From Almighty Scholar to Chief Scientist

After thinking about it in his heart, he began to move his hands.

Having thoroughly understood so many undergraduate books, he now has quite a lot of mathematical knowledge in his brain.

He also read a lot about the knowledge of Mersenne prime numbers. For example, there is a new Mersenne conjecture. This conjecture is about the fact that as long as two of the three given conditions are true, then the other one is also true.

In addition, there is another conjecture called Zhou's conjecture, which was proposed by Zhou Haizhong, a mathematician in Huaguo, in 1992. In the article "Distribution Law of Mersenne Prime Numbers", he made a conjecture about the distribution law of Mersenne prime numbers. Relatively accurate prediction, the content is: when 2^2^(n+1)>p>2^2^n, there are 2^(n+1)-1 of Mp that are prime numbers.

Although Zhou's conjecture did not help people find Mersenne prime numbers directly, it narrowed the scope of people's search for Mersenne prime numbers, so that it has also received considerable praise internationally, including the double winner of the Fields Medal and the Wolf Medal, Professor Atle Selberg, who has completed the elementary proof of the prime number theorem, also believes that Zhou's conjecture is innovative and has created a new method that is inspiring. In addition, its innovation is also reflected in the revelation of new laws.

However, it is quite difficult to prove Zhou's conjecture, and there is no proof or disproof so far, so it is still a worldwide mathematical problem.

For Lin Xiao, these conjectures are of no use to him for the time being, but they also have certain guiding significance for his research.

"If you say that, according to my method, it is possible to prove Zhou's conjecture?"

Thinking about this question in his heart, Lin Xiao took out a pen, found a draft paper and began to calculate.

For mathematicians, it is obviously the most convenient to use the most primitive pen and paper to solve mathematical problems, and it can also bring them a sense of psychological satisfaction as formulas appear in their pens.

After all, it allows them to mentally say, "Look, I'm doing the smartest job in the world."

……

【3, 7, 31, 127, 257...】

Lin Xiao's first job is naturally to list the first few items in front of the Mason number.

Due to the exponent items, after listing a few items randomly, the number is already quite large, but for Lin Xiao, a larger number does not affect his judgment on this number.

Now just write him a number of less than [-], and he can judge whether the number is a prime number within two seconds. As for more than [-] and less than [-], he can also judge in a relatively short period of time.

This is number sense.

In history, many geniuses have such cases, such as Euler, after he lost his eyesight, he directly calculated the Mersenne number 2^31-1 by mental calculation, which was the largest known prime number at that time; For example, Ramanujan, this one is even more heavyweight, and his number sense is also famous.

And sometimes, this kind of number sense is also very helpful for solving problems.

It is estimated that if Lin Xiao is asked to participate in the strongest brain, if he shows it a little bit, the people present will be amazed.

After writing a few steps, Lin Xiao found some problems.

"Because I don't have an exact expression for prime numbers, the relational expression for 'p' cannot be directly deduced to infinity... Do I have to assume that the Riemann Hypothesis is true?"

He scratched his head, a little speechless.

Although the Riemann conjecture is a problem in the complex variable function, it seems that it has nothing to do with the distribution of prime numbers, but the function on the complex plane after the analytical extension of the Riemann zeta function is equivalent to a certain function including π(x), π(x) is also the prime number counting function.

So assuming that after the Riemann conjecture is established, the distribution of prime numbers can be found directly, then he can use it directly.

However, all the inferences that assume the Riemann conjecture is true, or the inferences that assume the Riemann conjecture is not true, their proponents are obviously flustered, although most mathematicians believe that the Riemann conjecture is true, after all, in the computer The verified number has reached ten trillion zeros.

For Lin Xiao now, there is no need for him to do such a thing. Moreover, he will give a report at the Mathematicians Conference. Will the Mathematicians Conference accept a report that assumes the Riemann Hypothesis is true?

He doesn't think so.

In this way, he might as well just bring the things he sorted out and talk about them. Although there is nothing innovative, considering his age, I believe no one will say anything by then.

"Hmm... that won't work. I need to find a new relation to form a connection with the Mersenne prime number, otherwise I have to give up."

And this means that he has to expand his new method again.

He couldn't help recalling some knowledge about prime numbers in his mind.

Suddenly, he thought of Dirichlet's theorem.

[If r and N are relatively prime, then lim(x→∞)π(x;N,r)/π(x)=1/φ(N)]

"By the prime number theorem of arithmetic progression, it seems that the relationship between the two can be found."

Lin Xiao thought silently in his heart, and his strong number sense made him think of (4x+3).

"It seems that Mersenne prime numbers are all numbers like 4x+3?"

比如3，就等于4＊0＋3，而7，就等于4＊1＋3，再比如一个大一点的数字，比如欧拉心算出来的2^31－1，其等于2147483647，同样可以转换为（4x＋3）的形式。

This is what Lin Xiao saw directly.

His eyes lit up and he began to prove.

With this relationship, he put the Mersenne prime number on his transformation constructor, and there is no problem.

Chapter 75 Set off in July

Of course, Lin Xiao can see it directly, which shows that it is not difficult to come to this conclusion.

As for how to prove this conclusion, it is also not difficult for Lin Xiao, but after thinking about it, he directly wrote:

[Observing 4n+3 and Mp, we can easily find that Mp is a number in the form of 4n+3. 】

For some unimportant steps in the thesis, the bosses usually just use words such as "obvious" and "easy to get" and skip them directly. For Lin Xiao, although he does not think he is a boss, he uses a few It is still no problem to use.

"Well, it's done here, and now we can substitute 4x+3 into the previous relationship."

Lin Xiao continued with the next steps.

However, although there is 4x+3, there are still many difficulties in the next steps, and it is still difficult to actually complete it.

And time passed slowly like this. With Lin Xiao's current brain development rate of 3%, it is still difficult to face such a problem. After all, relatively speaking, it is more difficult to discuss the distribution of Mersenne prime numbers than the Fibonacci he studied before. The Nach sequence is more difficult.

……

[For positive integers a, b, we define a Mersenne prime number (polynomial) about F2 as an irreducible polynomial of the form 1+x^a(x+1)^b.In this case: GCD(a,b)=1 and (a or b is odd)...

For S∈F2[x], it is expressed as: -S is a polynomial obtained by replacing x with x+1 in S: S(x)=S(x+1)...]

"That's entering into the realm of polynomials."

In Lin Xiao's transformation constructor, it is necessary to enter the polynomial, so as to realize his statistics on nonlinear polynomials.

However, the Mersenne number is different from the Fibonacci sequence after all. We can list an infinite number of Fibonacci sequences, but the Mersenne number is always limited by the number of the largest prime number we have found so far.

Although everyone knows that prime numbers are infinite, it is very troublesome to decompose the prime factors of a large number, which is why things related to prime numbers are widely used in cryptography.

At this moment, there was a knock on Lin Xiao's door, and it was Sun Yu who knocked on the door.

Hearing that there was no response inside, Sun Yu was helpless. Lin Shen must have learned to be a demon again.

However, Lin Xiao had told him before that if there was no response to knocking on the door, he could just go in, so Sun Yu opened the door directly and walked in.

Seeing Lin Xiao really sitting in front of the table, with a pile of draft paper stacked beside him, Sun Yu quietly walked up, took a look, and suddenly remembered that this thing would make him unstable, and he was almost blind on the spot.

He quickly moved his eyes away, patted Lin Xiao and said, "Shen Lin, it's time to have dinner, we have to go to the Romanian embassy to get a visa later, don't forget."

Lin Xiao finally came to his senses, and after hearing Sun Yu's words, he responded, "I see."

Looking down at his current progress, he shook his head, it was still not ideal.

He is now starting from the tangent circle polynomial to carry out his own bridging work, but it seems that there are still problems, and now he can only wait until later to continue to look at it, anyway, submit the report before July 7.

However, solving mathematical problems is also like this. You have to do it slowly and step by step. Problems are inevitable. Even trial and error is a process.

So there is no need to be discouraged. What's more, Lin Xiao's research is a worldwide problem in the field of prime numbers. After he researches it, others can say that he can't do it?

Just kidding.

Seeing Lin Xiao shaking his head, Sun Yu next to him couldn't help asking: "God Lin, do you still have any problems?"

Lin Xiao nodded.

"Damn, there are still questions that can stump you?"

Sun Yu looked astonished, but Lin Xiao has solved even the world's problems, how can he still encounter such difficult problems?

He said: "Let me be healthy!"

Even if it is a heavenly book, he still has to read it today.

"Look."

Lin Xiao put the draft paper he had used in front of him, and then packed his things, passport, ID card and so on. After all, he was going to the Romanian embassy in China to get a visa later.

The Romanian visa does not take as long as it does in the United States. Short-term visas can be obtained within two days, or one day if time is tight.

And they are going to participate in the IMO competition, basically they can get it done after they go there, after all, special things are special.

As for Sun Yu, who was looking at Lin Xiao's draft paper over there, his face was only bewildered, as if he had read something, and he seemed to have seen nothing at all.

"Lin Shen, what are you writing?"

Lin Xiao glanced at it and said, "Mersenne primes, don't you know?"

Sun Yu: "... seems to know, is there any guess?"

"Whether there are infinitely many Mersenne prime numbers, and the distribution law of Mersenne prime numbers. There is also a Zhou's conjecture about the distribution law, and there is also a new Mersenne prime number conjecture."

Sun Yu: "...Then which problem do you want to solve?"

"Let's predict the distribution law. If it is predicted, it is equivalent to solving whether there are infinitely many Mersenne primes... so it is also a problem."

Sun Yu: "...Sorry to bother you."

It turned out that the problem of being able to stifle Lin Shen can only be a world-level problem. He was thinking about farting just now.

Lin Xiao: "Then let's go."

"okay."

Then they left the dormitory.

……

In the next few days, the enthusiasm about Lin Xiao getting the first place in the college entrance examination still hasn't subsided, but for him himself, he really can't feel such enthusiasm.

After all, the Huaqing Daily only interviewed him once last time, so we can't do it again now, can we?

As for the ordinary media... or the same sentence, the Chinese Academy of Sciences has enough security forces, and the ordinary media want to go in?

There are no doors.

Of course, Lin Xiao received a lot of rewards. For example, a real estate agent in Jincheng City directly offered him a house worth less than 150 million.

Since he is still in Beijing and will go to IMO in the future, he will have to wait until he returns to Jincheng in July before giving it to him.

In addition, some entrepreneurs in Jincheng also gave him some scholarships, with a total amount ranging from 20 to 300. All in all, these rewards from enterprises, except for the house, There are a total of about [-] yuan, and including the rewards from the city and Beijing University, it really exceeds [-] million after all of them are added up.

It is completely a wave of fat.

Moreover, the house that was originally planned to spend 100 million to buy, now does not need to spend money.

Of course, Lin Xiao only shared the joy with his family. After all, his wealth is not revealed, and he is here in Shangjing, so he continues to do his own research every day, and he goes to training courses if he has nothing to do.

And just like that, it's July.

The day to go to Romania for the IMO campaign is coming.

The six most outstanding mathematics geniuses of the year from Huaguo will go to the place where the heroes gather with the ardent expectations of the motherland and their own dreams, and compete for a top spot.

Chapter 76 Arrival in Romania

"You are mathematical geniuses who stand out from tens of thousands of people across the country. You don't need to use too many words to describe your excellence."

"Relax, you will definitely become the best team in the world."

"The motherland is looking forward to your outstanding performance. We are here and wish you a triumphant return!"

On July 7, at the Beijing International Airport.

In front of the security checkpoint for international departures, Gong Zhou, secretary-general of the Huaguo Mathematical Society, sent his last blessings to the six teenagers before the expedition.

For the Huaguo Mathematical Society, as long as Lin Xiao and the others can get the team No. 1 in the IMO, it is of course a good performance. They can get the rewards from above, so they pay more attention to this matter every year.

Of course, they are relatively relieved about the IMO national team this time.

The six players this time, probably due to the existence of Lin Xiao among them, formed a kind of catfish effect among geniuses, so that the other five players have worked hard for so many days.

Although Sun Yu, Ouyang Sheng and the others called Lin Xiao "Lin Shen" all day long, they were also studying hard. They slept for seven or eight hours a day, and the rest of the time was for studying besides eating.

Even if you can't get full marks casually like Lin Xiao, and it's even more impossible to directly study world-class problems, everyone is fighting for the goal of winning glory for the motherland.

No one will say that they saw someone so awesome that they exploded, and then they quit.

Therefore, judging from the past situation, the probability of winning the group No.1 this time is very high.

"Thank you, teacher!"