Chapter 74 Mersenne Primes

Mersenne number refers to a positive integer in the shape of 2^p-1, where p represents a prime number, 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's number is to commemorate the research of Mersenne, a famous French mathematician in the 17th century, on prime numbers of the shape 2^p-1.

In fact, the history of research on numbers like 2^p-1 can be traced back to more than 2,300 years ago.

After Euclid proved that there are infinitely many prime numbers, he proposed that a small number of prime numbers could be written in the form "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 another 1, a new prime number may appear.

This seems to be a very coincidence, but it also hides the charm of numbers. Therefore, the study of Mersenne prime numbers is also very famous in the mathematics community.

At this time, in Lin Xiao's opinion, he seemed to be able to use his own method to figure out the distribution pattern of Mersenne prime numbers.

"Try it."

After thinking this in his mind, he started to move his hands.

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

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

In addition, there is also a conjecture called Zhou's conjecture, which was proposed by Chinese mathematician Zhou Haizhong in 1992. He made a prediction about the distribution law of Mersenne prime numbers in the article "The Distribution Law of Mersenne Prime Numbers" A relatively accurate prediction, the content is: when 2^2^(n+1)\u003ep\u003e2^2^n, 2^(n+1)-1 of Mp are prime numbers.

Although Zhou's conjecture did not help people directly find Mersenne primes, it narrowed the scope of people's search for Mersenne primes, so that it also received considerable praise internationally, including double winners of the Fields Medal and Wolf Medal. Professor Atler Selberg, who completed the elementary proof of the prime number theorem, also believed that Zhou's conjecture was innovative and created an inspiring new method. In addition, its innovation was also reflected in revealing new laws.

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

For Lin Xiao, these conjectures are of little use to him for the time being, but they also have certain guiding significance for his research.

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

Thinking about this problem in his mind, Lin Xiao took out his pen, found some draft paper and started to calculate.

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

After all, this way they can say in their mind, “Look, I’m doing the smartest work in the world.”

…

【3, 7, 31, 127, 257...】

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

Since there are exponential items, after just listing a few items, the number is already quite large. However, for Lin Xiao, a larger number does not affect his judgment of the number.

Now just write him a number within ten thousand, and he will be able to judge within two seconds whether the number is a prime number. As for the number above ten thousand but within one hundred thousand, he can also judge in a relatively short time.

This is number sense.

In history, many geniuses have such examples, such as Euler, who directly relied on mental arithmetic to calculate that the Mersenne number 2^31-1 was a Mersenne prime number after he became blind, which was the largest known prime number at the time; For example, Ramanujan was a heavyweight and his number sense was also famous.

And sometimes, this kind of number sense can be of great help in solving problems.

It is estimated that if Lin Xiao participates in the most powerful brain show, everyone present will be amazed.

After writing a few steps, Lin Xiao discovered that there were some problems.

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

He scratched his head and was speechless.

Although the Riemann Hypothesis is a problem in complex functions, it does not seem to have anything to do with the distribution of prime numbers. It is just that the function on the complex plane after the analytical extension of the Riemann zeta function is equivalent to a function including π(x). π(x) is also the prime number counting function.

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

However, all inferences that assume that the Riemann Hypothesis is true, or inferences that assume that the Riemann Hypothesis is not true, are obviously panicked by their proposers, although most mathematicians believe that the Riemann Hypothesis is true. After all, in computers The number verified has reached ten trillion zero points.

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 assuming that the Riemann Hypothesis is true?

He doesn't think so.

In this case, he might as well just bring what he compiled and talk about it. Although there is nothing innovative, considering his age, I believe no one will say anything by then.

"Well... this won't work. I need to find a new relationship to form a connection with the Mersenne prime numbers, otherwise I will have to give up."

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

He couldn't help but recall some of the knowledge about prime numbers in his mind.

Suddenly, he thought of Dirichlet's theorem.

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

"Through the prime number theorem of arithmetic series, 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 in the shape of 4x+3?"

For example, 3 is equal to 4*0+3, and 7 is equal to 4*1+3. Another example is a larger number, such as 2^31-1 calculated by Euler mentally, which is equal to 2147483647, and can also be converted to (4x+3) form.

Lin Xiao saw this directly.

His eyes lit up and he began to prove it.

With this relationship, he set the Mersenne prime number to his own transformation constructor, and there was no problem.

Thanks to Yang Kun’s trumpet for the 600 starting coin reward, and thanks to the Bloody Hummer for the 500 starting coin reward.

Thank you for your support!