I Am the President of the University

"I'm the president of the university ()"!

[Recursive sequence of elliptic curve integer points and quadratic residual method solution]

【Abstract】The congruence property of a special type of recursive sequence is studied. Using these properties and the methods and techniques of elementary number theory such as quadratic residues, it is proved that all integer points of the elliptic curve y2=x3-30x+133 proposed by Zagier are (x, y)=(-7, 0), (-3, ±14), (2, ±9), (6, ±13), (5143326, ±11664498677). 】

[Key words: Diophantine equation; elliptic curve...]

Lu Zhou tapped the keyboard steadily with both hands, and saw more and more words on the computer screen, and soon filled the first job in the document.

Number theory is one of the branches of pure mathematics, which mainly studies the properties of integers. Integers can be solutions to equations (Diophantine equations). Some analytic functions (like the Riemann zeta function) include the properties of some integers and prime numbers, and through these functions, some problems in number theory can also be understood.

The source of this mathematical paper is that when he was studying number theory before, he noticed the integer point problem of elliptic curves. Due to the diversity of curve types, there is no unified method or algorithm that can solve any given within a finite number of steps. Integer points of the elliptic curve.

So how to solve this integer point problem?

Lu Zhou's head flashed, and he transformed the integer point problem of elliptic curves into the number-theoretic properties of recursive sequences, and finally used the elimination method to find all integer points.

Although the recursive sequence method is an elementary method, it can solve some difficult Diophantine equation solving problems by combining it with the quadratic residue and using certain skills.

"In order to prove Theorem 1, we need to study the congruence property of the recursive sequence. After calculation, we know that 649+180√13 is the Pel equation..." Lu Zhou whispered while writing the paper.

The roommate Lao Wang on the side did not know where he came forward, bowed his waist slightly, looked at the computer screen with great curiosity, and asked suspiciously, "Elbow, are you doing your homework or what?"

"Write a paper." Lu Zhou replied without looking away from the computer.

"Are the thesis and thesis to be submitted?" Pharaoh said with wide eyes and a little stumbling.

"Of course I want to submit it, otherwise why would I write it?" Lu Zhou paused for a moment, and then the sound of the keyboard sounded again.

"Submit! This is a legendary paper submission,

What a great feeling! "Pharaoh's eyes flashed with gold stars, and he always said to a sophomore student that he had never been exposed to the thesis, which is usually a course paper for homework, which is completely different from this real paper.

Maybe most undergraduates come into four years, and only come into contact with this when they write their graduation thesis.

"Isn't this a math paper?" Old Wang took a few glances, and then thought that Lu Zhou had won the Higher Education Club Cup of the National Undergraduate Mathematical Contest in Modeling, and he was very talented in mathematics.

"Yes, it's mathematics."

"It's too good!"

Pharaoh said with some admiration, then leaned in and stammered over the topic, "The recursive sequence of integer points on an elliptic curve... What is this, why can't I understand it at all."

Pharaoh's eyes suddenly blurred, let alone the content, just the title of a paper has already made him stunned.

"It's normal to not understand." Lu Zhou said casually.

Old Wang got up and stood behind Lu Zhou, looking at Lu Zhou's back with envy.

There is only envy, no envy or hatred.

Living together every day, he knows how hard Lu Zhou is studying and how much sweat he has put in.

It is only natural to have achieved success now.

Talented and so hardworking, such a person will be damned if he doesn't succeed.

"I really want to be as motivated as Lu Zhou!"

The idea that suddenly popped into his mind startled him.

But immediately, he was moved.

"Want to try it?" he asked incredulously.

Thinking about it, Lao Wang returned to his seat, took out the book and began to review.

A roommate next to him just went to the toilet and noticed Lao Wang's abnormality. He couldn't help but tease: "Yo, why is Lao Wang still learning, isn't it the end of the term?"

"Let's go, don't disturb my self-study, please ask me to learn Bawang in the future!" Old Wang said confidently.

ten minutes later.

Lao Wang started to fight up and down his eyelids, and the frequency of yawning has reached seven or eight times in one minute.

"Pharaoh, am I the king of pesticides? If one of 4 is missing, I'll be waiting for you. Will you come?" the roommate shouted.

"Don't fight... I'm studying." Old Wang said with a yawn.

"It's really not coming, there is a sister paper!" The roommate smiled.

"No, the book has its own golden house, and the book has its own Yan Ruyu!" Old Wang said firmly.

"Cut, you are only hot for three minutes. If you can last for a day, I will serve you."

After seeing no results, the roommate directly pulled a savage and started.

fifteen minutes later.

Old Wang turned his head to the roommate he had invited before, hesitated, and asked, "A Chun, are you finished?"

"What are you doing, aren't you looking for your Yan Ruyu?"

"Cough cough, Yan Ruyu doesn't seem to be at home today. I'm studying, it's not a big deal later. I'll play games with you before I go to study." After saying that, Lao Wang threw away the book and pulled a chair to sit next to his roommate. .

Seeing this scene, Lu Zhou couldn't help shaking his head.

As for people, most of them are three minutes of enthusiasm, or fake diligence.

Learning is not an easy task, and it is a long-term task that must be persevered, but most people give up halfway.

Lu Zhou is very firm in his beliefs. He wants to see where his limits are and how vast the world is.

...

Pulan Cafe at Guanghua University.

At 8 o'clock in the evening, it was already dark outside. Fortunately, the street lights on both sides of the road were on, so it didn't seem so lonely.

"Principal." X3

The three voices sounded one after another, Chen Hao nodded slightly, and said with a smile: "Have the group meeting at night, didn't it cause you any trouble?"

"No, but principal, you are too dedicated..." Li Qiubai said with emotion.

The principal of the evening hasn't gone back yet, so he stayed at the school, and there are a few universities in other people's homes.

"Fortunately, life is full, and it's good to be busy."

Just in time the coffee came.

The waiter put down the four cups of coffee and politely said, "Hello, your four cups of coffee, please enjoy it slowly."

Looking at the coffee on the table, Chen Hao waved his hand and said, "I don't know what you like to drink. Anyway, I ordered American, cappuccino, latte, and mocha. You're welcome, just take it."

Hearing the end, the three looked at each other, but Song Qingtong was the first to reach out and choose a cup of coffee.

After Wei Shu finished the selection, it turned out that there was only one American style left.

Several people drank coffee and started chatting.

"Tonight, I asked you to come out for a group meeting, mainly because the course guidance documents for the experimental class have also come out. Do you have any comments after reading it?" Chen Hao took a sip of coffee and asked with a smile.

The three of you look at me, I look at you, and Li Qiubai was the last to speak.

"Principal, after I read the documents, the professional courses and elective courses are fine, but isn't this general education too much?" Li Qiubai said a little depressedly.

There are 12 professional courses and 10 elective courses, but as a result, there are 14 general education courses.

This is very wrong. How can there be more general education courses than professional courses?

"14 lessons is okay, if it wasn't for the fear of giving you too many lessons, there might be more than 14 lessons." Chen Hao said with a chuckle.

Li Qiubai:  …

"Principal, can I take a little less of this general education course? I'm only interested in computers, as you know." Li Qiubai looked at Chen Hao and said in a consultative tone.

"I picked all three of you out, and of course all of them have potential in the main computer." Chen Hao said, changing the subject, "General education courses are still very useful, you can combine your general education courses with your own majors. Linking up the fields is a process of checking and filling gaps, and it is also a process of pursuing a higher level.”

"Principal, is there really no way to change it?" Li Qiubai smiled wryly.

Originally, he studied computer hard, but now suddenly 14 general education classes have popped up, making Li Qiubai's plan unable to keep up with the changes.

"Of course not, this is a set rule, you can't make an exception at will." Chen Hao shook his head.

The reason why he formulated this general education course is to cultivate students' all-round ability, and to expand their thinking ability. When thinking about things, they can stand from one angle and have multiple understandings.

"Okay...then I'm fine." Li Qiubai sighed silently.

With the remaining two, Wei Shu was very quiet, Song Qingtong looked at it, and there was only himself left.

After thinking for a while, Song Qingtong asked, "Principal, it's already early November. If you choose another course, will it be too late?"

As soon as these words came out, Li Qiubai's eyes also shone with a ray of light, and it was obvious that it went into his heart.

"This is not a problem. Regarding the curriculum of the students in the experimental class, when you choose a major, after the statistics are completed, we will open a separate course for you according to the courses the students want to take." Chen Hao put down the coffee lightly, looked at the three people and said.

"Is it like this? What if the school doesn't have the course that a classmate wants to choose?" Song Qingtong suddenly thought of this and asked curiously.

"If the school doesn't have this course, then recruit people! The experimental class is a reform of the school's system, so it will attach great importance to the training of the experimental class. You can rest assured." Chen Hao explained.

The three looked at each other, no matter what, at least they could feel the sincerity of the school.

Not to mention that you can take courses alone for the students of a class, and because students create new courses, people have nothing to say.

"The Q\u0026A is also over. I also have some opinions on the arrangement of your courses. If you think it is inappropriate, you can communicate with them." After speaking, Chen Hao formulated the courses for the three of Li Qiubai in the past two days.

"Freshman year: first semester (general history 5, elective 1), next semester (general education 5, elective 1)

Sophomore year: first semester (major 3, elective 2, general education 2), next semester (major 3, elective 2, general education 2)

Junior year: first semester (major 3, elective 2), next semester (major 3, elective 2)”

"In this way, 36 courses are just completed in 3 years, and the credits are more than enough. I don't recommend that you take too many courses, no more than 7 courses in a semester." Chen Hao reminded earnestly, "Learning lies in mastery, in essence, and in It's not that there are too many, I hope you all can study the course thoroughly."

The course arrangement was discussed for a while.

The group meeting started at 8:00 pm and didn't end until after 9:00 pm.

...

The next morning, the capital.

Beihang University, an angry shout sounded from the secretary's office.

The sound was so loud that it even attracted the attention of many passing school employees.

"Professor Fang, I disagree!"

ps: I haven't finished it yet. I'll make up the remaining one at noon tomorrow. It's too sleepy to write until four o'clock, and the efficiency is not good. I'm sorry everyone, alas, blame me for not being efficient.

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