"Positive integer factorization with polynomial algorithm solution proof! 》

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

He clicked quickly on the file, downloading it and reasserting his authority.

"Did you testify?"

Fingers quickly tapped a few times on the keyboard of the nine-square grid, and a short message was sent.

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

After the message was sent here, Liu Jiaxin's message over there also came back.

"Well, this method should solve the problem of positive integer factorization, but I'm not sure if there are any flaws in it, I want you to help me see it."

Xu Chuan quickly replied, "Printing, I'll look at it right away."

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

"It's okay, don't worry, you are busy with your business first, and the paper is not in a hurry."

The message on the other side was quickly replied, but Xu Chuan didn't care anymore.

He got up and pulled out the computer from his backpack, opened it quickly, 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. This kind of top-notch math paper, he can't wait to see the specific content.

When I opened it, the main topic of the paper came into view.

"Positive integer factorization with polynomial algorithm solution proof! 》

The title of the paper is very straightforward, that is, the first question in the p=Np?, which is also the difficult problem he and Liu Jiaxin discussed before.

But for p=Np? The problem, he does not understand very deeply.

As one of the 18 major mathematical unsolved problems of the 20th century, the mathematician Smale chose the following Np complete problem derived from traditional mathematical problems as "p=Np?" Representatives of the problem.

"That is, given k polynomials on n variables on Z?, ask if there is an algorithm for polynomial time to determine that they are in (Z?) There is a common zero point on n. This description was largely influenced by Brownwell's algorithm for determining Hilbert's zero-point theorem."

To put it simply, let f1,···,fk be a complex coefficient polynomial of n variables, according to Hilbert's zero point theorem, f1,···,fk does not have a common zero point on the complex number field if and only if there are n variables of the complex coefficient polynomial g1,···,gk satisfies k∑i=1· GiFi=1。

If it is difficult to understand these specialized mathematical languages, p=Np? The problem can be described in more layman's terms, but it can be divided into two parts.

'P Category Questions' and 'Np Category Questions'.

Of course, here are two concepts that are simplified to help understanding, and are simplifications made by a concise and clear understanding that puts aside mathematical rigor and complexity.

p stands for a class of problems that computers can solve very quickly. This speed has nothing to do with the computer hardware, but only depends on the convenience of the solution itself.

Np represents another kind of problem, they have an optimal solution, but for many of these problems, the computer does not have a quick way to find the optimal solution, and even, it can only be stupid and violent, try all possible combinations, and then find the optimal solution.

The most difficult type of NP problem is called NPC, or NP complete problem.

If this is still not specific enough, I believe you can understand it more concisely with a small story as an example.

Let's say you're attending a big banquet and want to know if there's anyone you know inside.

At this time, the host of the banquet said to you that you must know the lady who was standing in the right corner of the dessert table, and you immediately swept there and found that he was right, and you did know her.

So, through

After the information of the host of the banquet, you can easily judge that Ms. A you know.

But if he doesn't tell you that, you'll need to look around the hall and examine everyone before you know anyone.

Through the hint of the host of the banquet, finding Ms. Xiao A is a type P problem;

And you follow his prompts and find out that you know Ms. A, and it is easy to check that Ms. A is the Np problem.

In the reasoning of "The Devotion of Suspect X" by a certain island writer, Ishigami and Yukawa discussed which 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 where it is, and it tells everyone that it usually takes more time to generate a solution to a problem than it does to verify a given solution.

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

But if someone tells you that there are 500 atoms in the world, you can quickly prove him wrong. It's easy to verify, but it's not easy to solve, and this is an Np problem.

P-class problems are a class of problems that can be solved and verified in polynomial time; NP problems are a type of problem that can be verified in polynomial time but are not sure whether they can be solved in polynomial time.

Obviously, all p-class problems are Np-type problems, but it is not possible to determine whether Np is equal to p.

And since "p=Np?" Since it was proposed, many attempts have been made in both the mathematical community and the computer field.

The most obvious way to prove p=Np is to give an algorithm for the polynomial time of a complete problem of Np.

But over the past few decades, a large number of mathematicians and programmers have done a lot of work on algorithms to find polynomial times for the complete problem of Np, without success.

Of course, there is also a large group of people who are trying to give p≠Np? Even in today's mainstream mathematics 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 transformed into p, that is, every problem can eventually be turned into a simple proposition, which can be solved quickly by the computer.

This means that the current mathematical system, computer system, common sense of mankind.... And so on all kinds of things will be subverted.

If eventually p=Np is proven, we can turn any Np problem into a p problem. Problems that seem difficult right now can be easily solved.

For example, Go has the ultimate solution, the genetic code can be easily cracked in the biological field to manipulate the gene sequence at will, many mathematical conjectures can be calculated and deduced by computers, and a large number of problems are solved.

At the same time, such as p=np, this will lead to the complete failure of all encryption algorithms in the next short period of time, your bank card, mobile phone password, social accounts will no longer be secure, hackers can easily enter your computer, bitcoin, blockchain, these concepts that have been very popular in recent years will become a field that no one cares about.

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

If such a key really exists, it probably already exists in this universe.

For example, humans may already have the ability to see everything once, or a creature may be born without having to fight to survive because their algorithms are so good that they can survive in the most efficient way possible in any environment.

But intuitively, philosophically, religiously, and scientifically, it is difficult for people to believe that such a shortcut to the universe exists.

To be honest, Xu Chuan doesn't believe that there will be such a 'all-purpose' key in the universe, but when it comes to p=Np? Even if it is phased, he will devote the most concentrated energy to deal with it.

.......

The papers on the computer screen kept flipping, and lines of mathematical formulas and interpretations crossed Xu Chuan's eyes.

At that moment, a doorbell rang outside the room.

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

"Professor, here's what you want."

handed over the paper with Yu Wen and the fragrance of ink, and Tang Sijia added: "There is a stack of unused A4 paper under the paper, which can be calculated for you."

Although I know that Xu Chuan usually carries a pen and some manuscript paper with her, there is no doubt that what can be printed out as quickly as possible is extremely important.

Therefore, she was worried that the amount of manuscript paper she carried with her was not enough, so she directly pulled a stack of blank A4 characters from the printing room and sent it over.

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

"Great, thank you!"

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

On the other side, he didn't hear what his little assistant said clearly, so Xu Chuan waved his hand anxiously, and quickly returned to the study room of the hotel room with the paper and manuscript paper, and didn't even bother to close the door.

Outside the door, the smile on Tang Sijia's face stiffened, and then silently closed the door, and turned to leave while blessing in his heart.

Although she couldn't read the printed paper, out of curiosity, she searched for the title of the paper on her mobile phone during the free time of printing.

And the title of this paper seems to involve p=Np, one of the seven millennial problems? Guess.

As Xu Chuan's assistant, although she is not a mathematics major, she also knows some things in the field of mathematics, and is well aware of the weight of each millennial problem and its impact on the country and the world.

The solution of any millennial problem can greatly promote the development of mathematics, and even other disciplines, and even society as a whole.

Just like the NS equation, although she can't read the proof, or even understand the meaning of the NS equation, she knows very well that the solution of controlled nuclear fusion technology is based on the NS equation.

I hope the professor can also solve the p=Np smoothly this time? Puzzle.

Looking at the back that turned to enter the study, Tang Sijia silently prayed in her heart.

......

In the study, Xu Chuan didn't know that the little assistant outside still had so many thoughts, and at this moment his attention was all focused on the paper in his hand.

Rather than reading papers on a computer screen, he prefers this kind of knowledge that can be weighed by hand.

[Interpretation: This paper gives a method for determining or solving a P-class problem in polynomial time with a deterministic algorithm and its polynomial time determination algorithm. The upper bound of the number of terms of the Gi in the Boolean polynomial (1) of the complex Boolean polynomial (1) of the complex solution algorithm of the system of decision equations f1=0,····,fk=0 is given.......】

「..... This is aimed at exploring the barrier between the complexity categories of p and Np, and in a previous paper [1], we have shown that satcNF problems can be polynomially reduced to problems that find special coverage of a set under a special decomposition of that set, and vice versa."

「..... Definition 1: G= is a labeledultistagegraph if the following conditions are met:

is a vertex set, V=VUnUVu... UV，VnV=0，0≤ij≤L，i≠j。 If uV, 0≤i≤L, the level where u is located is called level i, and you are also called the vertex of level i. L is called the level of G.

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. Calls (u,v,l) the edge of the lth order of G.

3. AND contains only unique vertices. The only vertex in is called the source point, which is denoted as S, and the only vertex in is called the sink point, which is denoted as d......"

4........

.......

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

The factorization of integers is an easy and clear problem, but it is not a simple problem.

Comparatively speaking, factorization of smaller integers is an elementary school arithmetic problem, but once a large enough number, such as a 50-digit integer factorization problem, is a super math problem.

If you use the 'trial division' method that you learned in elementary school (e.g., 7((4^2)xp^2)÷(7^2), the result is 4p^2), even if you use an electronic computer, you will not be able to do it for the rest of your life.

Even if we assume that human beings have used electricity from generation to generation to use computers to decompose this integer by trial division, even after centuries since the invention of computers, this 50-digit number still cannot be decomposed.

Therefore, finding a polynomial to 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 to push one step further on this road, he is extremely looking forward to it.

「..... That is, these questions are polynomially equivalent."

"In this paper, we demonstrate that all these algorithmic processes have the time complexity of the polynomial relative to the length of the input data, and find a polynomial factorization algorithm that can handle the large positive integer factor."

When the last sentence came into view, Xu Chuan, who was sitting at the desk for an unknown amount of time, finally put down the paper in his hand, relieved the turbidity in his chest, and rubbed his somewhat sore lumbar spine.

Although the proof of this top-level conjecture is not something that can be completely determined by looking at it once, judging from the paper of the first time, judging from his mathematical intuition, Liu Jiaxin has done it!

........