Chapter 54 Accidentally Proving a Mathematical Problem
"The difficulty of the first three questions is a bit low. I hope the final question of this competition tomorrow will not disappoint people."

After Xu Yuan finished writing the last mathematical symbol on the test paper, he stopped writing, sat up straight, took a breath, and thought secretly in his mind.

At the same time, break away from the state of concentration.

The question types in the first day's exam were set number theory and geometry.

For him, all the problems can be solved easily. As soon as he finishes reading the question, he will have a solution in his mind, and it is absolutely correct without the need for the progress bar that automatically appears in front of him to confirm it.

To be honest, he was a little disappointed.

I thought that participating in international mathematics competitions and competing with elite peers from all over the world would make me more competitive and allow me to challenge more difficult competition questions.

As a result, he now only wants to say two words.

That's it.

However, if the other candidates knew what Xu Yuan was thinking at this moment, they would probably want to curse him.

It can only be said that the questions of the International Mathematical Olympiad are simple, but Xu Yuan is already at another level.

Once you break through a certain level, it will naturally be very easy to go back and do the previous questions.

Xu Yuan was too lazy to stay in the examination room any longer. He checked the personal information on the test paper and decisively handed it in.

Candidates from various countries in the same examination room raised their heads and looked towards Xu Yuan when they heard the noise.

There was surprise in his eyes.

For those who can sit here and take the exam today, handing in their papers early is basically a common thing for them, so it shouldn't be surprising.

It is really unbelievable that it could be possible so much in advance.

If the Olympic Mathematics Competition did not have strict rules, they would probably have suspected that they had obtained the answers in advance, otherwise why would they answer the questions so quickly as if they had copied the answers verbatim?

Xu Yuan didn't care about the candidates from other countries. After handing in his paper, he stood up and left the examination room.

-

The team leader will attend the meeting in advance to discuss the test questions and official answers, so he or she will live in the hotel arranged by the conference throughout the competition and is prohibited from having any form of contact with the participating candidates.

However, the deputy team leader can accompany the team members and check the test situation through the monitor in a special rest room.

Hu Zhiwei noticed that Xu Yuan finished the paper so quickly, and he couldn't stop showing joy on his face.

He was very clear about Xu Yuan's test-taking style. Since he chose to hand in the paper early, it meant that he was very confident in his answers.

"It seems that this year we have a guaranteed gold medal with full marks."

Just as he was thinking to himself, he suddenly heard an English voice in his ear, congratulating him.

"Congratulations, Professor Hu, your team has an excellent player."

Hu Zhiwei quickly turned his head and found that it was Professor Feng Zuming, the Asian coach of the American national team.

Although the two are of the same origin, they represent two different national teams after all. Facing the praise from his competitors, Hu Zhiwei showed no intention of keeping a low profile for the first time.

"Let me thank Professor Feng for his praise on behalf of this student." He thanked the other party first and then said in agreement: "Xu Yuan is indeed the best among our contestants."

The voice just fell.

The coaches from the Bulgarian and Goose teams also gave their affirmation.

Students who are selected for the national team will never give up answering the questions just because they are too difficult.

So the fact that Xu Yuan was able to hand in the paper early means that he had completed all the answers.

This speed of solving problems must be based on a solid mathematical foundation and very quick thinking ability.

It is indeed excellent and worthy of their praise.

for the rest of the time.

Then other contestants began to hand in their papers early and go to the lounge.

When the last remaining candidates were faced with a large number of empty seats around them, they felt a lot of pressure invisibly.

It wasn't until the bell for handing in the papers rang that everyone felt relieved.

During the meal, Hu Zhiwei did not ask Xu Yuan how he performed in the exam. Instead, he mentioned what happened in the room where the deputy team leader was.

He didn't hide the smile on his face at all when he spoke. The other team members, including Liu Yang, didn't show much surprise when they heard it.

Since when, Xu Yuan's strength in mathematics competitions has become a tacit acknowledgement by everyone.

Xu Yuan didn't take this little incident seriously and stayed in his room studying with his teammates in the afternoon.

The next day everyone arrived at the examination venue on time.

Take the final exam.

Xu Yuan got the test paper and quickly answered the first two questions, then focused all his attention on the last question.

As this is the most difficult question in theory in this mathematics competition, he is relatively looking forward to it.

"Let p be a prime number. Prove that there exists a prime number q such that for any integer n, the number np is not a multiple of q."

“This question is quite interesting.”

After reading the last question silently in his mind, Xu Yuan became quite interested.

Decide on the solution in your mind before you start calculating.

“Proof: p-1/p-1=1+p+p+…+p-”

……

"From Fermat's Little Theorem, we know that nq-1≡1(modq)"

……

After solving the problem according to Fermat's Little Theorem, Xu Yuan did not intend to stop writing.

Instead, he continued to calculate on the draft paper, with a hint of excitement on his face.

"I didn't expect this question to give me inspiration. Maybe I can take this opportunity to further explore the problem of the intervals between infinite Carmichael numbers."

During this period, he did not stop proving Carmichael numbers, and even tried to combine the weighted sieve method used by Chen Jingrun in his research on 1+2, and proved the problem of infinite Carmichael number intervals that had not been solved by three overseas mathematicians including Pomerance by improving the sieve method.

The progress bar originally generated for this issue not only stagnated but sometimes even decreased.

This shows that there was something wrong with his research direction.

But today, the problem he was solving involved Fermat's Little Theorem, which gave him a new idea.

Maybe it will make the progress bar go up.

So he didn't care about the exam he was taking and was completely immersed in his own world of argumentation.

"If we consider prime combinations such as kd+1 and kd'+1, we may be able to prove the existence of Carmichael numbers in [x/2, x]."

……

As time passed, Xu Yuan had already written mathematical formulas on both sides of the draft paper.

It's so densely packed that it's dizzying.

At the same time, he discovered that the progress bar on the panel in his field of vision was rising at a speed visible to the naked eye.

This further strengthened his confidence.

“n+n/（ln n）/2+δ，ln n/……”

It was not until the bell for handing in the paper rang that he finished writing the last mathematical symbol and a smile appeared on his face.

"When n>3300, there is always a Carmichael number between n and 2n."

"I just accidentally proved it."

In addition, as my eyes moved to the virtual panel, I suddenly discovered that the progress bar had successfully reached 100%.

-

Task: Carmichael number proof
Subject: Mathematics

Progress: 100%
Result: Completed
However, just when Xu Yuan was immersed in this joy, Hu Zhiwei saw that he did not hand in the paper early like yesterday, and he was quite worried that something might have gone wrong.

(End of this chapter)