Chapter 127 Proof of the Twin Prime Conjecture
As soon as Xu Yuan entered the office, he saw Professor Tang and Zhang Yitang sitting on the sofa drinking tea and chatting. They seemed to be quite familiar with each other.

"Why are you standing at the door? Come in." Tang Shihong noticed Xu Yuan at the door, put the teacup on the table and greeted him quickly: "Professor Zhang came to see you this time. He has been waiting for a while."

"professor Zhang."

Following Professor Tang's words, Xu Yuan politely greeted Zhang Yitang and walked over to sit on the sofa next to him.

Zhang Yitang smiled and waved his hand, saying, "Just call me Lao Zhang. It's more intimate this way."

Xu Yuan coughed lightly.

"Lao Zhang" was how he addressed Zhang Yitang in the email. The main reason was that the other party strongly requested that they discuss number theory methods as peers. But it was obviously inappropriate to call him that in front of Professor Tang, so he changed the subject in a second after his mind worked quickly.

"Is there something urgent you want to talk to me about?"

"Originally, I wanted to come to you directly, but Old Tang said it would be more convenient this way. I don't know if I have wasted your time." Zhang Yitang said a little embarrassedly.

Hearing this, Tang Shihong smiled and added: "Xu Yuan often comes to my place, it's not a big deal."

"That's indeed true." Xu Yuan nodded in acknowledgement.

After a few brief exchanges, Zhang Yitang no longer kept the topic in suspense, and after organizing his words a little, he stated his purpose.

"To be honest, I have been studying twin prime numbers since the last time we discussed the application of sieve method in prime numbers in an email."

"By improving the sieve method you used to prove the interval between infinite Carmichael numbers and applying it to the problem of twin primes, I obtained the result that there are infinite pairs of twin primes whose difference is less than 4200 million."

"I would like you to help verify the proof process."

Zhang Yitang said this calmly, then leaned over and took out a printed copy of his paper from his briefcase.

Little did he know that Tang Shihong next to him had his eyes wide open with shock on his face.

He stood up from the sofa and asked, "Professor Zhang, are you saying that you have proved the twin prime conjecture, the ultimate problem in number theory?"

There was an expression of disbelief in his tone.

The twin prime conjecture was proposed by Hilbert at the International Congress of Mathematicians a hundred years ago. It is, along with the Goldbach conjecture, the ultimate problem in number theory and is well-known in the mathematical community.

But no one has ever solved it.

If Zhang Yitang's proof that there are infinite pairs of prime numbers whose difference is less than 4200 million is true, it would be a big step forward on the important issue of the twin prime conjecture.

It’s no wonder that Tang Shihong was so surprised by this incident.

The main reason is that Zhang Yitang had no mathematical research results before.

It is inevitable to feel a little surprised when a well-known problem like the twin prime conjecture is suddenly solved.

"Whether the results are valid still needs to be verified," Zhang Yitang replied modestly.

His ability to prove such a result was closely related to the sieve method used by Xu Yuan to solve the problem of infinite Carmichael number intervals, so it was most appropriate to ask Xu Yuan to help verify the result.

At this time, Tang Shihong figured out the situation, calmed down his excitement a little, sat down, and looked at Xu Yuan.

But he didn't see any surprise on Xu Yuan's face.

Still remain calm.

Just like before.

A question mark suddenly appeared in my mind.

In his opinion, not to mention students, even internationally renowned mathematicians would be moved to hear the news, but Xu Yuan remained very calm.

It was like he had known about this for a long time.

But considering the current situation, he didn't ask too many questions.

Xu Yuan was naturally not surprised that Zhang Yitang solved the twin prime problem. He just didn't expect that because of his paper, the other party would be able to solve it so many years ahead of time and directly prove 4200 million prime pairs.

He was naturally familiar with the twin prime conjecture.

For all natural numbers k, there are infinitely many prime pairs (p, p + 2k)

The case where k = 1 is the twin prime conjecture.

When discussing this issue with Lao Zhang previously, he tried to prove it but found it difficult to solve it completely.

The thought stayed here, and then he no longer hesitated. He reached out and took the printed manuscript of the paper from Lao Zhang and started reading.

In response, Tang Shihong and Lao Zhang sat in their original positions and waited patiently, and even their movements became much lighter.

Afraid of disturbing Xu Yuan's thoughts.

Speaking of which, Xu Yuan felt a little bit weird at the moment. Before, he had published papers for review by the editorial board of Annals of Mathematics, but now he was experiencing the feeling of being a reviewer.

After all, this was his first time reviewing a math paper.

The incident with Lu Chunming doesn’t count.

Soon Xu Yuan was completely immersed in the thesis, as if he was isolated from the outside world, automatically blocking out any noise coming from the surroundings.

It can be said that even if Tang Shihong and the others had a conversation, it would not have any impact on him.

that's it.

As time passes by.

Xu Yuan flipped through many pages of the printed manuscript of the thesis in his hand, with a solemn and serious expression on his face.

He had to admit that Lao Zhang had done in-depth research on prime number pairs.

Although the paper used his sieve method to solve the problem of the interval between infinite Carmichael numbers, it was further improved and innovated, otherwise it would not be sufficient to complete the proof of prime number pairs.

Another twenty minutes passed. Xu Yuan suddenly stopped what he was doing and looked at Tang Shihong next to him.

“Do you have any scratch paper and a pen?”

"I'll get it for you." Upon hearing this, Tang Shihong hurriedly walked to the desk, took out the draft paper from the drawer and handed it to Xu Yuan.

There can be no delay in the whole process.

After Xu Yuan got the draft paper and pen, he bent over and started to calculate to verify Lao Zhang's result.

As for this, Old Zhang stared at Xu Yuan intently, and couldn't help but worry for himself. Although he was very confident in the proof process in this paper, he was still afraid that he would get the news that the result was not valid in the end.

He has devoted a long time to the study of number theory and never gave up even though he had not made any progress.

It is generally recognized in the mathematical community that most difficult problems are solved by mathematicians when they are young.

If you have no achievements after you are forty, then there is a high probability that your research in mathematics has come to an end.

After all, as we age, our physical functions and mental activity will decline to varying degrees, and even the most talented will gradually disappear.

He is already over fifty years old. If he still cannot solve the twin prime numbers this time, there is no hope.

After the two of them had been waiting anxiously for nearly an hour, they saw Xu Yuan finally stop, put the pen on the table and breathed a sigh of relief.

"How is it? Is the result true?" Lao Zhang asked impatiently.

Although Tang Shihong didn't say anything, his eyes were on Xu Yuan, afraid of missing a word.

But Xu Yuan frowned and didn't know what he was thinking, so he didn't respond.

Old Zhang was scared when he saw this scene. He thought that there was a problem in his proof process that led to the invalid result. Before Xu Yuan could speak, he asked again.

"Isn't it true?"

Hearing this, Xu Yuan came back to his senses and shook his head, saying, "According to my verification results, it is true."

As soon as he finished speaking, Lao Zhang and Tang Shihong both breathed a sigh of relief, and their faces were instantly replaced by joy.

"Congratulations, Lao Zhang."

"Solving the twin prime conjecture is a great academic achievement for mathematicians."

"You are a late bloomer."

Tang Shihong then poured another cup of tea for Lao Zhang and sincerely congratulated him.

Thinking that he had only achieved some results in the twin prime conjecture when he was over fifty years old, it was very appropriate to say that he was a late bloomer. He smiled modestly and quickly took the teacup.

"This is all thanks to Xu Yuan for solving the Carmichael number problem, otherwise I wouldn't have these ideas."

I picked up the teacup and was about to taste it when I heard the voices of Tang Shihong and Xu Yuan talking.

"Since the result is established, why are you still frowning?"

Xu Yuan naturally did not keep the secret and answered directly: "Although the proof is valid, the value can be reduced by 4200 million prime numbers."

He wasn't saying this randomly.

The twin prime conjecture can be weakened to whether it is possible to find a positive number such that the difference between infinite pairs of prime numbers is less than this given positive number. In the twin prime conjecture, this positive number is 2.

The positive number that Lao Zhang found was 4200 million.

Although it cannot be said to be a true proof of the twin prime conjecture, it is an important milestone.

This marks a major step forward in the mathematical community's research on the twin prime conjecture.

When Xu Yuan was verifying the calculation just now, he combined Lao Zhang's method with his previous idea of ​​proving the infinite number of Carmichael numbers, and many inspirations came to his mind.

Perhaps through further calculations, we can successfully find a positive number less than 4200 million.

If it succeeds, then we will be closer to the final positive number 2.

Until the twin prime conjecture is completely proved.

As Xu Yuan's words reached Zhang Yitang's ears, he decisively put down the teacup in his hand and confirmed with Xu Yuan with great excitement.

"Are you telling the truth?"

"Can the number 4200 million be reduced?"

In studying the twin prime conjecture, his goal was naturally to find the positive number 2.

However, the calculation cannot go any further after reaching 4200 million.

If Xu Yuan can continue to reduce this number, the value of this paper will also increase accordingly.

Xu Yuan could naturally understand Lao Zhang's excitement at the moment. With the burst of inspiration, he also wanted to try to what extent he could reduce the value, so he smiled and nodded.

"We can give it a try."

Seeing Xu Yuan's confident look, Lao Zhang no longer hesitated and quickly stood up and walked over to sit next to Xu Yuan.

Prepare to try the calculation again with Xu Yuan, looking for a positive number smaller than 4200 million.

Tang Shihong, who had already voluntarily made way for this, looked at the old man and the young man who had already entered the state. After a little hesitation, he finally decided to wait outside temporarily, leaving his office for them to use, so as not to disturb their thoughts.

Since his main research area is differential geometry, he really can't help with the twin prime conjecture.

……

(End of this chapter)