Draw a circle on the "Chan's theorem".

Chen Zhou was thinking, maybe one day, maybe it won't be long.

The "Chan's theorem" will become the complete Goldbach theorem.

Of course, in a sense, Goldbach's theorem can also be called "Chen's theorem".

As for this "Chen", it is naturally Chen Zhou's Chen.

Retracting this rather distant thought, Chen Zhou's attention was once again focused on Goldbach's conjecture.

Judging from previous research, there are four ways to study Gochai.

These are primes, exception sets, the three-prime theorem for small variables, and the almost Goldbach problem.

A prime number is a positive integer with a small number of prime factors.

Let N be an even number, and while it cannot be proved that N is the sum of two primes, it is sufficient to prove that it can be written as the sum of two primes.

That is, A+B.

Among them, the number of prime factors of A and B is not too much.

That is, Chen Zhougang wrote it, Ge Guess's proposition.

The latest progress of the "A+B" proposition is Mr. Chen's "1+2".

As for the ultimate mystery of "1+1", it is far away.

Progress in the direction of primes is obtained by sieving the method.

However, Mr. Chen used the sieving method to the extreme, and only stayed on the "1+2".

Therefore, many mathematicians also believe that it is difficult to break through the application of Mr. Chen's sieve method in the current research.

This is also the biggest reason why research in this direction has been stagnant for so long.

Before we find a more reasonable, or rather able to further the role of the sieve method, we have not found a more rational tool.

The proof of "1+1" will never have a big breakthrough.

Chen Zhou also agrees with this point of view.

However, a tool that has been used to the extreme, how easy is it to break through again?

The introduction of new mathematical ideas can also become more difficult for a mature mathematical tool.

But fortunately, when Chen Zhou studied the Cramel conjecture, he came up with the distribution structure method more or less, intentionally or unintentionally.

The original distribution structure method is a tool that combines mathematical ideas such as sieve method and circle method.

Therefore, in Chen Zhou's thoughts, the key point for him to break through the limitations of the large sieve method is on the distribution structure method.

On the scratch paper, Chen Zhou wrote the distribution structure method on the right side.

The method of priming numbers is on the left.

Underneath the prime-number method is the set of exceptions.

The so-called set of exceptions refers to the number line, taking a large integer x.

Look further from x to the even numbers that make Goldbach's conjecture untenable.

These even numbers are also known as exceptional even numbers.

The crux of this line of thinking is that no matter how big x is, there is only one exception to the even number, as long as it is before x.

And this exception is an even number of 2, that is, only 2 makes the conjecture wrong.

And 2, everyone understands.

Then, it would be clear that the density of these exceptional even numbers is zero.

This proves that Goldbach's conjecture is true for almost all even numbers.

The study of this line of thought may not be so well-known in China.

But from a global perspective, as soon as Vinogradov's three-prime theorem was published, four proofs appeared at the same time on the way to the set of exceptions.

Among them, including Mr. Hua's famous theorem.

One thing that's interesting is that.

Folk scientists often claim to have proved the Goldbach conjecture right in the probabilistic sense.

But in fact, they "proved" that the exception even number is zero density.

As for this conclusion......

Mr. Hua Lao has really proved it as early as 60 years ago.

Therefore, sometimes I really can't listen to the nonsense of the people.

Take Chen Zhou himself, if he cares about the voices of the people.

Then, the emails from the people who filled the mailbox were really enough for him.

"If the Goldbach conjecture of even numbers is correct, then the conjecture of odd numbers is also correct......"

After the third research path, "the three-prime theorem of small variables", Chen Zhou began to think about it and write down the research ideas of this approach.

[Knowing the odd number N, it can be expressed as the sum of three primes, if it can be proved that one of the three primes is a very small ......]

On this path, the person who has been studying is also Mr. Pan Lao, a famous mathematician in China.

If the first prime number can be taken as a total of 3, then it proves Gochai.

It was along this line of thought that Mr. Pan began to study the three-prime theorem with a small prime variable when he was 25 years old.

This small element variable does not exceed the theta power of N.

The goal of the study is to prove that θ can take 0.

That is, this small prime variable is bounded, thus leading to the Goldbach conjecture of even numbers.

Mr. Pan first proved that θ can take 1/4.

Unfortunately, there has been no progress in this area.

Until the 90s of the last century, Professor Zhantao pushed Mr. Pan's theorem to 7/200.

This number, although it is relatively small.

But it's still greater than 0.

Judging from the research process of the above three approaches, the contributions of Chinese mathematicians in this area can be said to be outstanding.

It's just that no one can finally solve this problem that has plagued mathematicians for nearly 300 years.

Moreover, because of the research of these mathematicians, Goldbach's conjecture has extraordinary significance in the Chinese mathematical community, and even in China.

On the scratch paper, Chen Zhou wrote down his thoughts while sorting out his research ideas.

Chen Zhou already has an extraordinary idea of his distribution structure method.

This method, which combines many mathematical ideas, has also been placed more expectations on Chen Zhou.

After sorting out the approach of "the three-prime theorem of small variables", Chen Zhou glanced at the blank space on the scratch paper.

Fortunately, the previous horizontal line was drawn lower down.

These essences that have been sorted out and compressed can stand on this blank paper.

Stretching, Chen Zhou glanced at the time, it was only past 10 o'clock in the evening.

Since it's still early, let's go ahead!

Chen Zhou, who thought like this, began to sort out the way of "almost Goldbach problem".

The "Almost Goldbach Problem" was first studied by Linnick in a 70-page paper in 1953.

Linnick proved that there is a fixed non-negative integer k such that any large even number can be written as the sum of two prime numbers and the powers of k 2s.

Some people say that this theorem looks like a scandalization of Goldbach's conjecture.

But in fact, it has a very deep meaning.

It can be noted that integers that can be written as the sum of the powers of k 2s form a very sparse set.

That is, for an arbitrarily given x, the number of such integers before x will not exceed the k-power of logx.

Thus, Linnick's theorem states that although we have not yet proved Goldbach's conjecture, we can find a very sparse subset of the set of integers.

Each time you take an element from this sparse subset and paste it into the expression of these two primes, the expression holds.

k here is used to measure the approximation of almost Goldbach's problem to Goldbach's conjecture.

The smaller the value of k, the closer it is to Goldbach's conjecture.

It is obvious, then, if k is equal to 0.

Almost the power of 2 in Goldbach's problem no longer appears.

Thus, Linnick's theorem became Goldbach's conjecture.