Technology invades the modern world

Chapter 194 Claritch Hotel: Enlightenment Through Prime Numbers (5.4k)

Any mathematical problem that relates to Hilbert's 1900th century question is a hot topic in the field of mathematical research.

As mentioned earlier, in cutting-edge mathematical research, finding problems is far more important than doing them.

Finding suitable problems and gradually feeding them to young scholars, allowing them to advance step by step, and leveling up along the way in mathematical research is even more difficult.

Questions like Hilbert's "Question of the Century" can become the final boss, and related questions can be set up with this as the goal in mind.

This is why the question of the century is so popular.

This is especially true in Göttingen.

The century-old question left by Hilbert is a great treasure trove that the Göttingen school has contributed to the world of mathematics, and it is certainly right that everyone should explore it.

But the Göttingen school should be able to extract the richest part. Compared to other universities, Hilbert's original manuscripts and notes were all kept in Göttingen, and in the 2000s, Rüdiger Thiele even unearthed the 24th problem from Hilbert's original manuscripts and notes.

As a result, the University of Göttingen managed to uncover some treasures in the first half of the year, but found nothing in the second half.

Under the leadership of Siegel, the Göttingen School's main focus is on twin primes. All six professors present here have some knowledge of this problem, and Siegel has thought about it in depth.

The result, obviously, was that they had no ideas.

Hearing them say they'll resolve this within six days is truly a pipe dream.

“Randolph, I know you’re exceptionally talented, but shouldn’t you leave yourself a way out?” Siegel reminded him. “You know, when you give an academic presentation in Göttingen, there will definitely be a lot of reporters there, even if we don’t let them into the venue.”

Your on-site proof of the twin prime conjecture will be announced to the public by the students and professors present.

We can't get them to only talk about successes and not failures.

Do you want to reconsider?
"Returning to Göttingen to present an academic report as soon as I achieve results will also be a way of supporting Göttingen."

Siegel naturally had to consider Lin Ran's feelings; he truly regarded Lin Ran as his student and the successor to his academic career.

He was well aware that if a scholar who had never failed before were to produce such a monumental work and then fail, he would face ridicule from the outside world and internal turmoil.

Siegel doesn't believe that hardship helps you grow. Top mathematicians and top scientists face hardships from life, and they forge ahead in the academic field.

Even after Euler went completely blind, it did not affect his work speed. After becoming completely blind in 1766, he still produced a large number of highly original papers.

Needless to say, Gauss was another example. When Hilbert was young, Paul Goldan said that he was doing theology rather than mathematics, but in the end, his conclusion was proven to be correct.

According to Siegel, mathematical geniuses, especially young scholars who make outstanding contributions in their youth, should maintain this indomitable spirit, break through numerous obstacles to produce a large number of results, until they stop in front of an unprecedented problem and then slowly think about how to break through.

Siegel did not want to see Göttingen's genius fall victim to such arrogance.

Lin Ran smiled and said, "Of course, Professor, I'm not 100% sure."

I was also fully prepared for failure.

I made this decision after careful consideration, not just for my own sake, but also for the revitalization of Göttingen in the mathematical community.

If I succeed, I will have made a significant contribution to the history of the University of Göttingen. This is a moment worth writing about in the history of mathematics, and future generations will inevitably mention this event at the University of Göttingen when discussing the 20th century.

If I fail, so be it; the professor's first failure in his life was in Göttingen, a significant chapter in his life.

Apart from Siegel, the other five professors were moved to tears.

They could sense Lin Ran's deep affection for the University of Göttingen; he truly was a talent nurtured by Göttingen.

Doiling said, "Okay, I'll go back to Göttingen to prepare. Randolph, on behalf of Göttingen, I thank you for your efforts."

I'm ready to expect a miracle.

Since Lin Ran had said that, Siegel didn't refuse. He just sighed and said, "Randolph, you can think about it in advance. I'll still be in London for a while."

When I was younger, I also pondered the twin prime conjecture. Although I didn't solve it, I have some preliminary thoughts that might offer you some ideas.

He turned to Doyling and said, "Doyling, could you please tell your student in Göttingen to look for a thick notebook on the third shelf of my office bookshelf? It contains Goldbach's Conjecture. Have him send that notebook to London."

After saying that, Siegel continued to say to Lin Ran: "Randolph, Goldbach's conjecture and the twin prime conjecture are both related to the distribution and density of prime numbers."

Goldbach's conjecture focuses on the sum of primes, while the twin prime conjecture focuses on the specific intervals between primes.

Both rely on tools from analytic number theory, and I've been wondering if we can use a common framework to study their properties.

If the twin prime conjecture holds true, it could support Goldbach's conjecture because it suggests that primes are densely packed at certain intervals, which helps in constructing the desired sum of primes.

So I think this might give you some inspiration.

Siegel had a very strange feeling.

They will spend another five days together in London.

There are five days left until my speech in Göttingen.

His relationship with Lin Ran was one of teacher-student relationship first, and then teacher-student relationship in practice.

He first obtained his doctorate, and then, taking the opportunity of proving the twin prime conjecture in London, he provided Lin Ran with some guidance.

It's a feeling of being out of place in time and space.

The supervision period begins after the doctoral degree, initially in London, with the final defense taking place in Göttingen.

Yes, Siegel now feels that they went to Göttingen to defend their doctoral dissertations.

Thinking of this, Siegel couldn't help but laugh. For the wonder of fate, he no longer opposed the matter, but hoped to do everything possible to help Randolph solve the twin prime conjecture.

"Randolph, we only have five days, so I hope to tell you all my thoughts on the twin prime conjecture."

The next day, only Lin Ran and Siegel remained.

The twin prime conjecture posits that there exist an infinite number of pairs of primes whose difference is 2, such as 3 and 5, or 11 and 13.

Based on calculations, twin primes appear to keep appearing as the numbers get larger.

Furthermore, based on the probability that both numbers are prime, there is a heuristic argument. The heuristic suggests that the number of twin prime pairs up to x is approximately C multiplied by the integral of dt/(log t)^2 from 2 to x, where C is the twin prime constant.

I discussed this with Hardy when I was in Cambridge. He and Littlewood were very confident in the validity of this conjecture based on their work on the circle method, but this is not a proof; it is a conjecture, merely a probabilistic model they proposed.

I've done some deeper thinking about this since then. Brun's theorem states that the sum of the reciprocals of twin primes converges, which means that twin primes are relatively sparse compared to all primes, but it doesn't tell us whether they are finite or infinite.

The sieve method might be able to solve this problem. It can be used to prove that there are infinitely many integers n such that n and n+2 have very few prime factors, and then perhaps it can be refined to prove that they are prime numbers.

This is a reasonable direction, after all, sieves have been very successful in studying near-prime numbers, and Selberg's sieve, for example, was used to estimate the number of integers with certain properties.

However, applying this directly to twin primes is challenging because the twin prime conjecture requires both n and n+2 to be prime, which is a more stringent condition.

In recent years, I've been thinking about whether using analytical methods like the L-function would be more appropriate.

After all, the L function is also a powerful tool, especially in problems involving arithmetic series.

This is simply because it doesn't directly apply to twin primes. I think we could consider the Dirichlet series, which captures the distribution of twin primes. The circle method pioneered by Hardy and Littlewood might offer some insights, even if it doesn't provide a complete proof.

I need to elaborate on the circle method; you are also a master in the field of number theory, so you must be very familiar with these cutting-edge methods.

For Goldbach's conjecture, which concerns the expression of an even number as the sum of two primes, the circle method provides an asymptotic formula for representing the quantity under certain assumptions.

Similarly, for twin primes, one can try to calculate the number of primes p up to x such that p+2 is also a prime.

Although the error term in the circle method is usually too large to conclusively prove the conjecture for all x, it is a valuable tool for understanding expected behavior.

Moreover, even if you cannot prove the complete twin prime conjecture in six days, some of the results are still very valuable.

Even if it can be proven that there are infinitely many prime numbers p such that p+2 has at most k prime factors, this would still be a significant step forward.

We don't necessarily need to completely solve the twin prime conjecture in one go.

Even if we only achieve this much, in my opinion, it is a great achievement.

Don't put too much pressure on yourself.

"Take a look at my manuscript once it arrives. We can discuss any questions you may have." Lin Ran grinned. "Okay, Professor."

The special program about Lin Ran and Korolev's moon landing became one of the hottest news stories in the world after it aired.

Newspapers were analyzing the back-and-forth and subtext of the two men's interviews. The liberal camp unanimously cheered for Lin Ran, feeling that the professor's words were impeccable and had exposed the hypocrisy of Soviet Russia.

The Soviet bloc's attacks focused on America, dredging up the Bay of Pigs invasion, the Cuban Missile Crisis, the Berlin Crisis, and the death of Kennedy, attempting to wage a propaganda war from the perspective that "I am no good, but you are no good either."

Judging from the public opinion battle, it seems as if the participants in the program are not Lin Ran and Korolev, but America and Soviet Russia.

Similarly, this war of public opinion has made insightful people realize that peace is still a long way off.

Neither side highlighted Lin Ran and Korolev's remarks on peace and space cooperation in the program as a key part of their reporting.

Lin Ran was going back to the University of Göttingen to give an academic report, which would be a live proof of the twin prime conjecture, and it quickly became the hottest news in Göttingen.

After returning to Göttingen, Deulin made phone calls to invite masters in Europe and even the field of American number theory. His gimmick was that Lin Ran would talk about his thoughts on the twin prime conjecture.

He didn't say that Lin Ran needed to provide on-site proof, but only emphasized that you would regret it if you didn't come.

Because it is the New Year holiday, many scholars are unwilling to travel all the way to Göttingen, while many others are willing to come.

Attending an academic lecture by Lin Ran is a very worthwhile thing for scholars whose travel and accommodation expenses are reimbursed and who only need to pay time costs.

To the scholars in Göttingen, Doilin was telling them that Lin Ran would be proving the twin prime conjecture on the spot, so they should be prepared and not fall behind.

This academic lecture was leaked to the media by local scholars. Göttingen, as a university town, has a highly educated population, and many locals know about the twin prime conjecture.

It caused a sensation in the local area.

Not only did students give up their holidays to attend the academic lectures, but many residents also hoped to come and witness this historic moment.

Unlike the professors, most of these residents believed that Lin Ran could do it.

If they've already accomplished the moon landing, proving the twin prime conjecture will be a piece of cake.

Lin Ran is going to Göttingen. Who in the world is most anxious about this? It must be Professor Fox.

On the third day, after he had figured out the whole story through his connections in Göttingen, he made a transoceanic call to Lin Ran's hotel:

"Randolph, we can't let this opportunity slip away to the University of Göttingen!"
You're a professor at Columbia University; proving the twin prime conjecture live should be done at Columbia University!

Professor Fox was almost in tears.

Because he had witnessed Lin Ran explaining the proof of Fermat's Last Theorem, Fox was clearly more convinced of the idea that geniuses are omnipotent than Siegel.

The culture of genius worship is extremely serious in the mathematics community.

Siegel had his doubts, partly because he was worried about affecting Lin Ran, and partly because he had studied the issue himself.

Fox hasn't done that.

“Professor Fox, I’m not sure I can prove it yet,” Lin Ran explained.

Fox insisted, "No, Randolph, I believe you can."

Others may not be able to do it, but you definitely can.

Legendre spent 40 years trying to solve the problem of the circumference of an ellipse using the integral method in the 19th century, but he still couldn't find a solution.

At the age of 20, Abel first solved the problem of solving algebraic equations of degree higher than 4 that had plagued the mathematical community for 250 years, and then directly solved the problem of solving elliptic integrals with a paper entitled "On the General Properties of a Very Wide Class of Transcendental Functions".

In the field of mathematics, the gap between geniuses and ordinary people is far greater than in any other field. Randolph and the Germans didn't believe it because they were too far from the center of the world. The Americans were different.

I have witnessed your magic too many times, Randolph, and I completely believe you can do it.

Not only do I believe it, but as far as I know, mathematicians from Princeton, NYU, and our own university are already teaming up to witness a miracle.

I have only one request: could such a miracle happen in Colombia?

Lin Ran sighed: "Just this once, after all, I was born in Göttingen but I haven't made any contributions to Göttingen."

Fox sighed, "Okay, I understand. I'll arrange for the academic secretary to prepare the tickets. Our entire Columbia Mathematics Department will come to witness this historic moment."

Lin Ran shook his head. In fact, he had done very little preparation. He knew that Zhang Yitang had pushed the problem to the point that the difference between prime pairs was finite.

This doesn't mean the twin prime conjecture has been solved; it only means that the twin prime conjecture has been advanced to a new level, but it's still far from being solved.

Zhang Yitang's work is an improvement on the results of Goldston–Graham–Pintz–Yldrm.

Later, in 2014, through the efforts of other mathematicians, the gap was optimized to 246, which proved that there are infinitely many pairs of prime numbers whose difference is less than or equal to 246.

This still doesn't mean the twin prime conjecture has been completely solved.

Now, I am essentially standing on the foundation laid by future generations to completely solve this problem.

Is Lin Ran confident? Yes, but not by much.

The reason I made that statement was simply to push myself; pressure is the source of motivation.

"Let me see my true potential," Lin Ran thought to himself.

“Jenny, let’s go. I think I need to go to the hotel for afternoon tea now,” Lin Ran said.

Last time Lin Ran came to London, he stayed at Winfield Manor, which turned into a sieve due to KGB infiltration. So this time, the White House team chose the Clarridge Hotel.

Jenny, who was sitting quietly by the window reading the London tabloids, got up and picked up a black sun hat from the coat rack: "Come on, Professor, it seems you have no inspiration."

Or perhaps the grand promises you made put too much pressure on you?

The two chatted as they walked out of the room. The moment they stepped out, Lin Ran suddenly grabbed Jenny and gestured for her to look back.

Jenny turned around and only saw the door and the hallway. She asked, puzzled, "What are you looking at?"

Lin Ran said, "Originally I didn't have confidence, but now I'm full of confidence."

Can you tell me the room number?

Jenny asked, "257, what's wrong?"

Lin Ran said, "This room number is amazing. 257 is a prime number, and each of its digits, 2, 5 and 7, is also a prime number."

"God is telling me that I will definitely be able to solve the twin prime conjecture during this trip to Göttingen."

Jenny said helplessly, "Professor, I didn't expect you to be so superstitious."

Lin Ran explained, "No, this isn't superstition. Sometimes, solving problems requires a little bit of luck. Luck can give you a strong psychological suggestion, and that kind of suggestion is the most helpful."

When they arrived at the hotel restaurant, Lin Ran didn't order any food, but instead called over the lobby manager.

"Professor, hello, how can I help you?" The manager, dressed in a tuxedo, was very polite.

Lin Ran asked, "I would like to ask if the hotel has a room number 523?"

After a moment's thought, the lobby manager said, "Yes."

Lin Ran nodded: "Please help me make arrangements. I need to move to room 523 tomorrow."

Lin Ran didn't ask if anyone was in the room; the hotel should take care of such a small matter for him.

After making the arrangements, Lin Ran explained to Jenny in detail: "Jenny, I'll have to trouble you to change rooms with me tomorrow."

A three-digit number is a prime number, and each of its constituent digits is also a prime number. The prime numbers for single digits are 2, 3, 5, and 7. There are 15 three-digit numbers that meet this condition. However, if we prevent the constituent digits from repeating, then there are only 2 such numbers: 257 and 523.
Since the Clarridge Hotel has rooms with these two prime numbers, I will spend my last two days in London in these two rooms respectively.

In the future, this will be known in the history of mathematics as the Claritch Hotel's Enlightenment into Prime Numbers. I believe that every mathematician who comes to London to work on prime number problems will stay overnight in these two rooms, because they are about to acquire the legendary status I have bestowed upon them.

Lin Ran's eyes were bright, and he was completely different from the person who had been thinking about the twin prime conjecture just two days ago. Jenny felt the strongest confidence she had seen from Lin Ran in the past few days.

This was one of the very few times she had directly sensed the childlike nature of the man before her. She smiled and squeezed Lin Ran's hand: "Professor, after you succeed, I will help you record this legendary story in the New York Times."

(End of this chapter)