A genius? I just love studying.

Chapter 33: Inviting God (Updated at 3 AM)

Chengdu Mathematical Society,
Vice President's Office, 3rd Floor, Administration Building,

"Teacher, are the questions this year too difficult?"

A young man in his twenties was sitting on the sofa with a cup of coffee in his hand, talking to a middle-aged man in his fifties sitting opposite him.

The two had just finished discussing an academic issue and were both a little tired, so they decided to sit down and rest.

Ma Jingtang rubbed his temples and shook his head, "It's better to be difficult!"

He sighed in his heart that time is unforgiving. When he was young, his mind was so sharp, but now he felt a little overwhelmed after just an hour of discussion. He has passed the golden age of mathematicians from 30 to 50 years old to achieve great results.

"Last year, the exams were simple, but quite a few students scored full marks. It looked like a flourishing scene, but what happened?"

"In the end, for a city as big as Chengdu, there wasn't even a single contestant participating in the IMO!"

Speaking of this topic, Ma Jingtang also happened to have something to say. Over the past year, he has been teased by the old guys from other provincial mathematics societies, so this year he deliberately greeted them and made the questions more difficult.

Yang Han smiled, thinking of the scene where the teacher was teased.

Chengdu is considered a strong province in mathematics competition, but last year it did not even have any contestants enter the IMO. It is no exaggeration to say that it was a waste of talent, and it is indeed a great shame.

"The teacher actually made that question the final question,"

However, Yang Han still disagreed with the teacher's approach. "That question is quite difficult, even harder than some CMO questions. I'm afraid there won't be a single perfect score this year."

"It doesn't matter whether you get a full score in the provincial competition or not. What matters is that you can enter the IMO and win a gold medal at the IMO!"

Ma Jingtang waved his hand nonchalantly, "Besides, that last question was set by that person. If we can solve it, we might even catch his eye, which would actually be a good thing for those little guys."

"That's true."

Yang Han nodded in agreement.

"I just don't know if I can get a perfect score this year."

……

Li Bin walked down from the podium, came to Chen Hui's side, and looked at Chen Hui's test paper.

In such a short time, the blank space of the first big question was already filled with words, and the little guy was already writing the solution process in the blank space of the second question.

"So fast?"

Now Li Bin will not think that Chen Hui is writing nonsense.

You can just write some numbers for fill-in-the-blank questions, but the big questions require a process. If you don’t know how, you can’t even write them randomly.

Regardless of whether it is right or wrong, at least it shows that the child has good mathematical literacy.

"It seems that Chengdu has two very good seedlings this year!"

Li Bin was somewhat happy. Combining everyone's reactions, he knew that it was not because the questions this year were too easy, but because there were two geniuses among the candidates.

Although he does odd jobs in the mathematics society every day, he likes it here very much. If the contestants from Chengdu Province achieve good results, the Chengdu Mathematical Society will also be proud of it.

After taking a quick look at the problem-solving process and confirming that Chen Hui had no problem answering the first big question.

Li Bin took another step and walked towards the middle classmate on the right.

Along the way, most of the other students were still doing fill-in-the-blank questions 7 and 8. Of course, some students selectively gave up question 8 and started looking at the big questions.

And now half an hour has passed since the exam started!

Even though there were more than two hours left in the exam, Li Bin knew that the next four questions would be the hard part and would not be easy to tackle in two and a half hours.

Well, for the average person, of course.

For example, the person in front of you has also finished the first question and started to review the second question.

His speed is just a little slower than the guy in the first row from Rongcheng No. 2 Middle School.

Time passed quickly. After finishing the second big question and looking at the third one, Deng Leyan felt very tired.

Last year, when he was still in the third grade of junior high school, he participated in the provincial competition and even made it to the national finals. Of course, he only won the bronze medal in the end.

He got full marks in the provincial competition last year, so he didn't take this exam seriously at all. Only he knew how much he had grown in this year.

A genius’ year is different from an ordinary person’s year.

But obviously, this year's questions were much more difficult than last year's. Even a year later, he still felt very strenuous doing them, which made him feel the same sluggishness as when he was doing the CMO questions last year.

Especially that annoying invigilator, who kept hanging around, making him very annoyed. He wanted to find a chair for him and push him onto it.

Chen Hui was not affected at all. He had long been accustomed to studying in any environment. Once he concentrated on doing something, it was difficult for the outside world to influence him.

After quickly finishing the second plane geometry proof question, Chen Hui looked at the third big question.

[Let A and B be positive integers, and S be a set of positive integers with the following properties:

(1) For any non-negative integer k, A^k∈S;
(2) If a positive integer n∈S, then every positive divisor of n belongs to S;
(3) If m, n∈S, and m, n are relatively prime, then mn∈S;
(4) If n∈S, then An+B∈S. Proof: All positive integers coprime to B belong to S.

"Number Theory?"

Chen Hui frowned.

He was not good at number theory.

But he did not give up on himself. He converted the known properties and conclusions into the language of number theory and easily found his goal.

All we need to do is construct a number that is coprime with B, assume it is p, and then prove that p∈S.

According to Property 3, if pi and pj are mutually prime, then pi·pj∈S. According to the prime number decomposition theorem, every positive integer greater than 1 can be uniquely expressed as the product of several prime numbers, and the powers of these prime numbers are unique.

所以P可以写成p1^α1·p2^α2···pm^αm，其中p1到pm均为素数。

In other words, we only need to prove pi^k∈S (k is any non-negative integer) to prove P∈S.

Soon, Chen Hui had an idea. According to the question, if pi is divisible by A, then according to properties 1 and 2, it can be easily concluded that pi^k∈S.

But what if pi cannot divide A?
If it cannot be divided evenly, it means that pi and A are also coprime. At the same time, because Pi is the prime factorization number of P, P and B are coprime, then pi and B are also coprime.

Properties 123 have already been used, so property 4 will definitely be used next.

An+B∈S
How should this property be utilized?
Chen Hui racked his brains but was at a loss. This was the second time he encountered this situation after his insight improved. It reminded him of the question Zhang Anguo gave him in the mathematics competition team. He was in the same situation at that time.

Later he learned that there was a conventional solution to Zhang Anguo's problem, but he didn't know it at the time.

Therefore, this problem must have a solution, or a formula or theorem that I have not thought of!
However, Chen Hui had not studied number theory in depth, and there was no system of number theory in his mind. For a moment, he didn't know where to look for such a solution or formula theorem.

Solutions, formulas and theorems, to put it bluntly, are the ladders built by our predecessors.

Newton once said that he was able to achieve such success only by standing on the shoulders of giants.

Therefore, the solution must of course be sought from our predecessors!

Chen Hui's mind was working rapidly and he began brainstorming.

There are many mathematicians who are good at number theory, but Chen Hui currently only knows a few of them: Fermat, Euler, and Gauss.

Fermat's research is very imaginative, including Fermat's size theorem, amicable numbers, and prime number distribution. These theorems play an important role in number theory.

But he only played high-end games throughout his life, and always asked future generations to prove his solutions. It should not be Fermat's turn to solve high school students' problems, right?

Gauss mainly studied algebraic number theory, such as the quadratic reciprocity law and the arithmetic-geometric mean, which are obviously inconsistent with the tone of this question.

So, is it Euler?

After some analysis, Chen Hui set his sights on the King of Mathematics.

He was somewhat excited, as he actually knew more about Ola than the other two.

This is because I listened to Teacher An’s advice when I was studying Euler integrals.

Otherwise he would be groping in the dark.

Try to save a dying horse as much as possible. The choice without choice is the best choice.

Chen Hui began to recall the theorems on number theory proposed by Euler throughout his life.

He was not a stubborn person. If he could not find a solution to the problem from Euler, he would give up on it, go back to study number theory, and come back next year.

Euler published more than 1500 papers in his lifetime, and the theorems, formulas and theories he proposed are as vast as the sea of ​​stars.

The improved memory helped Chen Hui a lot. With the help of strong insight, although he only read Euler's life once, he remembered the important formulas and theorems proposed by Euler very clearly.

When we think of Euler, we naturally think of his famous Euler theorem in the field of number theory.

Euler's theorem!

Soon, a dazzling light appeared in front of Chen Hui's eyes.

found it!
He found it!

The key to solving the problem is indeed hidden in Euler!
Euler's theorem:

If a and n are positive integers and a and n are coprime (i.e., their greatest common divisor is 1), then the result of raising a to the power of φ(n) modulo n is 1, i.e., aφ(n)≡1(modn)
Chen Hui fell into an unprecedented state of excitement, and countless ideas emerged in his brain like springs.

[According to Euler's theorem, A^aφ(pi^k)·n+B≡n+b(modpi^k), then let a0=1, an=A^aφ(pi^k)·A^n+B, then an≡A^n+B(modpi^k), and because (pi, A)=1, (pi, B)=1, so when n is from 0 to pi^k, an can take the complete remainder system of pi^k, at this time there must be at=t·pi^k∈S, so pi^k∈S!

In summary……】

Proof completed!

This chapter is nearly three thousand words. Thank you all for your support. Without further ado, I will start writing, writing like crazy.

At the beginning of the month, I will ask for the guaranteed monthly ticket and the new book list. Thank you.

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(End of this chapter)