Chapter 195: The Mathematics Department’s Heirloom
Because Göttingen is a university town, it becomes deserted here around Christmas.
In the 1960s, the resident population here was around 120,000, with 30,000 students.
Among them, at least 70% were students from other places.
This year is different from previous years; this year’s Göttingen is livelier than in previous years.
Mathematicians from all over the world fly to Frankfurt first and then take the train to Göttingen.
If one says the whole city is livelier, then it is no exaggeration for the local newspapers in Göttingen and the regional television station in Lower Saxony where Göttingen is located to describe it as bustling.
In 1965, West Germany’s television broadcasting mainly consisted of the Public Broadcasting Alliance (ARD) and Second German Television (ZDF).
Among them, ARD was composed of multiple regional broadcasting companies, each covering one or more federal states, providing national and regional programs.
NDR’s third channel specifically served regions like Lower Saxony.
All news related to Lin Ran’s return to Göttingen was reported in detail as local news.
They were just short of putting up a banner saying Professor, welcome home.
“Fox, it’s good that you contacted me in advance; otherwise, it’s hard to find enough hotels in Göttingen now, especially rooms like numbers 223 and 227, which are being snapped up like crazy.” Döblin led the group from Columbia University to the student dormitory at University of Göttingen.
Fox, who had arrived in Göttingen early with his team, had never thought of finding a hotel; he directly contacted Döblin and asked them to arrange the student dormitory.
Because Fox knew very well that doing research, especially involving a breakthrough, would definitely mean forgetting to eat and sleep, and it was impossible to rest after lecturing for four hours like an ordinary lecture.
It would be 100% a non-stop marathon.
If Lin Ran didn’t stop, they as audience members certainly couldn’t stop either.
They had to listen from beginning to end.
In that case, whether the accommodation was good or bad didn’t matter; having a place to stay was most important.
What Fox thought was that the best place to stay was definitely the student dormitory at University of Göttingen.
As for two people sharing a room, that didn’t matter at all.
He hadn’t even planned to return to the dormitory; Fox and the mathematics professors he led each had a sleeping bag.
Everyone’s plan was to lay the sleeping bags directly on the ground at the site and sleep.
Having a student dormitory as a place to stay was just being prepared.
Hearing Döblin say this, Fox’s mathematician acuity was sufficient: “Because Randolph is going to prove the Twin Prime Conjecture, so rooms with prime number door numbers are being snapped up like crazy, right.”
He immediately spotted the characteristic of these two numbers: both were prime numbers.
Döblin smiled wryly: “Not only that.”
He told Fox about Lin Ran’s actions in London and the idea of prime number enlightenment at Claridge Hotel.
This was also relayed to him by Seagull who was still in London, and then Döblin spread it as an anecdote.
It was reported by the local media.
There were few rooms that met the conditions to begin with, and at this time point, 257 and 523 were booked up, and then rooms with other three-digit twin prime number door numbers repeating those 13 digits were also snapped up like crazy.
After hearing this, Fox laughed: “I feel this will become a mathematician tradition from now on.
Like next year’s International Congress of Mathematicians, when the organizers book hotels for everyone, everyone will definitely want rooms 257 and 523, and next 223 and 227.”
Döblin smiled wryly: “Exactly, and this is still premised on the professor not proving the Twin Prime Conjecture.
If he really succeeds this time, then everyone will firmly believe that twin prime rooms help with thinking.”
Fox laughed: “It seems the first thing I do when I return to Columbia is to change all the mathematics department door numbers to prime numbers, so everyone won’t fight over them.
It’s just that such prime numbers are limited, so eventually everyone’s office door numbers will get longer and longer.”
Döblin said: “It’s still the professor’s influence that’s too great; even the local newspapers in Göttingen are joking that in today’s Göttingen, one brick can smash a bunch of mathematicians.”
On the afternoon of January 4, 1965, Göttingen train station was crowded with people.
Lin Ran arrived in Göttingen after taking the train with two transfers from London, followed by him were Seagull and Jenny as well as West German senior officials, with security personnel leading the way in front and security personnel behind.
Police could be seen everywhere at the train station.
The security at Göttingen train station had never been so comprehensive.
Welcoming him were University of Göttingen President Otto Kumler, Head of Mathematics Department Döblin, and several old professors.
Outside the station, student volunteers held welcome signs, and reporters from all over West Germany and even Europe gathered, holding notebooks to record this historic moment.
“Professor, I look forward to witnessing your miracle.” Otto said after shaking hands.
Döblin continued: “Professor, the stage is set up, just waiting for your performance; the entire Göttingen can’t wait.”
The lecture was held in the main auditorium of University of Göttingen’s main building, this 18th-century classical building famous for its dome and carved columns, seating 500 people.
According to University of Göttingen history, the auditorium is often used for important academic events, such as Nobel Prize winner speeches.
On January 5, 1965, the auditorium was packed, extra audience filled the corridors, the university set up speakers in nearby classrooms for broadcast, and arranged temporary seats in the courtyard for students and scholars unable to enter to listen.
Besides these, the local television station in Göttingen set up cameras, planning to live broadcast the entire event.
Inside the auditorium, the center of the stage was covered with blackboards, only blackboards.
“Ladies, gentlemen, let us first welcome Randolph Lin back to Göttingen with warm applause.” Otto said. “Göttingen is the professor’s alma mater; we are proud and honored to have cultivated an outstanding student like Randolph Lin. Now, I hand over the time to Randolph.”
Lin Ran whispered to Seagull: “Professor, the recording is up to you.”
Seagull nodded, “No problem.”
Lin Ran walked onto the stage, and thunderous applause erupted from the audience.
After the applause subsided, Lin Ran said:
“Ladies, gentlemen, esteemed colleagues, dear friends, good morning!
Being back in Göttingen, this land that nurtured my mathematical dreams, is a great honor. Standing in this auditorium, I feel like I’m back in my student days, when I listened to Hilbert’s successors lecture on number theory here, stayed up all night studying Euclid’s proofs, trying to glimpse the mysteries of prime numbers.
Of course, back then I never imagined I could prove Fermat’s Conjecture, propose the Randolph Program, let alone that one day I would stand here, attempting to challenge: the Twin Prime Conjecture.
Since Professor Hilbert raised it as the 8th problem in his report at the 1900 International Congress of Mathematicians, it has been exactly 65 years.”
Lin Ran turned around, wrote “3, 5”, “5, 7”, “11, 13” on the blackboard, then turned back, his gaze sweeping over the audience, his tone becoming solemn.
“You all recognize these numbers.
They are twin primes, prime pairs differing by 2.
They seem simple, yet hide our predecessors’ conjecture: are there infinitely many such pairs?
This problem can be traced back to ancient Greece; Euclid proved the infinitude of primes, but for twin primes, he left us an unsolved mystery.
Fast forward to the 19th century, when mathematicians began to seriously consider this problem.
In 1849, Alphonse de Polignac proposed a more general conjecture, asserting that for any even number k, there are infinitely many prime pairs p and p’ such that p’ – p = k.
When k=2, this is our Twin Prime Conjecture.”
Lin Ran continued writing on the blackboard: p’ – p = 2
“This conjecture seems intuitive; number theory is always like that, very intuitive, anyone can understand the problem, but in the rigorous world of mathematics, it is like an unclimbable peak.”
Lin Ran’s speech rate was fast, using English, standard English that every scholar present could hear clearly.
Germans are not as insistent on the German language as French people are.
Lin Ran fell into contemplation, slowed his pace, hands clasped behind his back, gaze directed to the depths of the auditorium, as if tracing history.
“By the early 20th century, mathematicians began using more powerful tools to tackle prime distribution problems. In 1919, Norwegian mathematician Viggo Brun made a breakthrough.
He invented a technique called the Brun sieve, proving that the sum of reciprocals of twin primes converges.”
Lin Ran continued writing on the blackboard:
“What does this mean? Compared to the sum of reciprocals of all primes diverging, twin primes are so sparse that their reciprocal sum doesn’t even tend to infinity.
Brun’s theorem tells us that twin primes are not as common as ordinary primes. Their sparsity makes proving infinitude extraordinarily difficult. But isn’t this the charm of mathematics? When we face a seemingly impossible problem, our creativity is truly sparked.”
Randolph walked to one side of the podium, took a sip of water, his gaze sweeping the audience below.
Reporters in the corner discussed in low voices, trying to capture every word from Lin Ran.
The atmosphere in the auditorium shifted from tension to anticipation, as the audience was drawn into the world of primes by his narration.
“Although Brun’s work didn’t prove the conjecture, it pointed the way for us. Hardy and Littlewood later provided heuristic support using the circle method, estimating the number of twin prime pairs up to x as approximately 2C (x / (log x)^2), where C is the twin prime constant, about 1.32032.”
Lin Ran continued writing the formula on the blackboard.
“But these are all probabilistic predictions, far from a true proof.
Today, standing here, I am not repeating these predictions, but showing you a possible answer—a proof combining analytic number theory and sieve method, attempting to unveil the Twin Prime Conjecture.
Over the next six days, we will embark on this journey together.
From prime distribution to the ingenuity of sieve methods, to the profound tools of analytic number theory, I hope to convince you that this conjecture is no longer a conjecture, but a theorem.
Of course, I know many of you, especially Göttingen’s professors, will scrutinize my proof with the most rigorous standards.
That’s exactly what I expect! Let’s begin!”
The audience below applauded, Seagull too, but his thoughts differed from others; his feeling was even stranger.
Professor Seagull was certain this was Lin Ran completing the graduation thesis defense he hadn’t been able to do at University of Göttingen.
He sat up straight, thinking, “Randolph, let me witness your legend, prove with action that the Göttingen school has not perished, and because of you, it will become even more glorious.”
Lin Ran turned and wrote Day 1 on the blackboard.
From writing Day 1, the scholars present felt a sense of rapid progress.
Because Lin Ran’s speed was too fast.
Lin Ran first pulled out Zhang Yitang’s result, that there exist infinitely many prime pairs with difference less than 70 million, then pulled out Terence Tao’s improved result, shrinking that difference from 70 million to 246.
But he couldn’t directly use Zhang Yitang’s result.
Because Zhang Yitang’s paper was built on the GPY sieve method and the 4/7 level result of Bombieri, Friedlander, and Iwaniec on prime distribution in arithmetic progressions.
These two, the GPY sieve method only appeared on arXiv in 2005, and the paper by Bombieri, Friedlander, and Iwaniec was in 1987.
In 1965, to reproduce it, Lin Ran couldn’t directly use Zhang Yitang’s result; he had to first write out the prefix papers.
Therefore, on the first day
Formulas piled up continuously on the blackboard; Lin Ran said little, wrote a lot, constantly pacing back and forth.
After filling a blackboard, he pushed it aside.
Fill one and push it aside; what University of Göttingen prepared in advance was movable blackboards.
University of Göttingen was happy with this; they didn’t want to erase a single one.
If Lin Ran really succeeded in proving it, these would be the mathematics department’s heirlooms, more valuable the longer they were inherited.
“Good, I’ve outlined my core idea.
I start from admissible k-tuples.
These k-tuples, these integers, for each prime p, have at least one residue class uncovered, ensuring they can all be prime.
My goal is to prove there exists k such that there are infinitely many n where the tuple ({n+h_1, n+h_2, …, n+h_k}) has at least two primes. This will mean prime pair gaps are bounded.
I used a variant of the Selberg sieve method, constructing a weight function to detect cases with at least two primes in the tuple.
By optimizing parameters, I estimated the number of n satisfying the conditions. The key is ensuring the main term exceeds the error term.”
“Controlling the error term requires knowledge of prime distribution in arithmetic progressions.
We first allow average modulus up to x^{1/2}.
Then enhance it for smooth moduli, extending the distribution level; this step is to enable the sieve method to handle large k values.
With these tools, I prove that for sufficiently large k, there exists finite N such that there are infinitely many prime pairs with difference at most N.
Then we first find one N, then gradually shrink this N until it equals 2.”
After Lin Ran finished, the scholars below looked very serious.
Because the idea Lin Ran proposed wasn’t strange; it was very orthodox, no essential difference from past mathematicians’ thinking around this problem.
It was just that the methods Lin Ran mentioned would have some innovative points.
If it was just this idea alone, it obviously wasn’t enough to solve the Twin Prime Conjecture.
“Now we start the first step, beginning with analytic number theory; we first push forward from Mark Balban’s result.
First prove that for specific Q near x, if we ignore log terms, the average error can be as small as x^{1/2}.
Then extend this result, expanding the modulus from 1/2 to 4/7, making the prime distribution error term controllable under larger moduli, applicable to sieve problems in analytic number theory.”
Lin Ran began; he was very quiet when writing, only speaking during explanations.
He said very little.
As he wrote, the mathematics department professors from Princeton below were numb.
Because the result Lin Ran casually wrote was the major achievement the Princeton Institute for Advanced Study was going to publish this year.
x at 1/2 is called the Bombieri-Vinogradov theorem in mathematics; also known as Bombieri’s theorem, it is a major achievement in analytic number theory, related to prime distribution in arithmetic sequences averaged over a series of moduli.
This type of result was first obtained by Mark Balban in 1961, and the Bombieri-Vinogradov theorem is a refinement of Balban’s result.
This achievement was published exactly in 1965, solved by Enrico Bombieri of Princeton and Ascold Vinogradov, hence called the Bombieri-Vinogradov theorem.
It took them until 1987, more than twenty years later, to push this result from 1/2 to 4/7.
And now Lin Ran, on the spot, casually proved their result and was about to achieve far beyond it.
The more Lin Ran wrote, the darker the faces of the professors from Princeton.
Because Lin Ran’s proof at 1/2 was flawless, meaning his push to 4/7 was probably also correct.
This sense of frustration was like you busting your ass jumping around positioning and using ultimate moves to barely defeat a boss, and someone else one-shots it with a casual basic attack.
Faster than you, and with a more elegant pose.
“Good, everyone sees, we have completed the proof here.
Just proved that prime distribution in arithmetic progressions can reach the =4/7 level.
Specifically, it shows that for modulus ≤4/7, the distribution error term for primes in arithmetic progressions (a mod q) (gcd(a,q)=1) can be effectively controlled.
This result expands the modulus range, making the sieve method applicable over a larger range.
The main thinking here is actually overcoming limitations of traditional methods through bilinear form estimates and divergence technique, enhancing prime distribution analysis capability.
We have laid the foundation for the bounded gap in the overall idea of the subsequent Twin Prime Conjecture.”
