From Almighty Scholar to Chief Scientist

Chapter 74 Mersenne Primes
A Mersenne number refers to a positive integer of the form 2^p-1, where p represents a prime number and is often recorded as Mp. If a Mersenne number is also a prime number, it is called a Mersenne prime number.

The reason why it is called Mersenne number is to commemorate the research done by Mersenne, a famous French mathematician in the 17th century, on prime numbers of the form 2^p-1.

In fact, for numbers such as 2^p-1, the history of research can be traced back to more than 2300 years ago.

After Euclid proved that there are infinitely many prime numbers, he proposed that a small number of prime numbers can be written in the form of "2^p-1".

This is obviously a very magical thing, where p refers to a prime number, and then let it become the exponent of 2, and then subtract a 1, a new prime number may appear.

This seems to be a very coincidence, but it also hides the unique charm of numbers, so the research on Mersenne prime numbers is also very famous in the mathematics world.

At this time, in Lin Xiao's opinion, it seems that he can also use his own method to find out the distribution law of Mersenne prime numbers.

"Try it."

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, Mp has 2^(n+1)-1 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 arithmetic, 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 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.

　Thanks to Yang Kun for the tipping of 600 starting coins for the trumpet, and thanks to the reward of 500 starting coins for Scarlet Hummer.

　　Thank you for your support!

　　
　
(End of this chapter)