A genius? I just love studying.

Chapter 137 A Special Solution to the Cubic Diophantine Equation

When they first saw this title, everyone in the classroom was a little flustered. They had seen similar images on social media or elsewhere more than once. These images often had very alarming titles, such as "A problem that 95% of MIT graduates cannot solve."

In reality, these questions are either empty rhetoric, fall into the trap of misinterpreting concepts, or are just irrelevant riddles.

But this question is clearly not the case!

For the students undergoing intensive training in this classroom, bananas, apples, pineapples, and the like are naturally not obstacles; they have long since transformed the problems into mathematical symbols.

In fact, this question asks them to find the integer solutions to the equation a/(b+c)+b/(a+c)+c/(a+b)=4.

This is clearly a Diophantine equation problem.

Any mathematical researcher familiar with Diophantine equations knows that first-order Diophantine equations are quite simple, second-order ones are also well understood and can generally be solved using relatively elementary methods, third-order ones involve a vast ocean of profound theories and countless open-ended problems, and fourth-order ones are simply insurmountably difficult.

The problem provides a cubic Diophantine equation.

或许题目给的不太明显，但只要简单做一个变换，去掉分母，我们就能得到a^3+b^3+c^3-3(a^2b+ab^2+a^2c+ac^2+b^2c+bc^2)-5abc=0.
Deng Leyan buried himself in calculations. He had encountered Diophantine equations before and knew that solving these types of Diophantine equations required the use of elliptic curves, but he had no idea how to solve them specifically.

The same applies to Wang Xiao.

Li Zehan, meanwhile, was scratching his head in frustration. He had been focusing on learning elementary mathematics to prepare for the CMO exam, but he had only just begun to learn advanced mathematics.

He knew the Diophantine equation, but he was at a loss as to how to solve it.

As for the others, they were like headless flies, scrambling through drafts on paper, but they had absolutely no clue what to do.

However, in their view, since the teacher presented this problem, there must be a solution. They did not believe that they could not solve it. As geniuses, they had such confidence.

The previously enthusiastic atmosphere of answering questions suddenly changed. Not a single person raised their hand in the entire classroom. It was completely silent as the students frantically worked out calculations on their scratch paper.

Xu Zhiyuan glanced at the students who were deep in thought in the classroom, a mischievous smile appearing on his lips.

It's time to give these little guys a math shock!

Unfortunately, everyone was engrossed in their calculations and no one noticed his wicked grin.

Your math level has improved from 2 (99%) to 100%.

Your math level has increased to level 3, and you have gained 1 free attribute point.

Chen Hui, who was immersed in studying, suddenly saw a barrage of comments and felt a surge of joy. He had finally reached level 3 in math proficiency. Just as he expected, he gained another free attribute point after reaching level 3 in math proficiency.

without hesitation,

"Give me some creativity!"

More than half a year has passed since Chen Hui obtained his first free attribute point, and he has finally decided to invest it in this attribute that he has never paid attention to before.

Previously, the mathematics he had encountered all involved using the knowledge he had learned to solve problems and address routine issues.

But recently, he has come to realize the importance of creativity more and more.

Without the condensed matter mathematics proposed by Schultz, he could not have solved the difficulties his teacher encountered in proving the Langlands program; and without the series of conjectures proposed by Langlands, he could not have solved the differential geometric realization of fractional Chern classes.

Both condensed matter mathematics and the Langlands program are mathematical tools that have never appeared before.

At this stage, doing mathematical research and solving mathematical problems is no longer just about applying what you've learned. In some cases, you need to create new tools to solve problems!
[Host: Chen Hui
Insight Level 4: (4.1/5)

Judgment Level 1: (1.8/2)

Creativity Level 1: (1.5/2)

Memory Level 3: (3.6/4)

The increase in creativity did not bring about any unique changes, which is what puzzled Chen Hui about this data panel. It seemed that each upgrade and point allocation did not bring about earth-shattering changes, and he could not even feel the occurrence of such changes.

But perhaps these changes were already happening every minute of his learning, which is why he didn't feel them.

After finishing all that, Chen Hui noticed the question on the projector and Li Zehan scratching his head next to him.

"interesting!"

With just one glance, Chen Hui, who hadn't yet been influenced by social media and online pollution, understood the meaning of the question.

Although he had a mobile phone for a long time, he rarely paid attention to other irrelevant information except for searching for information online. Even if he did, he would actively search for it rather than passively accept it.

"Can the boss solve this?"

Li Zehan noticed the commotion, looked up, and stared intently at Chen Hui.

"You can try it."

Chen Hui originally thought that there would be nothing of value in this training camp, so he didn't listen at all, but this question was obviously very difficult.

He likes challenging problems!
Upon hearing this, the other members of the group also looked over.

Since they had no clue what to do, they simply stopped writing and looked at Chen Hui's draft paper.

"Class, has anyone figured out the answer yet?"

A few minutes later, Xu Zhiyuan checked the time. It was already 9:54, almost time for get out of class to end. He was never a teacher who would extend class time. Of course, he was also never a teacher who liked to solve puzzles. He would give the answer and then let the students go back and think about the solution themselves, leaving some suspense. If they wanted to know what would happen next, he would explain it again tomorrow.

At this point, someone raised their hand and asked, "Teacher, can we have non-positive integer solutions? For example, a=-1, b=1, c=0?"

"Of course not, it must be a positive integer solution."

Xu Zhiyuan shook his head. The essence of this problem lies in finding positive integer solutions, and the difficulty also lies in finding positive integer solutions.

If one has the understanding, then one can easily write out a whole bunch of such special solutions.

But finding an understanding of this problem is the first step to unlocking its secrets, which unfortunately these little guys don't know.

"Alright, the solution to this problem is..."

Xu Zhisheng waited for another two minutes, and when he found that no one else raised their hand, he walked back to the podium, picked up the chalk, and began to speak and write.

"Everyone can think about the solution on their own. I'll come back tomorrow to explain this type of Diophantine equation." Because the solution was so long, he finished speaking before he even finished writing the first solution.

Although this training camp was to prepare for the IMO, Xu Zhiyuan felt that cultivating children's interest in mathematics was far more useful than winning IMO gold medals.

Many IMO gold medalists are awarded every year, but only a few ultimately become outstanding mathematicians.

Some IMO gold medalists, due to their fixed mindset, are very good at elementary mathematics, but find it difficult to adapt when they come into contact with advanced mathematics.

For example, he knew of an IMO gold medalist who was admitted to Yanbei University through a special program, but spent six years studying for his undergraduate degree because he failed many courses.

Unlike those contestants who were admitted to Tsinghua and Peking Universities through the IMO but then played games, neglected their studies, and were expelled, he studied very hard. He was always either doing practice problems or on his way to do practice problems. However, he did not focus on the definitions and theorems of advanced mathematics. Instead, he used the methods and techniques of elementary mathematics to solve problems in advanced mathematics.

While this approach might solve some problems, it's clearly like trying to solve problems that airplanes and cannons should be solving with millet and rifles—inefficient and ultimately unsuccessful.

Therefore, he was stuck at the undergraduate level.

Or rather, he is trapped by his own way of thinking, and if he can't break free, this child's life may be ruined.

A person who can win an IMO gold medal cannot be without talent, and Xu Zhiyuan felt it was a great pity that this outcome occurred.

So, he decided to show these children what airplanes and cannons look like before they participated in the IMO, and give them a military equipment demonstration!
As Xu Zhiyuan typed the answers on his phone, his mind wandered. Suddenly, he heard a noise behind him.

A student stood up and said, "

"?"

Everyone in the classroom stopped what they were doing and looked up at Chen Hui.

Even though they already knew this guy was strong, hearing this answer still made them think he was an idiot.

The last time they heard someone recite so many numbers was when a guy with an exceptional memory memorized a thousand digits of π after the decimal point...

They couldn't solve the problem, but they felt the answer was too outrageous.

Therefore, everyone's attention turned to Xu Zhiyuan.

Xu Zhiyuan was also somewhat bewildered.

To be honest, he couldn't remember the answer either; there was no need for him to remember three positive integers of over eighty digits.

But he quickly realized that if this guy could say those numbers, it meant he had actually figured it out.
How can this be?

Xu Zhiyuan was very skeptical because he couldn't calculate the answer himself. He had calculated it using a computer, and the computational workload exceeded the limits of the human brain.

But he recognized the student, Chen Hui!
He had heard some stories about this CMO who had achieved a perfect score, including at the seminar at Yanbei University.

Perhaps he really can figure it out?

Xu Zhiyuan suddenly felt a sense of anticipation.

If he can indeed calculate it without using a computer, then can this method be generalized to solve this type of Diophantine equation?
If possible, that would be an exciting result!

However, he soon found his idea laughable; he had actually tried to get a high school student to invent a special solution to a cubic Diophantine equation.

"Could you explain your solution method to everyone?"

Although he had little hope, Xu Zhiyuan still decided to listen to Chen Hui's ideas.

"I was also inspired by that student."

Chen Hui looked at the student who had just raised his hand. He was also a little excited. No matter when, solving a difficult problem always makes people feel excited and full of a sense of accomplishment.

Therefore, he doesn't mind sharing his problem-solving approach with everyone. "We can easily find a set of rational solutions: a=-1, b=1, c=0. Having rational solutions means that the equation we are looking for is actually an elliptic curve!"

"?"

The student being stared at by Chen Hui looked completely bewildered, his eyes revealing a clear sense of naiveté. "Did I even think that way?"

"Oh, here, an elliptic curve refers to a smooth projective curve of genus 1 over a field. For a field whose eigenvalue is not equal to 2, its affine equation can be written as y^2 = x^3 + ax^2 + bx + c. An elliptic curve over the complex field is a Riemann surface of genus 1. Model proved that an elliptic curve over a global field is a finitely generated commutative group, which is a prerequisite for the famous BSD conjecture. Abelian varieties are a higher-dimensional generalization of elliptic curves..."

Considering that the people in the classroom were all high school students participating in the IMO, rather than the professors from Yanbei University at the time, Chen Hui made a special explanation.

But his explanation only made things worse; the more bewildered the students in the classroom became.

"It's as if you think we'll understand just because you explained it!" many people thought to themselves.

Chen Hui, however, did not notice the students' reactions. His eyes were bright and sparkling, as if countless numbers and symbols were jumping around. "With this consensus, we can now transform this elliptic curve into the Weierstrass form, which is y^2=x^3+109x^2+224x."

"By the way, some of you might be wondering, didn't the original equation have three unknowns? How come there are only two unknowns here?"

"Because this equation is homogeneous, it means that if (a, b, c) is a particular solution to the equation, then (7a, 7b, 7c) is also a solution to it. This means that the equation appears to be three-dimensional, but it is actually only two-dimensional."

In geometry, it corresponds to a surface. A ternary equation generally defines a two-dimensional surface. In general, k n-variable equations define a d-dimensional manifold, d=nk. This surface is formed by rotating a line passing through the origin, and can be understood through a single intercepted plane. Therefore, it can also be concluded that this is a projective curve.

Chen Hui seemed genuinely eager for his students to understand and learn, striving to make his explanations easy to comprehend. To further clarify his presentation, he even stepped out of his seat, went to the podium, and drew a diagram of the elliptical curve on the blackboard with chalk.

"As shown in the figure, the 'fish tail' on the right extends continuously to positive and negative infinity, and the closed elliptic curve on the left is the opportunity for us to solve the problem. Given any solution (x, y) to this equation, we can restore the required a, b, c through transformation. In this way, we have constructed a birational equivalence."

At this point, 99% of the students in the classroom were starting to feel dizzy, with only a handful, such as Deng Leyan and Wang Xiao, still managing to keep up.

But they had already frowned at this point.

Because the elliptic curve problem is a massive undertaking, even though a lot has been done, the problem doesn't seem to be solved.

Xu Zhiyuan looked at Chen Hui with great interest. If he had previously thought that Chen Hui was incapable of solving this problem, now he felt that the kid might have actually found some tricks.

"Returning to the constructed elliptic curve, we can easily find a good rational number point on it, x=100, y=260. This is not a positive integer solution, but don't worry, next we will use the chord-tangent technique to perform addition to generate other rational number points."

"By drawing the tangent line at point P, we find the point where it intersects the curve again, and thus increase the value of point P, we get 2P = (8836/25, -950716/125). In this way, we can get a new set of solutions for a, b, and c, but obviously, they are still not positive integer solutions."

But that's okay, we'll continue iterating, calculating 3P, 4P, and so on up to 9P. Finally, we'll get the positive integer solutions for a, b, and c!

After saying that, Chen Hui threw down the chalk and said with a smile, "The calculation is a little complicated, but the overall idea is still very simple!"

(End of this chapter)