Volume 1: Student Life Chapter 426 Four Ways
Draw a circle on "Chen's Theorem".
Chen Zhou was thinking, maybe one day, maybe it won’t be long.
"Chen's Theorem" will become the complete Goldbach's Theorem.
Of course, in a sense, Goldbach's theorem can also be called "Chen's theorem".
As for this "Chen", it is naturally the Chen in Chen Zhou's name.
Retracting this relatively distant thought, Chen Zhou once again focused his attention on the Goldbach conjecture.
Based on previous studies, there are four approaches to the study of Ge Gua.
They are almost prime numbers, exceptional sets, the three prime number theorem with small variables, and the almost Goldbach problem.
An almost prime number is a positive integer with few prime factors.
Assume N is an even number. Although it cannot be proved that N is the sum of two prime numbers, it is sufficient to prove that it can be written as the sum of two almost prime numbers.
That is A+B.
Among them, the number of prime factors of A and B is not too many.
That is the proposition that Chen Zhou just wrote down and that I guessed.
The latest development of the "a+b" proposition is Mr. Chen's "1+2".
As for the ultimate mystery of "1+1", it is still a long way off.
All progress in the direction of almost prime numbers has been achieved using the sieve method.
However, although Mr. Chen used the sieving method to the extreme, it only stopped at "1+2".
Therefore, many mathematicians also believe that it is difficult for current research to surpass Mr. Chen’s application of the sieve method.
This is also the biggest reason why research in this direction has stagnated for so long.
Until a more reasonable tool or a tool that can further enhance the role of the screening method is found.
There will never be a major breakthrough in the proof of “1+1”.
Chen Zhou also agrees with this view.
However, how easy is it to make further breakthroughs for a tool that has been used to its extreme?
For a mature mathematical tool, the introduction of new mathematical ideas will also become more difficult.
But fortunately, when Chen Zhou was studying Cramer's conjecture, he more or less, intentionally or unintentionally, came up with the distribution structure method.
The original distribution structure method is a tool that combines mathematical ideas such as sieve method and circle method.
Therefore, in Chen Zhou's opinion, the key point to break through the limitations of the large screening method lies in the distribution structure method.
On the draft paper, Chen Zhou wrote the distribution structure method separately on the right side.
The method for almost prime numbers is on the left.
And below the almost prime number method is the exception set.
The so-called exception set refers to the large integer x on the number axis.
Then look forward from x and look for even numbers that make the Goldbach conjecture invalid.
These even numbers are called exceptional even numbers.
The key to this idea is that no matter how big x is, as long as there is only one exceptional even number before x.
And this exceptional even number is 2, which means that only 2 makes the conjecture wrong.
As for 2, everyone understands it.
Then, it can be shown that the density of these exceptional even numbers is zero.
This proves that the Goldbach conjecture holds for almost all even numbers.
Research along this line of thought may not be so well-known in China.
But from a global perspective, as soon as Vinogradov's three prime number theorem was published, four proofs appeared simultaneously on the path of exceptional sets.
Among them is Mr. Hua’s famous theorem.
An interesting thing to say is.
Amateur scientists often claim that they have proved that the Goldbach conjecture is correct in the probabilistic sense.
But in fact, they just "proved" that exceptional even numbers have zero density.
As for this conclusion...
Mr. Hua had already proven this 60 years ago.
So, sometimes we really can’t listen to the nonsense of amateur scientists.
Take Chen Zhou himself for example, if he cared about the voices of amateur scientists.
Well, the emails from those amateur scientists that filled up his mailbox were really enough to give him a headache.
"If the Goldbach conjecture for even numbers is true, then the conjecture for odd numbers is also true..."
After writing down the third research approach, “The Three Prime Number Theorem of Small Variables,” Chen Zhou began to write down his research ideas for this approach while thinking about it.
[Given that an odd number N can be expressed as the sum of three prime numbers, if we can prove that one of these three prime numbers is very small...]
The person who has been conducting research along this path is Mr. Pan, a famous Chinese mathematician.
If the first prime number can always be 3, then the Goldbach conjecture is proved.
Following this idea, Mr. Pan began to study the three prime number theorem with a small prime variable at the age of 25.
This small prime variable does not exceed N to the power of θ.
The research goal is to prove that θ can be 0.
That is, this small prime variable is bounded, which leads to the Goldbach conjecture on even numbers.
Mr. Pan was the first to prove that θ can be taken as 1/4.
Unfortunately, there has been no progress in this area since then.
It was not until the 1990s that Professor Zhan Tao extended Mr. Pan’s theorem to 7/200.
This number is relatively small though.
But it is still greater than 0.
Judging from the research history of the above three approaches, the contribution of Chinese mathematicians in this area can be said to be outstanding.
However, no one has been able to finally solve this problem that has troubled mathematicians for nearly three hundred years.
Moreover, because of the research of these mathematicians, the Goldbach conjecture has extraordinary significance in the Chinese mathematics community, and even in China.
Chen Zhou sorted out his research ideas and wrote down his thoughts on the draft paper.
Chen Zhou already has some unusual ideas about his distributed structure method.
Chen Zhou also has high expectations for this method, which incorporates many mathematical ideas .
After sorting out the path of "Three Prime Number Theorem of Small Variables", Chen Zhou glanced at the blank space on the draft paper.
Fortunately, he drew the previous horizontal line lower.
These organized and compressed essences are able to stand on this piece of white paper.
Stretching, Chen Zhou looked at the time. It was just past 10 o'clock in the evening.
Since it’s still early, let’s continue!
Thinking in this way, Chen Zhou began to sort out the approach of "Almost Goldbach Problem".
The "Almost Goldbach Problem" was first studied by Linnik in a 70-page paper in 1953.
Linnik proved that there exists a fixed non-negative integer k such that any large even number can be written as the sum of two prime numbers and k powers of 2.
Some people say that this theorem seems to be a vilification of the Goldbach conjecture.
But in fact, it has a very profound meaning.
It can be noticed that the integers that can be written as the sum of k powers of 2 form a very sparse set.
That is to say, for any given x, the number of such integers before x will not exceed logx to the power of k.
Therefore , Linnik's theorem states that although we cannot prove the Goldbach conjecture yet, we can find a very sparse subset in the set of integers.
Every time we take an element from this sparse subset and stick it into the expression of these two prime numbers, the expression becomes valid.
The k here is used to measure the degree of approximation of the Almost Goldbach Problem to the Goldbach Conjecture.
The smaller the value of k, the closer it is to the Goldbach conjecture.
So, it is obvious that if k is equal to 0.
The power of 2 in Goldbach's problem almost never appears again.
Thus, Linnik's theorem becomes the Goldbach conjecture.
