Chapter 673 "Proof that the Factorization of Large Positive Integers Has a Polynomial Algorithm!"
《Proof that the factorization of large positive integers has a polynomial algorithm!》
Looking at the file sent by Liu Jiaxin on his phone, Xu Chuan was stunned for a moment, then reacted.
He quickly clicked on the file, downloaded it and opened WeChat.
"Have you proved it?"
His fingers quickly tapped on the keyboard of the nine-square grid, and a short message was sent out.
At the same time, he quickly sent the file to his assistant and sent a message: "Please print this file as quickly as possible and send it to my room."
After the message was sent, Liu Jiaxin's message came back.
"Well, this method should be able to solve the factorization problem of large positive integers, but I'm not sure if there are any defects in it. I would like to ask you to help me take a look."
Xu Chuan quickly typed back: "Printing, I'll take a look here right away."
After a pause, he added: "I'll go back tomorrow afternoon."
"It's okay, don't worry, you should do your thing first, don't worry about the paper."
The message from the other side was quickly replied, but Xu Chuan didn't care anymore.
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 computer screen was always larger than the mobile phone. He couldn't wait to see the specific content of this top-level mathematical paper.
Opening it, the main title of the paper came into view.
"Proof that the factorization of large positive integers has a polynomial algorithm!"
The title of the paper is very straightforward, which is the first question in the P=NP? problem, and it is also a difficult problem that he and Liu Jiaxin discussed before.
However, he did not have a deep understanding of the P=NP? problem.
As one of the 18 major unresolved mathematical problems in the 20th century, mathematician Smale 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 n variables, ask whether there is a polynomial time algorithm to determine whether they have common zeros on (Z)n. This description is mainly influenced by Brownwell's algorithm for determining Hilbert's zero theorem."
In simple terms, let f1, ···, fk be complex coefficient polynomials with n variables. According to Hilbert's zero theorem, f1, ···, fk do not have common zeros on the complex field if and only if there are complex coefficient polynomials g1, ···, gk with n variables that satisfy 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 in relatively popular terms.
'P-type problems' and 'NP-type problems'.
Of course, these are two concepts that are simplified to help understanding. They are simplifications made by putting aside the rigor and complexity of mathematics and making simple and clear understandings.
P represents a class of problems that computers can solve very quickly. This speed has nothing to do with computer hardware, but only 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, there is no fast way for computers to find the optimal solution. In fact, they can only foolishly and violently try all possible combinations to find the optimal solution.
Among NP problems, the most difficult type of problem is called NPC, which is an NP-complete problem.
If this is still not specific enough, I will use a small story as an example, and I believe you can understand it more simply.
Suppose you are attending a grand banquet and want to know if there is anyone you know there.
At this time, the host of the banquet said to you that you must know Ms. A who is standing in the right corner of the dessert table, so you immediately scan there and find that what he said is right, you do know her.
Therefore, through the information of the banquet host, you can easily judge that you know Ms. A.
But if he doesn't tell you this, you need to look around the hall and examine everyone before you know if you know anyone.
Finding Ms. A through the hint of the banquet host is a P-type problem;
And you follow his hints and find that you know Ms. A, and it is easy to check that Ms. A is an NP problem.
In the mystery novel "The Devotion of Suspect X" written by a certain island country writer, Ishigami and Yukawa once discussed which is more difficult, solving a proposition or judging whether a proposition is correct.
In fact, the mathematical community has already given the answer, P=NP? The problem is there, it tells everyone that it usually takes more time to generate a solution to a problem than to verify a given solution.
For example, if you are asked to calculate the sum of the number of all atoms in the world, this problem is very difficult or even unsolvable.
However, if someone tells you that there are 500 atoms in the world, you can quickly verify that he is wrong. It is easy to verify, but not easy to solve. This is an NP problem.
P problems are a type of problem that can be solved and verified in polynomial time; NP problems are a type of problem that can be verified in polynomial time but is not sure whether it can be solved in polynomial time.
Obviously, all P problems belong to NP problems, but it is impossible to determine whether NP is equal to P.
Since the proposition of "P=NP?" was put forward, many attempts have been made in both the mathematics and computer fields.
To prove that P=NP, the most obvious way is to give a polynomial time algorithm for an 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 none of them have succeeded.
Of course, there are also a large number of people trying to give P≠NP?, and even in today's mainstream mathematics and computer industries, 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 difficult problem can eventually become a simple proposition that can be quickly solved by computers.
This means that the current mathematical system, computer system, common sense, and other aspects of human beings will be overturned.
If P=NP is finally proven, we can transform any NP problem into a P problem. Those problems that seem difficult now can be easily solved.
For example, Go has an ultimate solution, and in the biological field, the genetic code can be easily cracked to manipulate the gene sequence at will. Many mathematical conjectures can be calculated and deduced by computers, and a large number of difficult problems have been solved.
At the same time, if P=NP, this will cause all encryption algorithms to be completely invalid in a very short time in the future. Your bank card, mobile phone password, and social account will no longer be safe. Hackers can easily enter your computer. Bitcoin and blockchain, which have been very popular in recent years, will become a field 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 the world.
If such a key really exists, it probably already exists in this universe.
For example, humans may have the ability to understand everything after seeing it once, or some creatures do not have to fight for survival when they are born, because their algorithms are extremely excellent and can survive in any environment in the most efficient way.
But whether from intuition, philosophy, religion, or science, it is difficult for people to believe that such a cosmic shortcut exists.
To be honest, Xu Chuan does not believe that there is such a "universal" key in the universe, but when it comes to P=NP? Even if it is a temporary proof, he will devote the most concentrated energy to deal with it.
The paper on the computer screen kept flipping, and lines of mathematical formulas and interpretations flashed through Xu Chuan's eyes.
At this moment, the doorbell rang outside the room.
Quickly getting up, Xu Chuan walked through the bedroom and opened the door. At the door, Tang Sijia, his life assistant who accompanied him on a business trip, was standing at the door, holding a thick stack of freshly printed documents in her hands.
"Professor, this is what you want."
Handing over the paper with residual warmth and ink fragrance, Tang Sijia added: "There is a stack of unused A4 paper under the paper, which can be used for calculations."
Although she knew that Xu Chuan usually carried a pen and some manuscript paper with him, it was undoubtedly extremely important for her to print out the things at the fastest speed.
Therefore, she was worried that the amount of manuscript paper he carried with him was not enough, so she directly took a stack of blank A4 paper 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 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, without hearing clearly what his assistant said, Xu Chuan waved his hand impatiently, holding the paper and manuscript and quickly returned to the study in the hotel room, without even closing the door.
Outside the door, Tang Sijia's smile on her face froze for a moment, and then she closed the door silently, turned around and left while blessing in her heart.
Although she couldn't understand the printed paper, out of curiosity, she searched 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? Conjecture, one of the seven millennium problems.
As Xu Chuan's assistant, although not a mathematics major, she more or less knows something about mathematics, and is very clear about the weight of each millennium problem and its influence on the country and even the world.
The solution of any millennium problem can greatly promote the development of mathematics, other disciplines, and even the entire society.
Just like the NS equation, although she could not understand the proof and even did not understand the meaning of the NS equation, she knew very well that the solution of controlled nuclear fusion technology was based on the NS equation.
I hope the professor can also successfully solve the P=NP? problem this time.
Looking at the back figure turning into the study, Tang Sijia silently prayed in her heart.
In the study, Xu Chuan did not know that the assistant outside had so many thoughts. At this moment, his attention was all focused on the paper in his hand.
Compared with reading papers on the computer screen, he prefers this kind of knowledge that can be weighed by hand.
[Explanation: This paper gives a method for a P-type problem that can be determined or solved in polynomial time by a deterministic algorithm and its polynomial time determination algorithm. The complexity of the algorithm for determining the existence of a complex solution for the equation set f1 = 0, ···, fk=0 is given. The upper bound of the number of terms in gi in the Boolean polynomial (1) is given]
“.This is aimed at exploring the relationship between the complexity classes of P and NP. In previous papers [1], we have proved that the sat CNF problem can be polynomialized into the problem of finding a special cover of a set under a special decomposition of the set, and vice versa.”
“.Definition 1: G = is called a labeled multistage graph if the following conditions are met:
1. V is a vertex set, V=VUЙUVu…UV, VnV=0, 0≤ij≤L, i≠j. If uV, 0≤i≤L, the level where u is located is called level i, and u is also called a 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. (u, v, l) is called the l-th level edge of G.
3. Both contain only unique vertices. The unique vertex in is called the source point, denoted by S, and the unique vertex in is called the sink point, denoted by D”
4
The paper in his hand flows through his eyes, and Xu Chuan reads every sentence, every mathematical formula, and even every punctuation mark without blinking.
The factorization of integers is an easy-to-understand and clear problem, but it is not a simple problem.
Relatively speaking, factoring a smaller integer is an elementary school arithmetic problem, but once a sufficiently large number, such as a 50-digit integer, the factoring problem becomes a super mathematical problem.
If you use the "trial division method" learned in elementary school (such as 7M((4M^2）×P^2)÷(7M^2), the result is 4MP^2), even with an electronic computer, a person will never be able to do it in his lifetime.
Even if it is assumed that humans have used computers to use trial division to decompose this integer from generation to generation since their birth, even after several centuries from the invention of computers to the present, 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 one step forward on this road, it is extremely anticipated.
".That is, these problems are equivalent in polynomials."
"In this paper, we prove that all these algorithmic processes have polynomial time complexity relative to the length of the input data, and find a polynomial decomposition 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 an unknown period of time, finally put down the paper in his hand, took a deep breath, and rubbed his aching lumbar spine.
Although the proof of this top conjecture is not something that can be completely confirmed after reading it once, from the first reading of the paper, according to his mathematical intuition, Liu Jiaxin did it!