Technology invades the modern world

Chapter 197 Göttingen Mathematical Marathon (5k)

As time progressed into the third day, the third day came to an end.

The mathematicians present had become accustomed to Lin Ran's style: extensive blackboard writing, essentially writing papers on the spot, minimal explanation, and a few words when there was a breakthrough.

So their division of labor was very clear: the PhD students were responsible for keeping an eye on things during the day and copying down the content that Lin Ran had determined, while the others slept during the day, going to their dormitories in Göttingen to sleep.

In the evening, everyone would go out together, have a leisurely dinner in the dormitory, and then come to the auditorium to see the results of the students' copying during the day, and then direct their own students to make some coffee.

In short, PhDs are just there to be used.

As for what to do if you don't have a PhD, you can direct other mathematicians' PhDs.

It's just a matter of making two more copies of the manuscript and brewing two more cups of coffee.

The university town of Göttingen has no shortage of students.

Here, two-legged oxen and horses are much easier to find than real four-legged ones.

That night, everyone stayed up all night discussing and judging whether Lin Ran's paper was correct.

Everyone has already discussed it:
"If Randolph can write papers on the spot, we can naturally review them on the spot."

“That’s right. No matter what the final result is, we can extract the correct parts of Randolph’s academic report, integrate them into a paper, and send it to an academic journal so that they can publish it directly as a special issue without review.”

"Yes, it's like all of us mathematicians here are editors, writing and reviewing on the spot."

"Randolph wants to bring glory to Göttingen, and similarly, we are contributing to the revival of the Göttingen School, making this story even more legendary."

The signatures recommending the paper's publication are all from top-tier contemporary mathematicians; they've collectively run this marathon for mathematicians.

The paper's author is Randolph Lin, and the joint authorship of the reviewers includes Siegel, Pierre, Deulin, Andre, Atiyah, Harvey, and others. Any mathematics journal would be terrified upon receiving it.

"Yes, this is truly an unprecedented event in the mathematics world. It's so worthwhile to witness it firsthand."

This is unprecedented, and it's unlikely anyone will ever do so again.

As the third day progressed, and with the ideas Lin Ran had proposed on the first day and the necessary tools gradually becoming complete, everyone realized that the possibility of witnessing a miracle was increasing.

They say they sleep during the day, but in reality, they don't sleep for long before rushing to the Great Hall of the People.

During Lin Ran's rest time that evening, everyone was even more excited.

Because time is limited, there are only six days. If there are any problems, you need to tell Lin Ran as soon as possible so that Lin Ran can adjust his thinking.

That's why they say it's a marathon for Lin Ran, and it's also a marathon for them.

"Yes, to find a problem of similar magnitude to the twin prime conjecture, you only have six days, enough number of mathematicians are willing to witness it, and you actually have to be able to solve it. It's too difficult."

The mathematicians were just as excited as Lin Ran.

Doyle suggested, "There is an International Mathematical Olympiad for high school students now. Shouldn't we also create an Olympiad for university students and doctoral students?"

Let's use the Göttingen Mathematical Marathon format.

The same problem was solved by everyone in six days.

Those who solve the problem together will receive a medal.

The International Mathematical Olympiad, or IMO, was first held in Romania in 1959, with seven Eastern European countries participating. Since then, the IMO has been held continuously except for 1980.

Upon hearing this, Pierre nodded and said, "Good idea, but the biggest challenge is finding the right questions to feed the students."

Indeed, the bigwigs present believed that the biggest challenge in organizing such a competition was not finding participants, nor was it the venue or funding; these issues could be resolved by Göttingen.

The most difficult part is the problem.

Problems that undergraduate and doctoral students can potentially solve within 6 days.

Such questions cannot be too simple; if they are too simple, they are meaningless and everyone can win a medal.

It's too difficult and pointless; no one can do it anyway.

It needs to be suitable enough.

This is extremely difficult and tests the skill of the question setter.

Siegel laughed and said, "Isn't that precisely its significance?"

You think the biggest reason why mathematics emphasizes inheritance is that mathematicians with inheritance can obtain enough suitable problems from their predecessors, and these problems can help them grow?

However, for many students in underdeveloped areas, they do not have this opportunity. They can only rely on publicly available papers to find problems to work on, because their supervisors may not have enough accumulated knowledge.

If we had a program like the Göttingen Mathematical Marathon, it would be like having established mathematicians provide young students with a suitable problem to think about.

Each year's problems serve as excellent practice material for new mathematicians.

Isn't this more meaningful than the outcome of the match itself?

A host of mathematical luminaries, including Andre Weil from Princeton, Harold Davenport from Cambridge, and Jean Pierre from the École Normale Supérieure in Paris, thought the idea was excellent.

The only question is, why hold such a prestigious competition in Göttingen?
Princeton, Cambridge, and the École Normale Supérieure in Paris are all better than Göttingen these days, aren't they?
"It was because of Randolph, who started this unprecedented mathematical marathon in Göttingen, that we had the idea for this mathematical marathon in tribute to Randolph."

Moreover, Göttingen has a long history and a rich cultural heritage. Doiling was a natural choice on this issue.

I can tolerate everything else, but I really can't stand it if the math marathon isn't held in Göttingen.

Just as they expected, the Göttingen Mathematical Marathon, in this time and space, became the best proof for new mathematicians to enter the mathematical community.

This competition, similar to the IMO, is by invitation only. Mathematicians who have given a public lecture at the main venue of the International Congress of Mathematicians have one nomination slot, which can be used to recommend two students to participate in the Göttingen Mathematical Marathon.

Initially, it was held annually, but later Göttingen felt the pressure was too great, so it became a biennial event.

Later in Göttingen, when people mentioned marathons, the first thing that came to mind was the Göttingen Mathematical Marathon, not the marathon itself.

The trophy is designed to resemble a young Chinese man standing in front of a blackboard, deep in thought. It is jokingly referred to as "the professor's mantle," and winning the award is said to be like "receiving the professor's mantle in Göttingen."

The vast majority of Chinese mathematicians who are able to inherit the professorship will choose to do two years of postdoctoral research in Göttingen after graduating with their doctorates.

Many mathematicians in other countries can also do this, but the proportion is not as high as that of mathematicians in China.

To elaborate further, if the problem can be solved within six days, then everyone who solves it will receive a gilded trophy.

If no one can solve it within six days, the problem will be made public globally. Those who solve it within a year will be able to go to Göttingen to receive an award and a silver trophy.

Those who solve the problem a year later will receive a bronze trophy.

In short, the Göttingen Marathon has become a Göttingen tradition and the beginning of Göttingen's revival in this era.

Of course, staying in a room with a Randolph number before participating in a competition is a long-standing tradition.

On the day the Göttingen Mathematics Marathon begins, all the hotels in Göttingen will change their room numbers to prime numbers prepared in advance.

If you simply tell them you want to participate in the Göttingen Mathematics Marathon, the hotel reception will change your room number to Randolphian numbers when they take you there, to avoid limited rooms and impacting business.

As for the two rooms at the Clarity Hotel that offer the "Primal Enlightenment" course, they need to be booked at least six months in advance.

Back to the scene in Göttingen.

It wasn't until the third day that Lin Ran found the number: 70000000.

After Lin Ran finished speaking, the mathematicians in the audience all stood up and crowded in front of Lin Ran, trying to see the contents on the blackboard more clearly.

“Randolph, the rest of the time is up to us. We will review your content within the remaining seven hours.” Siegel patted him on the shoulder.

Lin Ran nodded and said, "Thank you, Professor. My annotations are clear enough. Let's talk again after I've rested."

The mathematicians present gathered in front of the blackboard, trying to digest the groundbreaking results. Paul Erdős, the mathematician known for his obsession with prime numbers, had moved his table and chair to the blackboard, just like the previous two days, and began to re-derive the results in his notebook according to Lin Ran's ideas.

Jean-Pierre thought to himself, "If Randolph is right, then the next step is to move from seventy million to two."

Andre Will sat a little further away, lost in thought.

He pondered the possible connection between this result and the distribution of the zeros of the Riemann function, thinking, "This might provide a new perspective on deeper questions about the prime number theorem."

“Alright, Randolph’s going to rest now. We need to review his conclusions as soon as possible.” After Lin Ran left, Siegel took his place in the middle of the group.

Everyone had different thoughts.

I'm considering which areas Lin Ran's results can be applied to.

That's the benefit of fame.

Everyone believes first and then questions.

Just like Shinichi Mochizuki in later generations, because he made great achievements, the mathematical community will not directly say that his proof of the ABC conjecture is wrong.

Siegel said, "Harold, Randolph's application of the sieve method draws on Selberg's work, but applying it to prime number gaps is entirely new. I'm worried that his estimate on secondary arcs might be too optimistic, so I'll leave that part to you!"

Harold nodded: "I also think the key lies in controlling the error term. His method seems very clever, especially his handling of integrals. But I still need to think about it carefully."

Siegel added, "You should carefully examine Randolph's lemma section, especially the part involving exponents and sums."

Then Siegel said to Paul Erdős, who was sitting in the corner, "Paul, you are responsible for reviewing his weight function construction to ensure that his method does not fail to control the error term like Vinogradov's method."

Siegel, based on his impressions of the mathematicians, assigned the review work.

Most of the people present were assigned a small task.

This is because Siegel is Lin Ran's mentor, and also because Siegel is the oldest here and has achieved equally remarkable accomplishments. It is no problem to say that he is among the most accomplished mathematicians present.

It's not even necessary to add one more.

He's so great himself, and the students he teaches are even more great; they have no say in his commands.

"How far do you think Randolph is from proving the twin prime conjecture?"

"Can we reduce the 70 million to 20 million? It might be resolved in the remaining three days, or it might take six months, a year, or even longer. I'm not sure."

Randolph has completely changed my understanding of mathematics.

"That's right. A new result is like opening a door, behind which lie countless possibilities. Randolph's results have now opened that door for us; all that's left is to keep moving forward along it."

It's good that Randolph can prove it, but if he can't, we might have a chance to beat him to it. After all, besides being a mathematician, the professor spends most of his energy on NASA. Maybe I can achieve a new breakthrough by combining his improved sieve method with my probability methods.

The mathematicians gathered in small groups, chatting as they worked on the tasks assigned to them by Siegel.

Pierre and André were no exception, but compared to other mathematicians, they were more concerned with a range of effects.

“Andre, this result reminds me of the impact of your introduction of probability models in algebraic geometry. If Randolph’s method can be generalized to other sequences of prime numbers, it will change the face of analytic number theory.”

“Indeed, mathematics always finds connections in unexpected places. His method reminds me of certain aspects of Chebotarev’s density theorem, and perhaps we can re-examine the distribution of prime numbers from an algebraic perspective.”

On the morning of the fourth day, Lin Ran washed up and walked out of the lounge. The professors outside did not go to rest; they all insisted on waiting for Lin Ran to come out.

When Lin Ran walked out of the lounge, the professors outside were applauding so loudly it felt like they were going to lift the ceiling off the auditorium.

“Randolph, after our initial review, your conclusions are correct. You have successfully found the prime gap, which is a remarkable achievement,” Siegel said.

While television viewers found it tedious, the reporters present didn't think so. They remained alert, with newspaper reporters working in three shifts to wait and capture the decisive moment.

Lin Ran pushed open the door, and the moment applause erupted outside was, in their eyes, just like that.

A reporter whispered to his colleague, "Does this count as completing a half marathon?"

"Roughly speaking, it's been six days, and three days have passed. Judging from the professors' expressions, it seems like they've made a breakthrough," my colleague replied.

Lin Ran stretched out his hands and pressed them down, "Everyone, there are three days left. Our next task is to turn 7000 million into 2 million."

Time is short and the task is arduous.

I hope to hear your enthusiastic applause only after I have completed this arduous task, and then we can celebrate properly.

On the fourth day, Lin Ran pushed the number from 7000 million to 246, which was something that mathematicians had accomplished a year after Zhang Yitang submitted his proof in the original timeline.

“Extraordinary proof”.

"Randolph's progress is too fast; it's beyond my understanding of human capabilities."

"The professor isn't human, haven't we known that from day one?"

"We are witnessing a miracle!"

"This is already a huge leap forward for the twin prime conjecture. I feel that the professor could further lower the limits of the sieve method's construction."

"The professor's methods have entered entirely new fields, and his achievements can be applied not only to twin primes, but also to Diophantine equations of primes, prime triples, and even the study of transcendental numbers."

"The most difficult stage has arrived. The result of going from 7000 million to 246 is already amazing enough, but I think going from 246 to 2 will be even more difficult."

The mathematicians were abuzz with discussion, their emotions completely ignited by Lin Ran's astonishing achievement.

More and more mathematicians have been arriving in Göttingen these days.

The last time such a grand occasion was before the Göttingen Mathematical Institute was dissolved by NAZ Germany, when Hilbert personally wrote to mathematicians, inviting them to attend the final funeral of the Göttingen Mathematical Institute.

Doiling and Siegel remarked, "As expected of a professor, as long as he stays in Göttingen, the reconstruction of Göttingen as a mathematical center will be complete."

Siegel said, "If we can run the Göttingen Mathematics Marathon well, the reconstruction of Göttingen can also be completed."

Doyle sighed, "Alright, I know what you mean. We really have to rely on ourselves. Göttingen can't keep the professor, and Germany certainly can't. If we keep him, the White House will probably go crazy."

Laurent Schwarz, Henry Kattan, Gauss Rao, Alan Beck—besides the mathematicians who had arrived before, since the figure of 7000 million was announced, all the well-known mathematicians from across Europe have gathered in Göttingen.

Originally, everyone was going to go to the dormitory to rest.

Nobody wants to go to the dorms anymore; nobody wants to waste a single minute.

So the entire auditorium at the University of Göttingen, which used to have chairs with small tables, now has all the chairs gone, leaving only individual desks and matching tables.

There was space next to it for a sleeping bag, so you can lie down next to it when you're tired.

In the cold winter of January, you won't die if you don't wash up for two days.

"Everyone's hoping for a miracle, aren't they?" Pierre grumbled to Siegel while munching on a baguette.

Siegel immediately understood what he meant. This was a miracle. If Lin Ran could really solve the twin prime conjecture, then it would be a divine miracle. Only a god could do such a thing.

On the last day, as midnight drew ever closer, the hearts of the mathematicians present were pounding with anxiety.

Lin Ran's movements became faster and faster. With one hour left until midnight, he wrote down the last symbol, not N=6, but N=2:

"The key to the problem lies in our initial EH conjecture, which provides a strong distribution estimate of prime numbers in the arithmetic series, allowing us to increase the distribution level from N^{1/2+\epsilon} to N^{1-\epsilon}. This significantly reduces the error term in the sieve method."

Based on an improved version of the GPY sieve method, I introduced a multidimensional weighting function to optimize the counting of prime pairs, ensuring that the principal term exceeds the error term.

Lin Ran circled a formula on the blackboard: "This is the key formula."

This helped me prove that S > 0, which means there exist infinitely many n such that both n and n+2 are prime numbers. He paused for a moment, looking around the room: "In other words, the twin prime conjecture has been definitively proven."

　Based on everyone's feedback, this chapter has significantly reduced the amount of technical content described.

　　
　
(End of this chapter)