Why do you, a sports student, come into the academic circle?

When Sun Mingbo returned home, he greeted his parents and told them that the mathematical conjecture he had studied had been inspired, and that he must take the opportunity to check it to prevent them from thinking wildly.

As for reporting on today's college entrance examination, let's wait until dinner.

It's also said that today is the day of the college entrance examination, and if they return to their rooms without a sound, I don't know how they are worried.

Sun Mingbo didn't waste time, he immediately returned to his room, closed the door, and entered the system subspace training ground with his laptop.

Why do you want to go to the system subspace

? Do you still need to ask

? In the system subspace, the flow rate of time is 2 to 1, and there is a system bonus to the efficiency of learning and working in it, where can you find such a good thing?

If you don't make full use of resources, a fool will do that.

He entered the system's subspace training ground, went to the athletes' lounge, turned on the computer, and saw his fingers tapping rapidly on the keyboard.

So much so that when I hit the fingers of the keyboard, there was an afterimage because of the speed of the hand.

It can be seen how eager Sun Mingbo is to input all the ideas in his brain about Zhou's conjecture and the proof steps into the computer.

The laptop screen immediately displays:

"On the Distribution of Mersenne Prime Numbers - Proof of Zhou's Conjecture"

Abstract

This article will prove Zhou's guess in detail.

Zhou's guess is a guess about the number of Mersenne primes, and this guess gives the number of Mersenne primes in a specific range.

The proof will be carried out in the following steps, first, when p<2^2, Mp has only one prime number.

Then, I'm going to increase the value of k by increasing the value of k, and for each increase in k, the number of primes in Mp will increase by a certain amount.

Next, I will form a "set" of every four primes to understand the law that every increase in the value of k will increase by one prime. Finally, I will use the above steps as a basis to complete the complete proof of Zhou's guess step by step.

Introduction

In number theory, Mersenne primes are a subset of prime numbers.

They are defined as prime numbers of the form p=2^k -1, where k is a positive integer. Zhou's guess is a guess about the number of Mersenne primes, and this guess gives the number of Mersenne primes in a specific range. Although this speculation has been put forward for a long time, it has not been proven and disproved.

This article will give a method of proving the validity of Zhou's conjecture.

Methods

: A step-by-step approach is used to prove Zhou's conjecture.

First, it is observed that when p<2^2, Mp={p} has only one prime number.

Then, by employing increasing k-values, it is observed that for each increase in k-value, the number of Mp primes increases by a certain amount.

In order to better understand and find this rule, every four primes will be formed into a "set", and each additional k value will create a set of four primes.

The first number of these prime numbers corresponds to a composite number, while the other three numbers correspond to all prime numbers.

Thus, the conclusion that each increase in the value of k corresponds to the increase in a prime number is confirmed.

First

, it is observed that when p<2^2, Mp={p) corresponds to only one prime number.

Secondly, when 2^2≤p<2^3, there are two primes correspondingly.

When 2^3≤p<2^4, there are three primes.

For 2^4≤p<2^5, there are four primes.

By analogy, these results suggest that each increase in the value of k increases by a prime number.

In order to better find and understand this law, every four prime numbers are formed into a "set".

When k=1, there is only one set, and there are only four prime numbers in the set: 1, 3, 5, 7.

When k=2, there are only four sets, and in each set there are four primes: 1, 3, 5, 7; 9, 11, 13, 15; the first number in one of the sets is a composite number.

When k = 3, there are only eight sets with only four primes in each set: 1, 3, 5, 7; 9、11、13、15; 17, 19, 21, 23; the first number in two sets is a composite number.

When k = 4, there are only sixteen sets, and there are only four prime numbers in each set: 1, 3, 5, 7; 9、11、13、15;...； The first number in the three sets is a composite number.

By analogy, in a similar way, it is possible to gradually approach the range of Zhou's guess.

As k increases, more and more numbers in the set are composite numbers, and the other three numbers are prime.

Therefore, it can be concluded that when p<2^2n+1, 2n+1-1 in Mp is prime.

In addition, when p

can be proved, and similarly, when M(M+k)=4k2+2k-1=(2k-1)(2k+1) primes can be proved.

Thereinto.............

The time soon reached half past seven in the evening, the table was already set, fragrant meals, after dinner, and reported to my parents about today's exam.

"I feel that all the questions in today's exam are very simple, I finished answering them early, and I checked them repeatedly, and there were no mistakes. Sun Mingbo reported to his parents a little proudly.

"How many points do you think you can get in these two subjects?" asked my mother with concern.

"Mathematics must be 150 points, and it is difficult to say that there is an essay in Chinese, so I estimate that I will get 145 to 148 points, which depends on the mood of the teacher. Sun Mingbo said with a grin.

"Don't worry, the college entrance examination is just like that, this time I will definitely win you a champion. In the end, Sun Mingbo swore and said.

Seeing Sun Mingbo so confident, my parents smiled happily.

At the same time, he once again instructed Sun Mingbo not to be careless and take it lightly because he felt that the test questions in this college entrance examination were simple.

Sun Mingbo didn't refute it, he knew his own affairs best, and his parents' instructions were also for his good, which was also another kind of happiness.

After chatting for a while, he quickly returned to his room, entered the system subspace training ground and continued to write his thesis.

After nearly 20 hours of struggle in the system space, the proof paper of Zhou's conjecture was finally written.

He stretched, moved his muscles, looked at the paper on the screen, he laughed, completed the system task, and victory was in sight.

A sense of satisfaction arose, and he even had a feeling that as long as he was serious, nothing would be difficult for him.

This is Sun Mingbo, if you are replaced by others, even if you know the entire proof process and steps, that is, those formulas are calculated, let alone 20 hours, you are 200 hours, and you may not be able to write it completely.

This is only the first step after the thesis is written, and there are more important things to do next.