 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 too obvious. .
Lan Jie even suspected that Qiao Yu deliberately showed his endless desire for money to lure adults who were interested in him to invest in him as much as they could.
So it’s better to be later.
It’s only been a year...
Let’s talk about it when Qiao Yu wins the gold medal in the World Series and is qualified to be recommended to Huaqing and Yanbei!
Thanks to Wanlai Tianyuxue and Laoli Space for their rewards and encouragement!
Continue to seek subscriptions and monthly tickets, brothers!
(End of this chapter)