A genius? I just love studying.

Chapter 123 Fermat's Little Theorem

The awards ceremony was just as long as the opening ceremony, but it wasn't as noisy as the cultural performance. Chen Hui was too lazy to change the location to study. After summarizing what he had learned in the past two days, he opened the next paper.

The papers Yuan Xinyi prepared for him were arranged in a deliberate order, progressing from simple to complex, with a clear structure. Chen Hui read through them without encountering any particularly difficult points. After finishing them, the full picture of the Langlands program naturally came to mind.

This is the advantage of having a teacher. If Chen Hui had to find the information on his own, not to mention the time wasted searching for materials, learning wouldn't have been so easy.

Chen Hui remained calm and composed, but the professors from Yanbei and Tsinghua Universities sitting in the back row of the lecture hall were getting impatient, especially with so many competitors around them. Even those who had long held high positions couldn't help but feel restless.

At this moment, Shi Potian, who had been sitting in the corner, suddenly stood up, walked calmly to where Chen Hui and the others were, patted Li Zehan on the shoulder, and said, "Hello, classmate, I am Shi Potian, a professor in the School of Mathematics at Yanbei University. I would like to say a few words to classmate Chen Hui. May I have room for you?"

Li Zehan looked up at Shi Potian, staring at him with wide eyes. In the end, he got up with a wronged expression and moved to the side.

"Hello Chen Hui, I am a full professor in the School of Mathematics at Yanbei University, a Changjiang Scholar, and I have published one paper independently in the top four journals. My research focuses on group theory, which is your specialty. My current research project is on the representation and classification of infinite-dimensional Lie groups and their applications in quantum field theory. Would you be interested in joining our research group?"

Shi Potian went straight to the point, saying, "I can provide you with a direct PhD program."

"?"

Li Zehan, who had just reluctantly moved to the side, widened his eyes.

This old man is actually so strong?
The slight resentment he felt from being forced to give up his seat vanished instantly.

Although he focuses on math competitions, he also has some understanding of the mathematics community. Mathematicians who can publish in the Big Four accounting firms are already among the top tier of people in the mathematics community.

The only ones who can truly surpass them are winners of major awards, such as those who deliver a 45-minute report at the Congress of Mathematicians—masters who are halfway to the Philippine Prize.

Yet this very person took the initiative to invite Chen Hui.

What he just said sounded like Chen Hui was interviewing him.

Rebel against Tiangang?

That's fine, but he also got a perfect score, so why did he invite Chen Hui but not him?
Everyone got full marks, so why the discrimination?
However, what he found even more incomprehensible was Chen Hui's reply, "Sorry, I already have a teacher."

Shi Potian was also taken aback. "Can you tell me who your teacher is?"

"Yuan Xinyi".

"It's him?"

Shi Potian suddenly realized, "No wonder!"

He had originally intended to try and win the bid, but he gave up on that idea.

Of course he knows Yuan Xinyi; Yuan Xinyi is currently on an exchange program at Yanbei University.

The teacher is Tian Yang, a semi-Fine Arts Award winner and an academician.

Even without considering his teachers, he still doesn't have an advantage over Yuan Xinyi, who has two papers published in the top four journals.

More importantly, Yuan Xinyi is not yet forty years old, while he is already in his sixties.

Youth is full of endless possibilities!

"I see you have a great talent for group theory. If you are interested in my research, you can contact me anytime. Our team has achieved some results in this area. Even if we can't be mentor and student, we can still collaborate. Mathematical research is no longer the era of lone heroes."

He didn't know how Yuan Xinyi had persuaded Chen Hui, but he knew he didn't have any leverage to win Chen Hui over.

Since that's the case, we might as well forge a good relationship.

"Okay, if I do any research in this area later, I will definitely contact Professor Shi."

Chen Hui didn't refuse. The professor was respectful and humble, so he didn't need to be arrogant.

Of course, he has no plans in that regard for the time being.

"Well then, it's a pleasure working together. I won't disturb your studies any longer."

Shi Potian laughed heartily and reached out, glancing briefly at the papers on Chen Hui's desk.

Goodness gracious, "Sur un problème de compatibilité local-global localement analytique" is indeed a paper related to the Langlands Programme, and it's a cutting-edge research result.

This shows that Chen Hui was not lying. These papers should have been selected for him by Yuan Xinyi. However, he was very puzzled as to whether this young man could understand such cutting-edge papers.

"Happy cooperation!"

Chen Hui politely extended his hand to shake hands with Shi Potian and returned the gesture with a smile.

Without any hesitation, Shi Potian got up and walked straight out of the lecture hall.

"Wow, you scoundrel! You have no sportsmanship!"

Seeing Shi Potian and Chen Hui chatting and laughing, and even shaking hands, Cao Bo, sitting in the back row, gritted his teeth in hatred.

They originally planned to extend an olive branch to Chen Hui after the awards ceremony, but unexpectedly, Shi Potian beat them to it.

Cao Bo couldn't help but get up and head towards Chen Hui's side.

But before he could get up, Xing Jiguang pressed him down. "The awards ceremony is still going on. It's fine if one or two people go, but what if everyone goes to Chen Hui's side? What kind of scene would that be?"

"Are you just going to watch that scoundrel get there first?"

Cao Bo was furious, but he hadn't lost his temper yet, and he didn't break free from Xing Jiguang's grip.

"Of course we should go find him!"

Xing Jiguang gestured towards the outside of the lecture hall, then got up and led the way out.

Seeing his actions, the other professors stopped pretending to be reserved and got up to follow him.

As soon as they stepped out of the lecture hall, they saw Shi Potian leisurely waiting there in the lobby.

Shi Potian's confident demeanor sent a chill down the spines of all the professors who had followed him out, making them realize that they were likely out of the running.

The professors here are all of similar ability and resources. Shi Potian is already at an above-average level among them, and many people are not even as good as him.

If Chen Huizhen and Shi Potian have reached an agreement, the difficulty for them to poach him will be hellish.

"Old dog Shi, you have no martial ethics, you are despicable and shameless!"

Cao Bo, who already harbored old grudges, went all out. He might not be able to win the fight, but taking the opportunity to vent his old hatred was a good idea.

Not only Cao Bo, but the other professors also had their complaints and glared at him angrily.

Shi Potian remained calm and composed, not even glancing at Cao Bo, but continuing to look at the lecture hall door as if waiting for something.

"Are they waiting for Chen Hui to come out?"

The other professors followed Shi Potian's gaze, their smugness infuriating.

At this moment, the last professor also walked out of the lecture hall.

Shi Potian then stood up and looked at everyone, "Chen Hui already has a teacher."

"?"

The other professors were dumbfounded; they had never seen such an arrogant person.

"Chen Hui's teacher is not me, but Yuan Xinyi."

Before the other professors could turn their anger into action, Shi Potian spoke again, then turned and walked out of Zhihua Building.

He was only waiting here to share the news with everyone, so that no one would embarrass themselves in the lecture hall.

Not Shi Potian?
The hall was silent as the professors looked at each other in bewilderment. Only after Shi Potian had left did they begin to talk, like insects awakening from hibernation.

"Yuan Xinyi?"

"It's him!"

Cao Bo looked at Xing Jiguang beside him.

"What a fine Yuan Xinyi!"

Xing Jiguang also sighed softly, shook his head and walked out of Zhihua Building.

If his opponent is Shi Potian, he is prepared to give it his all.

But if Chen Hui's teacher were Yuan Xinyi, he didn't think he had any chance of winning.

Leaving aside Yuan Xinyi himself, his teacher Tian Yang is a long-established academician of Yanbei University, whose academic resources are far beyond what Yuan Xinyi can compare to.

This was without his knowledge of Yuan Xinyi's latest breakthrough.

There's no comparison, absolutely no comparison. If he were Chen Hui, he would also choose Yuan Xinyi.

"You're just leaving like that?" Cao Bo caught up with Xing Jiguang and asked, his face full of resentment.

"if not?"

Xing Jiguang retorted irritably, "Who told you you don't have as good taste as them?"

They were professors in the grading group, and should have been the first to receive the news, but Yuan Xinyi actually finalized a contract with Chen Hui before them.

What could they do if the deal was finalized even before the CMO?
Cao Bo understood Xing Jiguang's meaning and began to face reality, but he couldn't understand it at all. "What do you think Academician Tian was thinking? Even if he wanted to take care of his students and bring geniuses under his wing, he should have recruited them from Yanbei. What's the point of letting that little guy go to Jiangcheng with Yuan Xinyi?"

Xing Jiguang stopped and turned to look at Cao Bo. "You're asking me? How am I supposed to know? If you're so capable, go ask Academician Tian."

……

At first, everyone maintained basic respect in the lecture hall, but as the awards ceremony progressed, the students started playing with their phones.

Many students also left their seats to go to the toilet and then never returned.

Some students quietly slipped to the back row and chatted in hushed tones.

There are quite a few people like Chen Hui who are engrossed in their studies. This is the CMO. Anyone who can come here is exceptionally talented and a master of academic excellence.

"The boss is the boss!"

"You looked so cool when you rejected the professor from Yanbei University!"

As soon as Shi Potian left, Li Zehan shifted his position and sat back down.

Before Li Zehan could even sit down properly, a slender hand grabbed his shoulder and pulled him aside. Then, a fragrant breeze wafted over, and a girl in a blue and white school uniform sat down in the spot where Shi Potian had just been sitting.

"Hello Chen Hui, could you explain the solution to question 6 of the CMO?"

The girl, with her bare face and bright, sparkling eyes, tilted her head and asked with eager curiosity. Her shoulder-length hair brushed against Chen Hui's hand, like a cherry blossom that had fallen by chance, bringing a faint fragrance of cherry blossoms.

"Why don't you ask the guy next to you? He got a perfect score too?"

Chen Hui withdrew his arm, rubbed his nose which was a little uncomfortable from the sudden fragrance, and looked at Li Zehan beside him.

But they found that the guy who had been jumping around was now looking down, not daring to look in their direction at all, looking like a quail.

He withdrew his gaze and looked at the girl's chest.

Shanghai High School!

Li Zehan should know this person.

The fact that Li Zehan acted this way suggests that this seemingly delicate young woman is not to be underestimated.

The girl had already written the sixth question on her draft paper. For people like them, memorizing the questions of an exam was not a difficult task.

The sixth question in this CMO is a number theory question. Number theory, as the jewel of theoretical mathematics, is very popular among question setters.

【求与数列an=2^n+3^n+6^n-1，n=1，2，3……每一项都互素的所有正整数】

"We can simply transform the original expression using Fermat's Little Theorem."

Chen Hui was too lazy to bother, so he raised his hand and wrote down the solution on the draft paper, wanting to get rid of the other party as soon as possible.

When n=1 and 2, we can calculate an=10 and 48 respectively. When n>3, according to Fermat's Little Theorem, we have...
2^(n-1)≡1 (mod n)

3^(n-1)≡1(mod n)
6^(n-1)≡1(mod n)
2，3，6的最小公倍数是6，所以在对上面三项分别乘以3，2，1，则有6*2^(n-2)+6*3^(n-2)+6*6^(n-2)≡6（mod n），所以有2^(n-2)+3^(n-2)+6^(n-2)-1≡0（mod n）。

In other words, for the sequence an, when n>3, there always exists an integer p that can divide an.

Since a1 and a2 are divisible by 2 and 3, the only positive integer that is coprime to every term of an can be 1. Q.E.D.

The girl stared at the draft paper for thirty seconds, her doubts disappearing, but new doubts quickly arose. "How did you come up with the idea of ​​using Fermat's Little Theorem to solve this problem? Can you explain your thought process to me?"

She understood the question in just thirty seconds, but in the exam room, she spent an hour and still couldn't solve it!
"?"

"You figured it out as soon as you saw the question?"

Chen Hui felt this was a bit tricky; the problem was similar to asking him to prove 1+1=2.

"Is not it?"

He glanced at Li Zehan, who was peeking up from the side.

Li Zehan shook his head vigorously, "Of course not."

"There are quite a few theorems in number theory: Euler's theorem, Fermat's little theorem, Wilson's theorem, and Huaxia's residue theorem. Just the four major theorems of elementary number theory require a lot of trial and error. There are also the quadratic reciprocity law, the simplified residue system modulo p, the elementary version of the prime number theorem, and theorems related to Fermat numbers..."

It took me some time to find the correct solution.

When math is mentioned, the chubby boy immediately becomes confident.

I've learned too many different things!
Chen Hui suddenly realized.

As a renowned school for competitive sports, Shanghai High School has extensive experience in cultivating students, so there won't be any gaps in knowledge caused by insufficient prior knowledge, resulting in students knowing theorems but not being able to apply them.

On the contrary, because I have too much knowledge in my head, I hate reading so much when I need it.

Ultimately, it comes down to not having studied things thoroughly enough.

Do you know the proof of Fermat's Little Theorem?

Chen Hui turned to look at the girl beside him, then turned to Li Zehan.

"Of course it's no problem!"

Li Zehan picked up a pen and began writing rapidly on the draft paper, and the girl did the same.

Anyone who can come to the CMO position must be able to prove Fermat's Little Theorem.

"anything else?"

Looking at the two who had written out a proof, Chen Hui asked.

"besides?"

Both of them looked at Chen Hui blankly.

Chen Hui remained silent, but picked up his pen and began to work out the problem again on the draft paper.

Both Li Zehan and the girl used the same dyeing method, indicating that they were taught by the same teacher.

Fermat's Little Theorem states that if p is a prime number and a is any integer relatively prime to p, then a^(p-1) ≡ 1 (mod p).
Proof: Let p be any integer coprime to a. Then p*a = a + 2a + ... + (p-1)a. Since p and a are coprime, a modulo p can only be 1 to p-1.

Assuming that the p-1 terms are congruent, the difference between the two terms modulo p is 0. Since p is an integer, the difference between the two terms must be equal to one of the p-3 terms, i.e. (q*a)≡0(mod p), which contradicts the given condition.

Therefore, the p-1 term is not congruent.

∴(1*2*……p-1)a^(p-1)≡(1*2*……p-1)(mod p)
∴a^(p-1)≡1(mod p).

"Can this still happen?"

Looking at the few simple lines of proof on Chen Hui's draft paper, and then at the two pages of proof they had written, Li Zehan and his companion began to question their own existence.

Simple and elegant!
In comparison, Chen Hui's proof seems to be more like mathematics.

But they still looked at Chen Hui with suspicion, wondering how this had anything to do with the previous topic.

"When you were learning Fermat's Little Theorem, you only focused on what it is and how to prove it, and then you went on to do practice problems, trying to master the knowledge point by doing practice problems, but you didn't delve into why it is!"

"So these formulas are like water without roots in your minds. Even if you memorize them, you won't be able to think of them when you need to use them. When learning mathematics, the important thing is not how many formulas and theorems you memorize, but whether you understand them."

"The proof process is crucial for thinking training. Mathematical induction, proof by contradiction, and other proof methods are not only manifestations of logical deduction, but also whetstones for cultivating 'mathematical intuition'."

If you've delved into the proof of Fermat's Little Theorem, you can often quickly identify congruence structures in problems through "number theory aesthetics," and like me, you can immediately tell that Fermat's Little Theorem should be applied to a problem.

Chen Hui didn't explain the concept of the complete residual system to them. The concept itself isn't important; understanding the concept is!
Li Zehan and the other man seemed to be deep in thought.

Your math level has increased from 75 to 76% (Level 2).

As soon as Chen Hui turned around, a barrage of comments popped up in front of him.

Chen Hui was somewhat surprised, but it was also within his expectations.

Being a teacher to others is a process of organizing what you have learned. If you can explain a knowledge point clearly to others, then you have truly mastered that knowledge point!

Discussions can spark inspiration, which is why so many accomplished mathematicians are willing to take on students; the teaching process is also a process of personal growth.

(End of this chapter)