Great Country Academician

"The factorization of large positive integers has the solution proof of polynomial algorithm!" 》

Looking at the file sent by Liu Jiaxin on his mobile phone, Xu Chuan was stunned for a moment and then reacted.

He quickly clicked on the file, downloaded it and at the same time opened up his authority.

"Did you prove it?"

His fingers quickly tapped on the Jiugongge keyboard a few times, and a brief message was sent.

At the same time, he quickly sent the document to his assistant and sent a message: "Help me print out this document as quickly as possible and send it to my room."

After the message here was sent, the message from Liu Jiaxin came back over there.

"Well, this method should be able to solve the factorization problem of great positive integers, but I'm not sure if there are any flaws in it. I would like you to help me take a look."

Xu Chuan quickly replied: "It's printing, I'll look at it right away."

After a pause, he added: "I'll go back tomorrow afternoon."

"It's okay. There's no need to rush. Just do your business first. Don't worry about the paper."

The message from the other side was quickly replied, but Xu Chuan no longer paid attention.

He stood up and took out his computer from his backpack, quickly opened it and uploaded the PDF paper to the computer.

Before the printed paper was delivered to him, the screen of the computer was always larger than that of the mobile phone. He couldn't wait to see the specific contents of this top-notch mathematics paper.

Open it, and the main topic of the paper comes into view.

"The factorization of large positive integers has the solution proof of polynomial algorithm!" 》

The title of the paper is very straightforward, it is the first question in the P=NP? problem, and it is also a difficult problem that he and Liu Jiaxin discussed before.

But for P=NP? The problem is that his understanding is not very deep.

As one of the 18 major unsolved problems in mathematics of the 20th century, mathematician Smail chose the following NP-complete problems derived from traditional mathematical problems as representatives of the "P=NP?" problem.

"That is: given k polynomials on Z with respect to n variables, ask whether there is a polynomial-time algorithm to determine that they have a common zero point on (Z)n. This description is mainly inspired by Brownwell's theory of H Effects of Algorithms Determined by Herbert’s Zero Point Theorem.”

To put it simply, f1,···, fk are complex coefficient polynomials with n variables. According to Hilbert's zero point theorem, f1,···, fk do not have a common zero point in the complex number field when and Only if there is a complex coefficient polynomial g1 with n variables,..., gk satisfies k∑i=1·GiFi=1.

If it is difficult to understand these professional mathematical languages, P=NP? The problem can be divided into two parts to describe it in relatively simple terms.

‘P-type problems’ and ‘NP-type problems’.

Of course, here are two simplified concepts to help understanding. They are simplifications made for simple and clear understanding without mathematical rigor and complexity.

P represents a class of problems that computers can solve very quickly. This speed has nothing to do with computer hardware, it just depends on the convenience of the solution itself.

NP represents another type of problem, which has an optimal solution. However, for many of these problems, when the computer seeks the optimal solution, there is no fast way, and even, it can only stupidly and violently try all possible combinations. , and then find the optimal solution.

Among NP problems, the most difficult type of problem is called NPC, which is NP-complete problem.

If this is still not specific enough, let me use a small story as an example. I believe you can understand it more simply.

Suppose you are at a big party and want to know if there is anyone you know there.

At this time, the host of the banquet told you that you must know the lady A who is standing in the right corner of the dessert table, so you immediately scanned there and found that what he said was right, you did know her.

Therefore, through the information about the host of the banquet, you can easily determine that you know Ms. A.

But if he doesn't tell you that, you need to look around the hall and examine everyone to see if there's anyone you recognize.

Finding Ms. A through the hint of the banquet host is a type P problem;

And if you follow his prompts and find that you know Ms. A, it is easy to check that Ms. A is an NP problem.

In the mystery novel "The Devotion of Suspect X" written by an island country writer, Ishigami and Yukawa once discussed whether it is more difficult to solve a proposition or to judge whether a proposition is correct.

In fact, the mathematical community has already given the answer, P=NP? The problem is there, and it tells everyone that generating a solution to the problem usually takes more time than verifying a given solution.

For example, if you were asked to calculate the total number of atoms in the world, the problem would be very difficult or even unsolvable.

However, if someone tells you that there are 500 atoms in the world, you can quickly verify that they are wrong. It is easy to verify, but not easy to solve. This is an NP problem.

P-type problems are a type of problems that can be solved and verified in polynomial time; NP-type problems are a type of problems that can be verified in polynomial time but it is not certain whether they can be solved in polynomial time.

Obviously, all P-type problems belong to NP-type problems, but it is impossible to determine whether NP is equal to P.

Since "P=NP?" was proposed, many attempts have been made in both the mathematics and computer fields.

The most obvious way to prove that P=NP is to give a polynomial-time algorithm for the NP-complete problem.

But in the past few decades, a large number of mathematicians and programmers have done a lot of work to find polynomial-time algorithms for NP-complete problems, but without success.

Of course, there are also a large number of people trying to give P≠NP? , even in today's mainstream mathematics community and computer industry, most scholars and researchers believe that P≠NP? .

The reason is simple. If P=NP, it means that every NP problem can be converted into P, that is, every difficult problem can eventually be turned into a simple proposition, so that computers can quickly solve it.

This means that humanity’s current mathematical system, computer system, common sense, and other aspects will be subverted.

If P=NP is finally confirmed, we can transform any NP problem into a P problem. Problems that seem difficult now can be easily solved.

For example, Go has the ultimate solution. In the biological field, the genetic code can be easily cracked to manipulate gene sequences at will. Many mathematical conjectures can be calculated and derived using computers. A large number of difficult problems have been solved, etc.

At the same time, if P=NP, this will lead to the complete failure of all encryption algorithms in a short period of time in the future. Your bank cards, mobile phone passwords, and social accounts will no longer be safe. Hackers can easily enter your computer and Bitcoin. , Blockchain, a concept that has been very popular in recent years, will become an area that no one cares about.

If P=NP, then in this universe, there must be a simple key that can solve all the problems in this world.

If such a key existed, it would probably already exist in this universe.

For example, humans may already have the ability to see everything once and for all, or some creatures may not have to fight for survival since they are born, because their algorithms are extremely good and can survive in any environment in the most efficient way.

But whether it is from intuition, philosophy, religion, or science, it is difficult for people to believe that such a shortcut to the universe exists.

To be honest, Xu Chuan doesn't believe that such a 'master' key exists in the universe, but when it comes to P=NP? Even if it is a staged proof, he will use the most concentrated energy to deal with it.

The papers on the computer screen were constantly flipping, and lines of mathematical formulas and explanations flashed across Xu Chuan's eyes.

At this moment, the doorbell ding-dong-ding-dong came from outside the room.

Quickly getting up, Xu Chuan walked through the bedroom and opened the door. At the door, Tang Sijia, the life assistant who was traveling with him on a business trip, was standing at the door, holding a thick stack of newly printed documents in his hands.

"Professor, this is what you want."

Handing over the paper with the lingering warmth and fragrance of ink attached to it, Tang Sijia added: "There is a stack of unused A4 paper under the paper, and I can do the calculations for you."

Although she knew that Xu Chuan usually carried a pen and some manuscript paper with her, there was no doubt that something that she could print out as quickly as possible was extremely important.

Therefore, she was worried that the quantity of manuscript paper that this person brought with him was not enough, so she directly took a stack of blank A4 words from the printing room and sent it over.

Sure enough, after hearing that there was a blank A4 paper attached to the paper, Xu Chuan's eyes lit up and he quickly took the paper and manuscript paper from his assistant Tang Sijia.

"very good, thank you!"

Tang Sijia smiled slightly and said, "You're welcome, Professor, if you have any other needs, just send me a message."

On the other side, Xu Chuan didn't even hear what his little assistant said clearly, so he waved his hands impatiently, and quickly returned to the study room of the hotel room with the paper and manuscript paper in hand, without even bothering to close the door.

Outside the door, the smile on Tang Sijia's face froze for a moment, then she closed the door silently, turned around and left, and said a blessing in her heart.

Although she could not understand the printed paper, out of curiosity, she searched for the title of the paper on her mobile phone during the free period of printing.

The title of this paper seems to involve one of the seven millennium problems, P=NP? guess.

As Xu Chuan's assistant, although she is not a mathematics major, she knows something about the field of mathematics. She is very aware of the weight of each millennium problem and its impact on the country and even the world.

Any solution to the Millennium Problem can greatly promote the development of mathematics, other disciplines, and even society as a whole.

Just like the NS equation, although she couldn't understand the proof or even understand the meaning of the NS equation, she knew clearly that the solution to controllable nuclear fusion technology was based on the NS equation. .

I hope the professor can successfully solve P=NP this time? problem.

Looking at the figure who turned around and entered the study, Tang Sijia silently prayed in her heart.

In the study room, Xu Chuan didn't know that the assistant outside had so much on his mind. At this moment, all his attention was focused on the paper in his hand.

Rather than reading papers on a computer screen, he prefers knowledge that can be weighed with his hands.

[Interpretation: This article gives a method that a P-type problem can be determined or solved in polynomial time using a deterministic algorithm and its polynomial time determination algorithm. The upper bound on the number of terms of gi in the Boolean polynomial (1) of the complexity of the complex solution algorithm for the system of decision equations f1 = 0,..., fk=0 is given]

". This is aimed at exploring the relationship between the complexity categories of P and NP. In a previous paper [1], we have shown that the sat CNF problem can be polynomialized into finding a special decomposition of a set under a special decomposition of the set. Coverage issues and vice versa.”

".Definition 1: G = is called a labeled multistage graph, if the following conditions are met:

1. V is a set of vertices, V=VUЙUVu...UV, VnV=0, 0≤ij≤L, i≠j. If uV, 0≤i≤L, the level u is at is called level i, and u is also said to be the vertex of level i. L is called the level of G.

2.E is a set of edges, and the edges in E are all directed edges, which are represented by triples (u, v, l). If (u, v, l)E, 1≤l≤L, then ueV-1vEV. Call (u, v, l) the l-th level edge of G.

3. Both sum and sum contain only unique vertices. The only vertex in is called the source point, denoted by S, and the only vertex in is called the sink point, denoted by D."

4

The paper in his hand flowed through his eyes, and Xu Chuan flipped through every sentence, every mathematical formula, and even every punctuation mark in an instant.

Factoring integers is an easy to understand and clear problem, but it is not a simple problem.

Relatively speaking, the factorization of smaller integers is a primary school arithmetic problem, but once the number is large enough, such as the factorization of a 50-digit integer, it becomes a super mathematical problem.

If we use the 'trial division method' we learned in elementary school (such as 7M((4M^2)×P^2)÷(7M^2), the result is 4MP^2), even if we use an electronic computer, a person will not be able to do it in his lifetime. come out.

Even assuming that humans have been using computers to decompose this integer using trial division from generation to generation since the birth of the computer, even after centuries since the invention of the computer, this 50-digit number still cannot be decomposed.

Therefore, finding a polynomial that can complete the factorization of large positive integers in a limited time is one of the ultimate dreams of mathematicians in the field of number theory.

Including Xu Chuan himself, he has always been looking forward to someone being able to complete it. Even if it is just a step forward on this road, he is extremely looking forward to it.

".That is, these problems are polynomially equivalent."

"In this paper, we show that all of these algorithmic procedures have polynomial time complexity with respect to the length of the input data, and find a polynomial factorization algorithm that can handle large positive integer factors."

When the last sentence came into view, Xu Chuan, who had been sitting at the desk for who knows how long, finally put down the paper in his hand, breathed a sigh of relief, and rubbed his sore lumbar spine.

Although the proof of this top-level conjecture is not something that can be completely determined after just one reading, judging from the first paper and his mathematical intuition, Liu Jiaxin did it!