Great Country Academician

Chapter 673 P≠NP?

Putting down the paper in his hand, Xu Chuan quietly looked at the title on the home page, recalling the entire reading process.

For people like him, seeing a good paper in a new field is no less than ordinary people eating a delicacy that they have never enjoyed before, which is enough to last a lifetime.

The polynomial decomposition problem of large positive integer factors undoubtedly meets this standard.

In fact, the problem of factoring large numbers is one of the most basic and oldest problems in mathematics, and it is one of the problems that people still pay attention to but cannot be completely solved.

Its importance and difficulty in the field of number theory are not weaker than the existence of the Yang-Mills equation in the field of partial differential equations.

Because large integers may be prime numbers or composite numbers, the prerequisite for solving this problem is to first judge the given large number, determine whether the given number is a prime number (i.e., the primality determination problem) and decompose the large composite number. Decompose large numbers into prime factors in two ways.

In mathematics, it is very similar to the qualitative detection problem, but qualitative detection has been completely proved to be solvable in polynomial time, while the problem of factoring large numbers remains unsolved.

Even, for hundreds of years, the problem of factoring large numbers has not been proven to be a P problem solvable in polynomial time, nor has it been proven to be an NP-complete problem.

However, in the paper in front of him, Xu Chuan saw a detailed answer, or in other words, a path leading to one of the ultimate questions in number theory.

After carefully reviewing the paper in his hand, Xu Chuan opened his eyes, dragged the computer from the corner of the desk, and clicked on the prestige chat box.

"I've read the paper once, it's very good!"

His fingers tapped lightly on the keyboard, and a compliment was transmitted thousands of kilometers across the computer screen.

This was not against his will, but a sentiment from the bottom of his heart.

Although he had known for a long time that she was very talented in mathematics and computers, he never thought that one day she would be able to enter this field.

In academia, or online, when people discuss a subject, if it has high research value and practicality in some aspects, is difficult enough to learn, and has certain difficulties in the job market, It will be called "Tiankeng Professional".

These majors are usually considered to be basic subjects, which are difficult to learn, and their employment prospects and salary packages are often not as good as other majors.

For example, the four most common "biochemical environmental materials" sinkholes.

However, many times, the most basic mathematics major in natural science is rarely recorded, or few people call it a sinkhole major.

It’s not that it’s not hard enough, it’s that it’s too hard.

If other majors are a sinkhole, you can see that there are many people (scholars) at the bottom of the sinkhole struggling to climb up.

The mathematics major is like a cliff, bottomless and shrouded in clouds and mist, so that if you throw something, there will be no response. You can't see how deep it is, nor how many people are inside. You can only see a few big cows flying around above the clouds and mist close to the top of the cliff.

In the words of the mathematical world, these giants flying above the clouds and mist are all gods in the mathematical world.

Xu Chuan himself is the one who flies the highest.

Now, after solving the polynomial algorithm problem of positive integer factorization, Liu Jiaxin has also leapt from the abyss of mathematics to the top of the clouds.

Although this is not a complete solution to P=NP? This millennium problem is just one of the phased results, but its difficulty and impact on the world are extremely great.

Because, in addition to being an important issue in mathematics and computing theory, any kind of proof will be of great significance to mathematics, cryptography, algorithm research, artificial intelligence, game theory, multimedia processing, and even philosophy, economics, and many other fields. have a profound impact on the field.

To switch to another area that arguably touches everyone: “Passwords!”

Nowadays, whether it is a mobile phone, computer, email, etc. that requires information exchange, or anything that involves account security, it all involves the existence of passwords.

In computer cryptography, currently, the most important public key algorithm is RSA.

It is the cornerstone of computer communication security, ensuring that encrypted data cannot be decrypted. RSA encryption is asymmetric encryption and can complete decryption without directly transferring the key.

Simply put, it is a process of encryption and decryption using a pair of keys, called public key and private key respectively.

Assumption: Party A and Party B communicate with each other. Party B generates the public key and private key. Party A obtains the public key and encrypts the information (the public key is public and can be obtained by anyone). Party A uses the public key to encrypt the information.

Only the private key can be cracked, so as long as the private key is not leaked, the security of the information can be guaranteed.

Therefore, it is widely used in various fields, and its security depends on the difficulty of decomposing large integers.

When all the factors of the composite number are very large, it is very difficult to obtain the specific factors using brute force, and this is the core of the RSA system theory.

However, after solving the problem of polynomial algorithm for the factorization of large positive integers, the algorithm of the RSA encryption system can quickly collapse into a 'solution' after finding a method.

What this means is naturally self-evident.

Of course, this is only theoretical. In fact, it is impossible to treat encryption algorithms such as RSA as nothing, even with this paper.

Perhaps when quantum computers mature in the future, and then cooperate with this paper, they will probably truly dominate the field of traditional computers.

As for now, it can only be said that it still needs time to ferment.

But one can imagine how much impact this paper will have on the entire world. Computer communication codes alone will undergo a complete transformation.

Those encryption methods based on traditional positive integer factorization may be abandoned and replaced by various countries.

After all, it's no longer theoretically safe.

Late at night, in the study, the click of authority sounded softly. After sending a message, Xu Chuan made a video call.

After waiting for a while, the video was connected. On the opposite side, Liu Jiaxin, who was also in the study, appeared on the phone, revealing a slender swan neck and pale white pajamas.

Looking at the senior sister on the opposite side of the video, Xu Chuan's eyes naturally fell on the exposed skin that was whiter than her pajamas. He was stunned for a moment and forgot to speak.

Although the two of them often interacted with each other because of company and mathematics matters, the two of them basically met during the day, and there was no such time as seeing each other in pajamas.

On the opposite side, Liu Jiaxin noticed Xu Chuan's gaze, and then realized that she was wearing pajamas at home. She pursed her lips and adjusted the buttons of her coat in embarrassment.

"Cough~"

Xu Chuan came back to his senses, coughed slightly and said: "I have read the paper in detail. So far, it is very excellent! Although I can't say for sure that you have completely solved this problem, after all, it has not yet been solved. It’s been peer-reviewed, but if you ask me to give my opinion, there’s no doubt that you did it.”

"Thank you." On the other side of the video call, Liu Jiaxin said with a smile: "Sorry to trouble you, I'm still asking you to help me so late."

"No, no, no, don't say that!"

Hearing this, Xu Chuan quickly shook his head and said: "This is not a trouble. If it is, then I hope there will be more trouble like this!"

For a mathematician, if he can see such a paper, let alone not sleeping, even if he is woken up by someone while sleeping, he will not have any opinions. He failed to read it in the first place. That would feel like a pity.

Of course, for a girl, this may not be a standard answer.

But it was obvious that neither of them were paying attention to anything other than academic matters at this moment. Both of their thoughts were focused on the paper in their hands.

". Make in-depth changes to the quadratic sieve factorization method, introduce the Hamiltonian graph determination method and the polynomial function algorithm, so that the problem of the existence of complex zero points can be converted into a linear equation system solution problem, and then given Determining the system of equations f1 = 0,..., fk=0 has the complexity of a complex solution algorithm."

".According to Fermat's little theorem, if p is a prime number, then a^(p-1)≡1(mod p) holds for all a∈[1, n-1]. So if in [1.n-1 ] Randomly pick one out and find that it does not satisfy Fermat's little theorem, then prove that n must be a composite number."

"."

During the video call, Liu Jiaxin explained that Dazheng integer factorization has the core and ideas for solving polynomial algorithm problems, while Xu Chuan asked some questions of his own from time to time across the screen.

Although the paper has completely described the proof process of the polynomial algorithm problem of the factorization of large positive integers, reading the paper alone and listening to the creator's explanation based on the paper are two completely different concepts.

If all the problems could be understood by reading the papers, then the mathematical community would not require provers to give lectures after solving these world-class conjectures.

Time ticked by in the middle of the night, and they didn't stop until after midnight.

In the study, Xu Chuan's eyes were bright with some thoughts. After pondering for a moment, he came back from his distraction, looked at Liu Jiaxin on the other side of the video call, and said with a smile:

"It is an excellent proof that sublimating the quadratic sieve factorization method, introducing the Hamiltonian graph determination method and the polynomial function algorithm while reversing the collapse of large integers, this can be said to be a new mathematical tool. Based on the previous work Yes, you did even better than I imagined."

On the opposite side, Liu Jiaxin pursed her lips and shook her head slightly, saying: "But I can't find a method that can transform NP problems into P problems, nor can I solve NP problems and NPC problems."

Looking at the senior student opposite, Xu Chuan smiled and joked: "Thinking of solving P=NP at once? Guess? You are too greedy."

After a slight pause, he continued: "In the P=NP? problem, the problem of polynomial decomposition of large positive integer factors is itself one of the two most difficult problems. If we can solve this, the remaining problems may not be far away. It’s not that far away.”

Opposite me, Liu Jiaxin thought for a while, hesitated and then said: "But I think this problem is still far away, maybe it will never be solved."

Hearing this, Xu Chuan paused, raised his eyebrows in surprise, and asked, "Do you think P≠NP?"

Although he has not studied this problem for a long time and with full concentration, he has naturally explored the few remaining conjectures among the seven millennium problems.

Although it is not very in-depth, to be honest, his view on this issue is not that P=NP, but P≠NP.

That is, there is no simple key that can solve all the problems in this world.

This can be regarded as his implicit mathematical intuition.

Even after reading the proof of the polynomial decomposition problem of large positive integer factors tonight, which showed that P=NP was a big step forward, he still retained his own opinion and felt that P≠NP.

Of course, Xu Chuan never believed that his opinion on an unresolved issue must be right.

After all, he is just a person who has learned a little more knowledge than ordinary people. He is not an omniscient and omnipotent god.

But in P=NP? In terms of difficult problems, or in terms of P-type problems and polynomial decomposition problems of large positive integer factors, the senior student in front of me should be one of the people who have gone the furthest so far, or in other words, the farthest.

What if she thinks P=NP? The conjecture may be incorrect. Combined with the views of most people in the mathematical community and his own intuition, perhaps P=NP does not exist.

That is, NP-type problems can never collapse into P-type problems.

Some people may wonder that since the polynomial decomposition problems of large positive integer factors have been confirmed, why is P not equal to NP? Shouldn't it be pushing one step further towards P=NP?

For this question, can we only say P=NP? The conjecture itself is not a fully defined mathematical problem.

Among the seven millennium problems of the Clay Institute of Mathematics, it is called the ‘Non-deterministic Polynomial problem, that is, the non-deterministic problem of polynomial complexity. ’

P=NP? In the conjecture, P and NP on both sides are not fixed, and it targets endless polynomial and non-deterministic problems. In this case, it is not easy to prove that P≠NP.

If P=NP, you need to ensure that every NP-type problem can collapse to a P-type problem. If P≠NP, then you need to prove that every potential algorithm will fail.

The algorithms and problems here not only refer to the present, but also include everything in the past and future.

So instead of saying P=NP? The problem is a mathematical conjecture, rather it is a way of thinking, a method of classifying and recognizing problems based on their inherent difficulty.

On the opposite side, Liu Jiaxin nodded and said softly: "Well, maybe this problem has no solution. We can neither prove P=NP nor P≠NP."

"I tried to solve an NP-complete problem in the past, but found that it was impossible to find an algorithm that could solve the problem in all situations. I could only try my best to achieve the best results."

Xu Chuan nodded and said with a smile: "It seems that we have reached a consensus."

Smiling, he leaned back in his chair and continued: "If we talk about the problem alone, it is not just the P=NP? problem, there are many problems that are the same, and often we cannot solve it directly. But Many times, the process of studying them is the most essential thing.”

"For example, now, the problem of polynomial decomposition of large positive integer factors has given us a general framework and tools, which helps us think about how to deal with difficult problems arising from actual needs, and can also help us better improve mathematics. and Developments in Other Sciences.”

"And these are the most important!"