I haven’t been exposed to number theory since junior high school. I was still a freshman in high school. Most students couldn’t understand what the instructor was writing on the blackboard. But just because they couldn’t understand, they didn’t realize how good they were. They could solve this kind of problem in just a few steps. No matter what they thought,

Operations that only gods can do.

Obviously very complex problems can be solved using simple processes. This is the power of human thinking, it is so incredible.

Of course, it is also because these students have never learned Euler's theorem and do not know how to calculate the answer. They think that Li Xuan and the coach are gods who can calculate this kind of problem. After learning it, they will find it very simple.

Okay, the so-called immortals just learned a little early.

Then after going through this question, Li Xuan discovered that most of the students around him were just getting started with number theory. It turned out that many students in the mathematics competition group had solved this question. They must have read books related to number theory, but seeing these students in the Chaoyang Cup

His performance cannot be regarded as a master of math theory.

Speaking of number theory masters, he naturally thought of Ouyang Zhe.

I heard that the national training team player Ouyang Zhe I met last time is a genius who is very good at number theory. He can solve the number theory problems of the CMO League. Such basic problems of Euler's theorem can be solved by verbal arithmetic.

In China, high school students are always very good at geometry and algebra, but not very good at number theory and combinations. Geniuses in number theory are very popular, but geniuses in combination are even rarer.

Sitting in the first row, Qiao Siling found that she couldn't understand what the coach was writing. She raised her hand and humbly asked for advice: "Coach, what is Euler's theorem?"

Lin Xuerui smiled. Qiao Siling's curious eyes reminded her of when she first learned number theory. As she spoke, she wrote on the podium:

"There are Euler's theorem in number theory and geometry. Euler's theorem in number theory is: if n and a are positive integers, and n and a are relatively prime, a^φ(n)≡ 1 (mod n)."

"Here φ(n) is called Euler's function, which is a number of positive integers that are less than n and relatively prime to n.

"For example, φ(8)=4, because 1, 3, 5, and 7 have 4 positive integers, which are relatively prime to 8."

"So, generally, 3^4 ≡1 (mod 8)"

"This question is to find the remainder of 3^83 divided by 100."

"According to Euler's theorem, 3^φ(100)≡ 1(mod 100).φ(100)=40,1,3,7,9...a total of 40 numbers and 100 are relatively prime."

"3^40 ≡ 1 (mod 100)."

"In other words, 3^80 ≡ 1 (mod 100 )."

"3^83≡3^80x3^3≡1x3^3≡27(mod100)."

…

Qiao Siling pursed her lips and silently copied down the writing on the blackboard.

Li Xuan didn't start writing. He had taught himself Euler's theorem a long time ago and could write it with his eyes closed. But when he saw the equations on the blackboard, he discovered one thing. When these real masters write mathematical problems, if they don't jump

The steps are really clear, simple and easy to understand, and easy to accept.

Yan Pengfei also copied this example. He has never been good at logical thinking and considers himself a novice in mathematics. He has just come into contact with elementary number theory. When he first saw Euler's theorem, he still didn't understand the meaning of Euler's function, which made him feel a little hurt.

, tell me this is elementary number theory?

If he can't learn elementary number theory, then what is he? Thinking about this immediately makes him feel very stressed.

Of course, it seems okay now and I can understand it for the time being.

In class, Lin Xuerui looked at her classmates. The content of Euler's theorem was not deep. She deliberately slowed down her pace in the first class. For the time being, most of her classmates could keep up with her pace.

I definitely can't procrastinate like this in the future. After all, we are all competitive students, so we will definitely speed up the pace.

Lin Xuerui wanted to hear what her classmates thought: "After completing this question, what do you think?"

Some students at the bottom started blowing as usual:

"Coach, I don't have any ideas. I just think Mr. Euler is awesome."

"Kexi, Gauss and other big guys are so awesome that they are trembling."

"No, no, I think why is this guy Euler so annoying..."

After complaining for a while, the classmates quickly calmed down and listened to what the coach said.

Lin Xuerui smiled, not hearing what she heard, and said: "There is a general formula for solving the Euler function in this theorem. If you encounter something you don't understand, just study it and ask everyone to figure it out for themselves after class."

"What are the four major theorems of elementary number theory? I will explain them one after another in the future. Students who are interested can also open "Elementary Number Theory" and read it first. There are many number theory books in the library. You can borrow them."

"Today I will first talk about what you should know to learn number theory."

"Mathematics has two main branches, one is algebra, which studies quantitative relationships and is dominated by orderly thinking, and the other is geometry, which studies spatial forms and is dominated by visual thinking."

"But no one studies classical geometry anymore because all the problems have been solved. And number theory is the purest branch of mathematics. It was originally called arithmetic. There are still some world-wide problems waiting for you to solve."

"Number theory is the study of the properties of integers and is called the queen of mathematics. Gauss called number theory the crown of mathematics. This is the most profound field of mathematics."

"In number theory, many theorems seem simple, but are extremely difficult to prove. For example, the well-known Goldbach's conjecture: Can every even number greater than 2 be written as the sum of two prime numbers? It has not yet been solved. This is

It is a field that only geniuses dare to enter, and many talented mathematicians have broken their heads in this field."

"Number theory comes from life. In the process of real life, human beings refined arithmetic and gave birth to the concepts of integers and the four arithmetic operations of addition, subtraction, multiplication and division. After having the concept of multiplication, humans discovered that in integers, all numbers can be synthesized with prime numbers, so integers

The basic elements are prime numbers, also called prime numbers, 2, 3, 5, 7, 11, 13... There are many theories in number theory, all of which are studying prime numbers."

"Number theory was once regarded as the most useless mathematical knowledge. Pure mathematics has no use in life and production. But now, number theory has become one of the foundations of modern cryptography."

"Before World War II, the theory of relativity, like number theory, was called innocent knowledge, which meant that it was of little use to the war and would be of no use to the future. Then Einstein came up with the atomic bomb, which silenced those who said this. The code for the evolution of number theory

Knowledge makes those who want to crack the code doubt life, which changed World War II to a certain extent.

"The later Gulf War was even called a mathematical war."

"What is the relationship between number theory and cryptography? Ciphers require asymmetry, and prime numbers just meet this condition: it is easy to multiply two prime numbers, but knowing the product, it is difficult to decompose them into prime numbers. Even the complex RSA password cannot be cracked by computers. Everyone needs to

Thanks to the unbreakable super-large prime number in the bank card password.”

"The largest prime number that humans can find so far is the Mersenne prime number. How big a prime number can be found can test the computer level of a country."

…

Lin Xuerui talked about a lot of common sense about number theory, and also talked about the famous conjectures in number theory, which have not been broken so far.

The students' hearts agitated when they heard it, and they aroused great ambitions, wanting to solve several major conjectures in number theory.

When it comes to number theory, the most famous ones are of course several conjectures, which are the crown jewels of mathematics.

●Goldbach’s conjecture: Can every even number greater than 2 be written as the sum of two prime numbers?

●Twin prime number conjecture: Twin prime numbers are pairs of prime numbers with a difference of 2, such as 11 and 13. Is there an infinite number of twin prime numbers?

●Is there an infinite number of prime numbers in the Fibonacci sequence?

●Is there an infinite number of Mersenne primes? (Referring to positive integers in the shape of 2^p-1 that are prime numbers, called Mersenne primes)

●Fermat's conjecture has now been proven, and Fermat's conjecture has become Fermat's last theorem.

●Riemann Hypothesis.

Li Xuan is also fascinated by these world-wide problems, especially the Riemann Hypothesis, which tests people's brain power and imagination.

The Riemann Hypothesis, the most important and most anticipated problem in mathematics today, holds that all prime numbers can be expressed as a function.

In fact, hundreds of years ago, mathematicians including Euler began to work hard to find the general formula of prime numbers. However, later generations finally found the general formula of prime numbers, but they all had great limitations.

In the Riemann Hypothesis, the functions mentioned have the most universal significance.

Riemann was far ahead of many other mathematicians of his generation. He published a few pages of paper at that time, revealing the secrets of the distribution of prime numbers. However, the text was too concise and the proof was omitted.

For him, it was simple, provable and obvious. However, this proof was omitted, which confused later generations of mathematicians and took decades of hard work to complete it. Even some of his conclusions, such as the Riemann Hypothesis, were proven.

It's still blank now.

A similar example is when Fermat wrote down Fermat's conjecture and said: "I found a really excellent proof, but unfortunately the margins are too narrow to write it down." Basically, Fermat regarded the wrong proof as the correct proof.

However, unlike Fermat, Riemann left behind research manuscripts that proved many of his conclusions. It was the Riemann Hypothesis that proved that he did not have time to do it. He wanted to study Riemannian geometry and other other work.

The paper says: This proof work is left to you.

Then there is no more.

Up to now, mathematicians are still in a state of hard thinking. They have tried to prove the Riemann Hypothesis countless times and suffered numerous blows.

Nowadays, many papers are correctly established based on the Riemann Hypothesis.

As long as someone can prove the Riemann Hypothesis, hundreds of conclusions can be upgraded to theorems. From this aspect, Goldbach's Hypothesis is far less important than the Riemann Hypothesis.

After this get out of class, after Lin Xuerui left, many students gathered together and discussed excitedly:

"I feel so passionate about mathematics. I will apply to the Department of Mathematics in the future to see if I can prove Goldbach's conjecture."

"I'm very surprised. Are these conjectures so difficult to prove? I think the twin prime conjecture seems quite simple."

"It looks simple, but proves complicated."

"I can understand the other conjectures, but I can't understand what the Riemann Hypothesis means. Does anyone understand what the Riemann Hypothesis is talking about?"

Liang Zhihui also shook his head after hearing this, frowned and said: "I have looked for college books to read. I can understand the calculus of Newton and Leibniz, and I know how to do the exercises. I can also do the linear algebra of Arthur Kelly. After class exercises

It can be done, but for Riemannian geometry, to be honest, I read that textbook three times and didn’t understand what he was talking about at all, and I didn’t know how to do any of the assignments.”

Yan Pengfei didn't understand calculus, and directly suggested: "Wisdom, listen to my advice, don't look at Riemannian geometry, you will go crazy, there is something wrong with Riemann's brain, even Einstein himself may not be able to figure it out"

With this kind of mathematics, only a powerful person like Einstein can understand what Riemann is talking about."

[ps: To explain, the historical liberal arts figures in this time and space are fictitious, and the names of important figures in the sciences coincide with our time and space. This is to facilitate everyone’s understanding. In fact, they should all be fictitious, or the background should simply be in our time and space, but if written this way, liberal arts will appear later.
Chapter completed!