Chapter 197: Göttingen Mathematics Marathon
As time entered the third day, the end of the third day.
The mathematicians on site had all formed a habit of Lin Ran’s style: massive blackboard writing, essentially writing papers on site, with minimal explanations, and just a few words when there was a breakthrough.
So their division of labor was very clear: the PhD students were responsible for watching during the day, and by the way, copying down the content that Lin Ran had confirmed. They slept during the day, going to the dorms in Göttingen to sleep.
In the evening, everyone mobilized together, leisurely eating dinner in the dorms, then coming to the auditorium after eating to look at the results that the students had copied during the day, and then commanding their own students to brew some coffee.
In short, the PhD students were there to be bossed around.
As for what to do if you didn’t have PhD students, you could command the PhD students of other mathematicians.
It was just a matter of photocopying a couple more manuscript copies and brewing a couple more cups of coffee.
What the University of Göttingen lacked the least was students.
Here, two-legged cattle and horses were much easier to find than actual four-legged cattle and horses.
In the evenings, everyone discussed all night, judging whether Lin Ran’s paper was correct or not.
Everyone had already discussed it:
“Randolph can write papers on site, so naturally we can review them on site.”
“Exactly, no matter what the final result is, we can extract the correct parts from Randolph’s academic report this time, integrate them into a paper, send it to an academic journal, and have them publish it directly as a special issue without review.”
“Yes, it’s like all us mathematicians here are editors, writing and reviewing on the spot.”
“Randolph wants to bring glory to Göttingen, and similarly, we are also contributing to the revival of the Göttingen school, making this story even more legendary.
I suggest that the signatures for the paper publication all be top contemporary mathematicians, all of us collectively finishing this marathon that belongs to mathematicians.”
The paper author is Randolph Lin, and the joint reviewer signatures are Seagull, Pierre, Döblin, Andrei, Atiyah, Harvey, and others; any mathematics journal receiving it would be scared shitless.
“Yeah, this is truly an unprecedented event in the mathematics community. Being able to witness it all firsthand is really worth it.
Unprecedented before, and probably no one after.”
As they entered the third day, after the tools needed for the approach Lin Ran proposed on the first day gradually became complete, everyone realized that the possibility of witnessing a miracle was increasing.
They said they slept during the day, but in reality, they couldn’t sleep long during the day before running to the main hall.
During the time Lin Ran rested in the evening, everyone was even more excited.
Because time was limited, only six days; if there was any problem, it needed to be told to Lin Ran immediately so he could adjust his approach.
So that’s why they said, for Lin Ran this was a marathon, but for them, wasn’t it also a marathon?
“Yeah, to find a problem on the level of the Twin Prime Conjecture, with only six days, and enough mathematicians willing to come witness it, and you actually being able to solve it—that’s too hard.”
The mathematicians’ state was as excited as Lin Ran’s.
Döblin proposed: “Now there’s the International Mathematical Olympiad for middle school students; shouldn’t we also create an Olympiad competition for university students and PhD students?
Using the Göttingen Mathematics Marathon mode.
Same problem, everyone solves it in six days.
Those who solve it all get a medal together.”
The International Mathematical Olympiad, or IMO, started in 1959, with the 1st IMO held in Romania, with 7 Eastern European countries participating. Since then, except for 1980, the IMO has never been interrupted.
Pierre, upon hearing this, nodded: “Good idea, but the biggest difficulty here is finding suitable problems to give to the students.”
Exactly, the big shots on site thought that the biggest difficulty in holding such a competition wasn’t finding participants, nor venues, funds, and such—those could be solved by Göttingen alone.
The hardest was the problems.
Problems that university students and PhD students might be able to solve within 6 days.
Such problems couldn’t be too simple; too simple and there’d be no meaning, everyone would get medals.
Too hard also had no meaning, no one could solve them.
They needed to be appropriately difficult.
This was very hard, a severe test of the problem-setters’ skill.
Seagull laughed: “Isn’t that exactly its significance?
Think about it, the biggest reason mathematics emphasizes inheritance is that mathematicians with inheritance can get sufficiently appropriate problems from their predecessors, and these problems can help them grow, right?
But for students in many underdeveloped areas, they don’t have this condition; they can only find some problems from public papers to work on, because their mentors may not have enough accumulation.
And if we have a way like the Göttingen Mathematics Marathon, it’s equivalent to mature mathematicians providing young students with an appropriate problem to think about.
Each year’s problems would be the best practice for new mathematicians.
This is more meaningful than the competition’s wins and losses, isn’t it?”
Andrei Weil from Princeton, Harold Davenport from Cambridge, Jean Pierre from Paris Normal School, and so on—a group of mathematics masters all thought this was a great idea.
The only question was, why hold such an awesome competition in Göttingen?
Princeton, Cambridge, Paris Normal School—aren’t they all more awesome than Göttingen right now?
“Because of Randolph. Randolph starting this unprecedented mathematics marathon in Göttingen is what gave birth to our idea of a mathematics marathon in tribute to Randolph.
Besides, Göttingen has a long history and sufficient heritage.” Döblin was uncompromising on this point.
Other things he could tolerate, but not having the mathematics marathon in Göttingen—that he really couldn’t tolerate.
Just as they thought, in this spacetime, the Göttingen Mathematics Marathon became the best proof for new mathematicians entering the mathematics community.
This competition was similar to the IMO, invitation-based. Mathematicians who had given public speeches at the main venue of the International Congress of Mathematicians had one recommendation slot, able to recommend two students to participate in the Göttingen Mathematics Marathon.
Initially one per year, later the Göttingen side felt the pressure was too great and changed it to every two years.
Later, in Göttingen, when mentioning marathon, everyone’s first reaction was the Göttingen Mathematics Marathon, not the marathon long run.
The trophy design was a young Chinese person standing in front of a blackboard in a thoughtful pose, jokingly called the professor’s mantle; winning it was called receiving the professor’s mantle in Göttingen.
Any Chinese mathematician who could obtain the professor’s mantle would mostly choose to do two years of postdoctoral research in Göttingen after getting their PhD.
Mathematicians from other countries often did too, but the proportion wasn’t as high as for Chinese mathematicians.
To say a bit more, if someone could solve the problem within six days, all who solved it would get a gold-plated trophy together.
If no one could solve it within six days, the problem would be made public globally; those solving it within a year could go to Göttingen to collect the award and get a silver trophy.
Those solving it after a year would get a bronze trophy.
In short, this Göttingen marathon became a tradition of Göttingen and also the beginning of Göttingen’s revival in this spacetime.
Of course, before the competition, staying in a room numbered a Randolph number was a tradition among traditions.
On the starting day of the Göttingen Mathematics Long Run, all local hotels in Göttingen would change to pre-prepared prime number room numbers.
As long as you said you were participating in the Göttingen Mathematics Long Run, the hotel front desk would take you there and help change the room number to a Randolph number, to avoid limited such rooms affecting business.
As for the two rooms at Claridge Hotel for prime number enlightenment, you had to book at least half a year in advance.
Back to the scene in Göttingen.
Up until the third day, Lin Ran found that number, 70000000.
After Lin Ran finished speaking, the mathematicians in the audience all stood up and crowded to Lin Ran, trying to see the content on the blackboard more clearly.
“Randolph, the remaining time is ours; we’ll review your content within the remaining seven hours.” Seagull patted his shoulder.
Lin Ran nodded: “Thank you, Professor. My annotations are clear enough. We’ll talk after I rest.”
The mathematicians on site gathered in front of the blackboard, trying to digest this achievement beyond the era.
Paul Erdős, the mathematician famous for his obsession with prime numbers, like the previous two days, had already moved a table and chair to the front of the blackboard and begun re-deriving on his notebook according to Lin Ran’s approach and results.
Jean Pierre thought to himself, “If Randolph is right, then the next work is to push from seventy million down to 2.”
Andrei Weil sat a bit farther away, lost in thought.
He was pondering in his mind the possible connection between this result and the zero distribution of the Riemann function, thinking: “This might provide a new perspective for deeper problems of the Prime Number Theorem.”
“Alright, Randolph has gone to rest; we need to review his conclusions as quickly as possible.” After Lin Ran left, Seagull took his place in the middle of the crowd.
Everyone had different thoughts.
Thinking about which fields Lin Ran’s result could be applied to.
This was the benefit of reputation.
Everyone believed first, then doubted.
Like Shinichi Mochizuki in later generations, same principle: because he had made great achievements before, the mathematics community wouldn’t directly say his proof of the ABC conjecture was wrong.
Seagull said: “Harold, Randolph’s sieve method application borrows from Selberg’s work, but using it for prime gaps is entirely new. I’m worried his estimate on the minor arcs might be overly optimistic; that part is yours!”
Harold nodded: “I also think the key is controlling the error term. His method looks very clever, especially his handling of the integral. But I still need to think carefully.”
Seagull added: “Randolph’s lemma part, especially the exponential and parts, remember to check carefully.”
Then Seagull said to Paul Erdős in the corner: “Paul, you’re responsible for reviewing his weight function construction, ensuring his method doesn’t fail to control the error term like Vinogradov’s method.”
Seagull presided, assigning review work according to his impressions of each mathematician.
Most people on site were assigned a small piece of work.
This was both because Seagull was Lin Ran’s mentor and because Seagull was the oldest here, with equally astonishing achievements; saying he was among the highest-achieving mathematicians present was no problem at all.
It could even be without “among.”
He was so awesome himself, and the student he taught was even more awesome; everyone was completely fine with being commanded.
“How far is Randolph from proving the Twin Prime Conjecture now?”
“Shrinking seventy million to two? It might be solved in the remaining three days, or it might take half a year, a year, or even longer; I’m not sure.
Randolph has already overturned my understanding of mathematics.”
“Exactly, a new result is like opening a door, with countless possibilities behind it. Now Randolph’s result has opened the door for us; the rest is just advancing through that door.
It’s good if Randolph can prove it, but if Randolph can’t, maybe we have a chance to beat him to it. After all, besides being a mathematician, the professor spends most of his energy at NASA; maybe with his improved sieve method combined with my probability method, I can have a new breakthrough.”
The mathematicians gathered in twos and threes, working on the tasks Seagull assigned while chatting idly.
Pierre and Andrei were no exception; just compared to other mathematicians, they were more focused on a series of influences.
“Andrei, this result reminds me of the impact when you introduced schemes in algebraic geometry. If Randolph’s method can be generalized to other prime sequences, 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 the Chebotarev density theorem; perhaps we can re-examine prime distribution from an algebraic angle.”
On the morning of the fourth day, after Lin Ran simply washed up, he walked out of the rest room; the professors outside hadn’t gone to rest, all insisting on waiting until Lin Ran came out.
When Lin Ran walked out of the rest room, the applause from the professors outside nearly brought down the ceiling of the auditorium.
“Randolph, after our preliminary review, your conclusion has no issues; you have successfully found a prime gap—this is an incredible achievement.” Seagull said.
The audience in front of the television found it boring, but the reporters on site didn’t think so; they stayed alert, with newspaper company reporters waiting here in three shifts precisely to capture the decisive moment.
The moment Lin Ran pushed open the door and applause erupted outside was, to them, such a moment.
A reporter whispered to his colleague next to him: “Does this count as finishing the half-marathon?”
“Probably, six days total, exactly three days passed; looking at the professors, it feels like there’s been a breakthrough achievement.” The colleague replied.
Lin Ran raised both hands and pressed down: “Everyone, there are still three days; our next task is to turn 70 million into 2.
Time is short, the task is arduous.
I hope to hear everyone’s enthusiastic applause again after I complete this arduous task; we’ll celebrate properly then.”
On the fourth day, Lin Ran advanced this number from 70 million to 246, which was also what mathematicians achieved one year after Zhang Yitang submitted the proof in the original spacetime.
“Extraordinary proof.”
“Randolph’s progress is too fast; this has completely exceeded my cognition of humanity.”
“The professor isn’t human; didn’t we know that from day one?”
“We are witnessing the birth of a miracle!”
“This is already a huge leap for the Twin Prime Conjecture. I feel the professor’s construction of the sieve method can lower the bound even further.”
“The professor’s method already involves entirely new fields; his achievement isn’t just applicable to twin primes—Diophantine equations for primes, prime triplets, even transcendental number research can use it.”
“The most difficult phase is coming; from 70 million to 246 is already astonishing, but I think from 246 to 2 will be even harder.”
The mathematicians’ discussions rose everywhere; everyone’s emotions were completely ignited by Lin Ran’s astonishing achievement.
More and more mathematicians came to Göttingen these days.
The last such grand occasion was before the Göttingen Mathematics Research Institute was disbanded by NAZI Germany, when Hilbert personally wrote letters to mathematicians inviting them to attend the final funeral of the Göttingen Mathematics Research Institute.
Döblin and Seagull complained: “As expected of the professor; as long as the professor is willing to stay in Göttingen, the reconstruction of Göttingen as a mathematics center is complete.”
Seagull said: “If we hold the Göttingen Mathematics Marathon well, Göttingen’s reconstruction can also be completed.”
Döblin sighed: “Alright, I get your meaning; indeed, we still have to rely on ourselves. Göttingen can’t keep the professor, Germany certainly can’t; if we keep him, the White House would probably go mad.”
Laurent Schwartz, Henry Cartan, Gauss Rao, Alan Baker—besides the mathematicians who had arrived before, ever since the 70 million number came out, all notable mathematicians on the entire European mainland gathered in Göttingen.
Originally, everyone was going to the dorms to rest.
Now no one wanted to go to the dorms; no one wanted to waste a minute.
So the entire University of Göttingen main hall, which previously had chairs with small desks, now had no chairs at all, just individual desks and matching tables.
Space next to them was left for sleeping bags; when tired, just lie down right there.
In the cold winter of January, not washing up for two days wouldn’t kill anyone.
“Everyone is expecting the miracle to appear, right?” Pierre chewed on his baguette and complained to Seagull.
Seagull immediately understood; now this was a miracle, but if Lin Ran could completely solve the Twin Prime Conjecture, that would be a divine miracle—only a god could do that.
On the last day, as time approached twelve o’clock, the hearts of the mathematicians on site were all suspended.
Lin Ran’s movements grew faster; an hour before 12, he wrote the final segment of symbols, not main=6, but N=2:
“The key to the problem lies in our initial EH conjecture, which provides a strong distribution estimate for primes in arithmetic progressions, allowing us to raise the distribution level from N^{1/2+\epsilon} to N^{1-\epsilon}. This significantly reduces the error term in the sieve method.”
“Based on the improved version of the GPY sieve method, I introduced a multidimensional weight function, optimizing the counting of prime pairs to ensure the main term exceeds the error term.”
Lin Ran circled a formula on the blackboard: “This is the key formula.
This helped me prove S > 0, which means there exist infinitely many n such that both n and n+2 are prime.” He paused for a moment, scanning the audience: “In other words, the Twin Prime Conjecture has been completely proven.”
Considering everyone’s opinions, this chapter has greatly reduced the description of professional content.
