Chapter 28 Carmichael Number Problem [Seeking votes]

"There are some number theory questions that just happen to match Carmichael numbers, which makes it quite interesting."

Xu Yuan looked at the formula he had written on the draft paper and pondered in his heart with great interest.

He had learned about the so-called Carmichael number when he read the materials.

Its definition is an exact number n. For all integers b that are coprime with n, b^nb is a multiple of n, so n must be a Carmichael number.

In addition, according to Fermat's little theorem, all prime numbers have this property.

Therefore, Carmichael numbers are also called pseudo-prime numbers.

In 1994, three overseas mathematicians proved that there are infinite Carmichael numbers and published their paper in the Annals of Mathematics.

But he encountered difficulties when he continued to try to prove the intervals between these infinite number of Carmichael numbers.

This issue remains unresolved to date.

There are also many people in the mathematics community who study Carmichael numbers, but unfortunately there has not been much progress.

I happened to have some free time right now, and I happened to come across Carmichael numbers, so I thought about trying to study and calculate them.

After all, only by challenging these difficult math problems can we constantly stimulate our thinking and discover our talents.

that's it.

As time passes by.

Xu Yuan's brows were rarely furrowed, as if he had encountered an obstacle.

He didn't stop writing and finish his calculations until there were five minutes left before the deadline.

I secretly estimated in my heart.

"It's a problem that even mathematicians find difficult to solve. It's really hard to solve it."

It took more than an hour, and the progress bar generated by the calculation process was still in single digits, which shows the complexity of this problem.

-

Task: Carmichael number proof
Subject: Mathematics

Progress: 8%
Result: Not completed
"At least the progress bar is still on an upward trend. It hasn't dropped, which means my current idea is feasible."

Coming out of the deep learning state, Xu Yuan muttered to himself as he looked at the panel that was disappearing in front of him. Just as he was thinking about having a chance to ask Qi Tongxiang for advice, he heard the bell for handing in the paper.

……

After handing in the paper and walking out of the examination room, Lin Haoming quickly followed.

The two of them went together to take a car back to the hotel.

On the way, no one, including other people, took the initiative to mention the exam situation until we met up with the team leader.

"How was your morning, Xu Yuan? How difficult were the exam questions this year?"

Team leader Zhang Yuhui, who had been waiting anxiously for the whole morning, hurried over to the team members when he saw them getting off the car. He looked at Xu Yuan and asked.

Xu Yuan said without hesitation: "It's within the normal range. The difficulty is not particularly high."

"That's good, let's go eat first." Zhang Yuhui shouted with relief.

But he didn't notice the reactions of other people around him, especially Lin Haoming and Li Mingliang who looked at each other.

Hesitant to speak.

Finally, he followed the team silently.

They racked their brains while doing the questions in the morning, just to squeeze out as many ideas as possible into their minds.

The difficulty may not be abnormal, but it is definitely not as easy as Xu Yuan said.

Obviously, Xu Yuan's difficulty is not limited to himself.

However, the first exam is over, and there is no point in discussing the difficulty. The most important thing is how to face tomorrow's exam.

The afternoon is the time for exam preparation, and candidates can stay in the hotel and communicate on their own.

After dinner, Zhang Yuhui asked Xu Yuan to explain his problem-solving process, so as to help others sort out their thoughts. By mastering the style and characteristics of this type of questions and the methods of solving them, it might be easier to answer questions tomorrow.

"These three questions this year are all classic. In addition to the trigonometric function sets and other topics involved in the previous part, the final number theory question is still about Carmichael numbers."

"If you want to accurately complete the proof, the key is to find the idea." "These are my ideas and detailed proof process. You can take a look and compare them first."

In the hotel room, Xu Yuan rewrote his answers and handed them to Lin Haoming and the others to circulate among themselves.

After the others read it carefully, their expressions varied.

Some showed joy on their faces while others frowned.

In their hearts, Xu Yuan's answer was the most correct one. Any difference only meant that they had made a mistake.

"My solution to the first question is the same as Yuan-ge's, so it should be safe."

“I used the wrong theorem.”

"If Brother Yuan hadn't told me, I wouldn't have known this was still a Carmichael number."

"I don't know how many points I will be deducted?"

Xu Yuan listened to everyone's discussion, but then noticed that Lin Haoming next to him looked a little pale.

He quickly asked, "What's wrong? Why do you look so bad?"

"Brother Yuan."

"I think I got the combination of the second question wrong." Lin Haoming answered with a slightly trembling voice.

As these words were spoken, even Zhang Yuhui frowned.

There were a total of three questions in the morning exam. If you got one wrong, it would be equivalent to losing a full 21 points, and your ranking would be far behind others.

If we don’t perform exceptionally well tomorrow, we’ll basically have no chance of winning the gold medal.

As fellow contestants, everyone understood Lin Haoming's feelings and started to comfort him.

"You don't necessarily lose all your points. If you only lose a few points, it's not a big deal."

"Don't think about it."

"Yes, Haoming, just do your best in the exam tomorrow."

But at this time, Lin Haoming obviously couldn't listen, and shook his head and said: "I can be sure that my thinking method is completely different from Brother Yuan's."

Xu Yuan was also surprised when he heard this. Based on his understanding of Lin Haoming's strength, he should not make such a mistake.

Besides, this set of questions is the simplest one.

The thought lingered here, and he said, "There is more than one solution to each problem. Write down your proof first."

……

"Okay Brother Yuan."

Lin Haoming came to his senses belatedly and quickly picked up a pen to start writing on the draft paper.

Contestants like them who made it to the finals had a very clear memory of the questions they had just done in the morning, and it took them about ten minutes to finish writing the entire process.

He handed it to Xu Yuan with a nervous look on his face.

Xu Yuan took the draft paper and looked over. His first reaction was that Lin Haoming's proof process was indeed very complicated.

However, after careful analysis, he found that although Lin Haoming's thinking was a bit different and complicated, he could still complete the proof in the question.

Still the correct answer.

The corners of his mouth slightly raised into a smile, giving the other person a reassurance.

"Don't worry, you just used a cumbersome method of proof."

“I did nothing wrong.”

Lin Haoming's expression instantly changed from negative to positive after hearing this: "That's great, I should be able to sleep well tonight."

"As long as you didn't make any mistakes, you should prepare well this afternoon and try to finish this final battle tomorrow."

Zhang Yuhui had been listening for a long time, and finally came over with a smile on his face, secretly relieved. Originally, this kind of thing should be his responsibility as the team leader, but because Xu Yuan was in the team, he was relieved a lot of worry.

Especially whenever he saw Xu Yuan's always confident expression, he felt more at ease than ever before.

I have a vague feeling that this year might be the year with the best results in the provincial mathematics competition.

　Please read and vote~
　　
　
(End of this chapter)