I Am the President of the University

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

【Abstract: The congruence properties of a special type of recursive sequence are studied. Using these properties and elementary number theory methods and techniques such as quadratic remainder, 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). 】

[Keywords: Diophantine equation; elliptic curve...]

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

Number theory is a branch of pure mathematics that studies the properties of integers. Integers can be solutions to equations (Diophantine equations). Some analytical functions (such as the Riemann zeta function) include some properties of integers and prime numbers. Through these functions, you can also understand some number theory issues.

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

So how to solve this integer point problem?

Lu Zhou had a flash of inspiration and transformed the problem of integer points of elliptic curves into studying the number theory properties of recursive sequences, and finally used the elimination method to find all the 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 remainder and using certain techniques.

"In order to prove Theorem 1, we need to study the congruence properties of the recursion sequence. After calculation, we know that 649+180√13 is the Pel equation..." Lu Zhou murmured in a low voice while writing the paper.

The roommate on the side, Lao Wang, came forward from nowhere, bowed his waist slightly, looked at the computer screen curiously, and asked with a suspicious look on his face: "Elbow, are you doing your homework or something?"

"Write a paper." Lu Zhou replied without taking his eyes away from the computer.

"Do you want to submit a thesis or paper?" Lao Wang said with his eyes widened and stuttering.

"Of course I have to submit it, otherwise why would I write it?" Lu Zhou's hand movements paused slightly, and then the continuous sound of the keyboard sounded again.

"Submit! This is the legendary paper submission. It feels so awesome!" Lao Wang suddenly saw stars in his eyes. He said to a sophomore student who has never been exposed to thesis at all. He usually just writes homework courses. Thesis is completely different from this real thesis.

Maybe after four years, most undergraduates will only come into contact with this when they write their graduation thesis. \b

"Isn't this a mathematics paper?" Lao Wang glanced at it for a few times and then remembered that Lu Zhou was a person who had won the Higher Education Cup of the National College Student Mathematical Modeling Contest and had extraordinary talent in mathematics.

"Yes, it's math."

"That's awesome!"

Lao Wang said with some admiration, and then he leaned closer and read the title again, "The recursive sequence of integer points of elliptic curves... What are these, why can't I understand it at all."

Lao Wang's eyes suddenly blurred. Let alone the content, just the title of the paper had already confused him.

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

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

There is only envy, no jealousy or hatred.

Living together every day, he knew how hard Lu Zhou worked in his studies and how much sweat he put in. \b

The achievements we have made now are natural.

People who are talented and work so hard will be damned if they don't succeed.

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

The idea that suddenly popped into his head startled him.

But then, he felt a little moved.

"How about giving it a try?" he asked unconfidently.

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

A roommate next to him happened to go to the toilet and noticed something strange about Lao Wang. He couldn't help but joked: "Hey, why is Lao Wang still starting to study? Isn't it the end of the semester?"

"Go aside, don't disturb my self-study, please ask me to learn from Bawang in the future!" Lao Wang said with confidence.

ten minutes later.

Lao Wang's upper and lower eyelids began to fight, and he yawned as frequently as seven or eight times in a minute.

"Old Wang, am I the King of Pesticides? 4 is short of 1, so I'm waiting for you, are you coming?" shouted the roommate.

"No... I'm studying." Lao Wang said with a yawn.

"I really won't come, there is a girl!" The roommate chuckled.

"If you don't come, there is a golden house in the book, and a beautiful face in the book!" Lao Wang said firmly.

"Tch, you only have three minutes to warm up. If you can last for a whole day, I will obey you."

When the roommate saw no results, he just grabbed a savage and started.

Fifteen minutes later.

Lao Wang turned his head to the roommate he had invited before, hesitated for a moment, and asked: "A Chun, have you finished playing?"

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

"Ahem, Yan Ruyu doesn't seem to be at home today. She's studying. It's okay to be late. I'll play games with you first and then go study." After finishing speaking, Lao Wang threw away the book and pulled up a chair to sit next to his roommates. .

Lu Zhou couldn't help but shake his head when he saw this scene.

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

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

Lu Zhou was very firm in his beliefs. He wanted to see where his limits were and how vast the world was.

…

Guanghua University Pulan Cafe.

At 8 o'clock in the evening, it was already dark outside. Fortunately, there were street lights on both sides of the road, which made it not so lonely.

"Principal. "X3

Three voices sounded one after another. Chen Hao nodded slightly and said with a smile: "We have a group meeting late at night. Is it not causing any trouble to you?"

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

It was late at night and the principal hadn't gone back yet. He stayed at the school. How many other universities could there be?

"Fortunately, it's good to have a fulfilling life and be busy."

Just then the coffee order arrived.

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

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."

After hearing the last word, the three of them looked at each other, but Song Qingtong was the first to reach out and choose a cup of coffee without ceremony.

After Wei Shu finished his selection, no one chose the American style.

A few people drank coffee and started chatting.

"Tonight, I invite you to come out for a group meeting. The main thing is that the curriculum guidance document for the experimental class has also come out. Do you have any opinions after reading it?" Chen Hao took a sip of coffee and asked with a smile.

The three of them looked at me, and I looked at you. The last one to speak was Li Qiubai.

"Principal, after reading the documents, I found that the professional courses and elective courses are fine, but is the general education too much?" Li Qiubai said a little depressed.

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

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

"Fourteen periods is okay. If I weren't afraid of giving you too many lessons, there might be more than 14 periods." Chen Hao said with a chuckle.

Li Qiubai:...

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

"I selected all three of you. Of course, you are all potential computer scientists." Chen Hao said, changing the subject. "General education courses are still very useful. You can combine your general education courses with your majors." Connecting the fields is a process of checking for omissions and filling them, and it is also a process of pursuing higher standards.”

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

Originally, he was studying computers hard, but now 14 general education classes suddenly 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 exceptions at will." Chen Hao shook his head.

The reason why he formulated this general education course is to cultivate students' all-round abilities and expand their thinking ability, so that they can stand in more perspectives and understand things in multiple ways when thinking about things.

"Okay... then I have no problem." Li Qiubai sighed silently.

Among the remaining two people, Wei Shu was very quiet. Song Qingtong looked at it and saw that he was the only one left.

After thinking for a while, Song Qingtong asked: "Principal, it's already early November. If we re-select courses, will we be out of time?"

As soon as these words came out, Li Qiubai's eyes shone with light, and it was obvious that the words had reached his heart.

"This is not a problem. Regarding the courses for students in the experimental class, after you choose your major, after the statistics are completed, we will open separate courses for you based on the courses the students want to take." Chen Hao put down his coffee gently and looked at the three of them.

"Is this like this? What if there are students who want to take a course that the school does not offer?" Song Qingtong suddenly thought of this and asked curiously.

"If the school does not 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 of them looked at each other. No matter what, they could at least feel the sincerity of the school.

Not to mention that you can create separate courses for students in a class, but you can also create new courses for students, which makes people speechless.

"The Q\u0026A is over. I also have some opinions on the arrangement of your courses. If you think anything is inappropriate, we can discuss it together." With that, Chen Hao laid out the courses for Li Qiubai and the others in the past two days.

"Freshman year: first semester (General History 5, elective 1), next semester (General History 5, elective 1)

Sophomore year: First semester (Major 3, Elective 2, General Studies 2), Next semester (Major 3, Elective 2, General Studies 2)

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

"In this way, you can complete 36 standard courses in 3 years, and the credits are more than enough. I don't recommend that you take too many courses. You can't take more than 7 classes in a semester." Chen Hao reminded seriously, "Learning lies in mastering and essence. It’s not about too much, I hope you all can learn the course thoroughly.”

We discussed this course arrangement for a while.

The group meeting started at 8pm and ended at around 9pm.

…

The next morning, Beijing.

At Beijing University of Aeronautics and Astronautics, an angry roar sounded from the secretary's office.

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

"Professor Fang, I disagree!"

ps: I haven’t finished writing it yet. I’ll make up the remaining bits at noon tomorrow. I’m too sleepy to write until four o’clock, and the efficiency is not good. I'm sorry everyone, alas, I blame my inefficiency.

\b

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