Chapter 194: Prime Number Enlightenment At Claridge Hotel
Any mathematical problem related to Hilbert’s 1900 century problems is the hottest of the hot in the field of mathematical research.
As mentioned earlier, in the field of cutting-edge mathematical research, finding problems is far more important than solving them.
Finding suitable problems, slowly feeding them to young scholars, allowing them to gradually level up on the path of mathematical research, is even more difficult.
While Hilbert’s century problems can become the final boss, intermediate related problems can be set with it as the goal.
This is why the century problems are so popular.
It’s even more so in Göttingen.
Hilbert’s century problems left behind, for the Göttingen school, are like the great treasure from the Age of Discovery that the Göttingen school contributed to the world mathematics community; it’s fine that everyone can come excavate it.
But the Göttingen school should be able to dig out the richest part. Compared to other universities, Hilbert’s original manuscript notes are all kept in Göttingen; by the year 2000, Rüdiger Thiele even dug out the 24th problem from Hilbert’s original manuscript notes.
As a result, in the first half of the century, the University of Göttingen could still dig up some treasure, but in the second half, they got nothing.
Under Seagull’s leadership, the Göttingen school’s main attack direction was the twin prime conjecture; on this problem, all six professors present know something about it to varying degrees, and Seagull has thought deeply about this problem.
The result, obviously, was no ideas.
Now hearing the other say he would solve this problem in six days, it really sounded like a pipe dream.
“Randolph, I know you have extraordinary talent, but shouldn’t you leave yourself some leeway?” Seagull reminded: “You know, when you give an academic report in Göttingen, there will definitely be many reporters on site, even if we don’t let reporters into the venue.
If you prove the twin prime conjecture on site, it will be announced externally by the students and professors present.
We can’t make them only talk about success and not failure.
Do you want to think about it again?
When you really achieve results in the future, coming back to Göttingen first to give an academic report would also be support for Göttingen.”
Seagull naturally had to consider Lin Ran; he truly regarded the other as his own student, as the successor to his academic legacy.
He was very clear that a scholar who had never failed, pulling off such a big stunt, if it failed, the external mockery, the inner turmoil.
Seagull didn’t believe in that nonsense about adversity helping you grow; top mathematicians or top scientists, their adversities come from life, but in the academic field, they charge forward relentlessly.
Euler, even completely blind, was not affected in his work speed; after going completely blind in 1766, he still produced a large number of highly original papers.
Gauss goes without saying; when Hilbert was young, Paul Gordan said he was doing theology not mathematics, but in the end, his conclusions were proven correct.
In Seagull’s view, mathematical geniuses, especially young scholars who make outstanding contributions when young, should maintain this relentless momentum, break through heavy obstacles to produce a large number of achievements, until they stop in front of an unprecedented difficult problem, then slowly think about a breakthrough.
Seagull didn’t want to see Göttingen’s genius fall to such arrogance.
Lin Ran smiled: “Of course, professor, I don’t have 100% confidence.
I’ve also fully prepared myself psychologically for failure.
My decision is based on thorough deliberation, not just for myself, but to revive Göttingen in the mathematics community.
If I succeed, then I’ve left a brilliant stroke in the history of the University of Göttingen, a segment worth special mention even in mathematics history; in the future, whenever people mention the 20th century, they can’t bypass this scene at the University of Göttingen.
If I fail, it’s the same; the first failure in the professor’s life given to Göttingen is also a brilliant stroke.”
Except for Seagull, the other five professors were all teary-eyed.
Because they heard deep affection for the University of Göttingen from Lin Ran’s mouth; as expected of the talent we cultivated in Göttingen.
Döblin said: “Good, I’ll go back to Göttingen to prepare right now. Randolph, on behalf of Göttingen, I thank you for your efforts.
I’m already prepared to witness a miracle.”
Since Lin Ran had said this, Seagull didn’t refuse; he just sighed: “Randolph, you can think ahead; I’ll still be in London for this period.
When I was young, I also thought about the twin prime conjecture problem; although I didn’t solve it, I have some phased ideas that should be able to give you some direction, more or less.”
He turned to Döblin: “Döblin, help me notify your student in Göttingen; in the third row of the bookshelf in my office, find a thick notebook on it about the Goldbach conjecture, and have him mail that notebook to London.”
After speaking, Seagull continued to Lin Ran: “Randolph, the Goldbach conjecture and the twin prime conjecture are both related to prime distribution and density.
The Goldbach conjecture concerns the sum of primes, while the twin prime conjecture concerns specific gaps between primes.
Both rely on tools in analytic number theory; I’ve always thought whether a common framework can be used to study their properties.
If the twin prime conjecture holds, it might provide support for the Goldbach conjecture, because it shows primes are dense at certain specific gaps, which helps construct the required prime sums.
So I think it can give you some inspiration.”
Seagull had a very wondrous feeling.
They still had to stay together in London for five days.
There were still five days until the Göttingen speech.
Between him and Lin Ran, it was first the master-student title, then the master-student reality.
He first had this PhD student, then this time in London, using the proof of the twin prime conjecture as an opportunity, he would give Lin Ran some guidance.
This was a feeling of spacetime dislocation.
The guidance time was after the PhD degree, and the guidance space was first in London, finally defending in Göttingen.
Right, Seagull now felt that their trip to Göttingen was for the PhD defense.
Thinking of this, Seagull couldn’t help laughing; for the wonder of fate, he no longer opposed the matter, but hoped to help Randolph solve the twin prime conjecture in every possible way.
“Randolph, we only have five days, so I hope to tell you all my thoughts on the twin prime conjecture.”
The next day, this time it was only Lin Ran and Seagull.
“The twin prime conjecture posits that there are infinitely many prime pairs differing by 2, like 3 and 5, or 11 and 13.
From computational checks, as numbers get larger, twin primes seem to keep appearing.
Additionally, 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 times the integral from 2 to x of dt/(log t)^2, where C is the twin prime constant.
When I was at Cambridge, I discussed this with Hardy. He and Littlewood, based on their circle method work, strongly believed in the correctness of this conjecture, but this is not a proof; it’s a conjecture, just a probabilistic model they proposed.
Subsequently, around this, I did some deeper thinking; Brun’s theorem shows that the sum of the reciprocals of twin primes converges, meaning twin primes are relatively sparse compared to all primes, but it doesn’t tell us if they are finite or infinite.
The sieve method might be used to solve this problem, using sieves to prove there are infinitely many integers n such that n and n+2 both have few prime factors, then perhaps refine to prove they are prime.
This is a reasonable direction; after all, the sieve method is very successful in studying almost primes, like Selberg’s sieve method used to estimate the number of integers with certain properties.
But applying it directly to twin primes is challenging, because in the twin prime conjecture, n and n+2 both need to be prime, which is a stricter condition.
These past few years, I’ve been thinking if using analytic methods like L-functions might be more suitable.
After all, L-functions are also powerful tools, especially in problems involving arithmetic progressions.
It’s just that for twin primes, they are not directly applicable. I think we can consider capturing the Dirichlet series for twin prime distribution; the circle method pioneered by Hardy and Littlewood might provide some insights, even if not a complete proof.
No need for me to introduce the circle method more; you are also a master in the number theory field, surely very familiar with these cutting-edge methods.
For the Goldbach conjecture, that is, representing even numbers as the sum of two primes, the circle method gives asymptotic formulas for the number of representations under certain assumptions.
Similarly, for twin primes, we can try to compute the number of primes p up to x such that p+2 is also prime.
Although the error terms in the circle method are usually too large to conclusively prove the conjecture for all x, it is a valuable tool for understanding the expected behavior.
And even if you can’t prove the full twin prime conjecture in six days, partial results would be very valuable.
Even proving there are infinitely many primes p such that p+2 has at most k prime factors would be a major advance.
We don’t have to pursue completely solving the twin prime conjecture all at once.
Even just achieving this step, in my view, would be a great achievement.
Don’t put too much pressure on yourself.
Wait until my manuscript arrives, then take a look; we can communicate anytime if there are questions.”
Lin Ran grinned: “Okay, professor.”
After Lin Ran and Korolev’s moon landing special program aired, it became the hottest news globally.
Newspapers were all interpreting the offense and defense and subtext in their interview; the free world unanimously cheered for Lin Ran, feeling the professor’s words were irrefutable, tearing off the Soviet Union’s hypocritical mask.
The Soviet camp’s attacks focused on America, dredging up the Bay of Pigs Invasion, Cuban Missile Crisis, Berlin Crisis, and Kennedy’s death to rehash old news, trying to conduct public opinion offense and defense from the angle of “I’m not a good thing, but you’re even worse.”
From the perspective of the public opinion war, it seemed like the ones on the program were not Lin Ran and Korolev, but America and the Soviet Union.
Similarly, such a public opinion war also made discerning people realize that peace is still far away.
Neither side focused their reports on Lin Ran and Korolev’s statements about peace and space cooperation in the program.
And Lin Ran returning to the University of Göttingen to give an academic report, with the content being an on-site proof of the twin prime conjecture, quickly became the hottest local news in Göttingen.
Because after Döblin returned to Göttingen, he called one by one to invite masters in the number theory field from Europe and even America; the gimmick he used was that Lin Ran would share some of his thoughts on the twin prime conjecture.
He didn’t say Lin Ran would prove it on site, just emphasized that they would regret not coming.
Because it was New Year’s holiday, many scholars were unwilling to travel all the way to Göttingen, but many were willing to come.
Attending a Lin Ran academic lecture, for these scholars where travel and accommodation could be reimbursed with only time cost, was a very worthwhile thing.
For local Göttingen scholars, Döblin said Lin Ran would prove the twin prime conjecture on site, telling everyone to prepare and not fall behind the pace.
This academic lecture was leaked to the media by local scholars; as a university town, Göttingen residents have high quality, many local residents know what the twin prime conjecture is about.
In a short time, it caused a sensational effect locally.
Not only students not on holiday wanting to attend the academic lecture, many residents hoped to come on site to witness this historic moment.
Unlike the professors, most of these residents believed Lin Ran could do it.
Even the moon landing was achieved; proving the twin prime conjecture should be a piece of cake.
Lin Ran going to Göttingen, who in the world was most anxious? It must be Professor Fox.
On the third day, after figuring out the whole story through his connections in Göttingen, he made a transoceanic call to the hotel where Lin Ran was staying:
“Randolph, we can’t just let this opportunity go to the University of Göttingen for nothing!
You are a professor at our Columbia University; proving the twin prime conjecture on site should be done at Columbia University!”
Professor Fox was almost in tears.
Because he had witnessed Lin Ran explaining the Fermat’s Conjecture proof process on site; compared to Seagull, Fox obviously believed more in the idea that geniuses are omnipotent.
The genius worship culture in the mathematics community is extremely serious.
Seagull was skeptical, first because he was worried about affecting Lin Ran, second because he had researched this problem himself.
Fox hadn’t worked on it.
“Professor Fox, I might not even prove it.” Lin Ran explained.
Fox insisted: “No, Randolph, I believe you definitely can.
Others maybe not, but you definitely can.
In the 19th century, Legendre spent 40 years using integral methods to solve the elliptic perimeter problem and still couldn’t solve it.
Abel at 20 first ended the 250-year puzzle of solving algebraic equations of degree higher than 4 plaguing the math world, then with a paper ‘On the General Properties of an Extremely Extensive Class of Transcendental Functions’ directly solved the elliptic integral problem.
In the field of mathematics, the gap between geniuses and mortals far exceeds any other field; Randolph, Germans don’t believe it because they are too far from the world’s center; Americans are different.
I’ve witnessed too many of your miracles, Randolph; I completely believe you can do it.
Not only I believe; as far as I know, mathematicians from Princeton, New York University, our own school, are already teaming up to witness the miracle.
I have just one plea: can such a miracle be held at Columbia?”
Lin Ran sighed: “Just this once; after all, I came from Göttingen but haven’t contributed to Göttingen.”
Fox sighed: “Okay, I understand; I’ll arrange for the academic secretary to prepare flight tickets right now; our Columbia University mathematics department will go en masse to witness this historic moment.”
Lin Ran shook his head; actually, he had done very little preparation; he knew Zhang Yitang had advanced this problem to bounded gaps between primes.
This couldn’t be said to have solved the twin prime conjecture, only that the twin prime conjecture had been advanced to a new level, still some distance from solution.
Zhang Yitang’s work was an improvement on the Goldston–Graham–Pintz–Yıldırım result.
Later in 2014, through efforts of other mathematicians, the gap was optimized to 246, proving there are infinitely many prime pairs with difference less than or equal to 246.
This still couldn’t be said to fully solve the twin prime conjecture.
And now, he essentially had to stand on the shoulders of successors to completely solve this problem.
Did Lin Ran have confidence? Yes, but not much.
The reason for making the bold claim was completely to force himself; pressure brings motivation.
Let me see my true potential now, Lin Ran thought.
“Jenny, let’s go; I think I need to go to the hotel for some afternoon tea now.” Lin Ran said.
Last time Lin Ran came to London and stayed at Winfield Manor, it ended up infiltrated by the KGB like a sieve, so this time the White House team chose the Claridge Hotel.
Jenny, sitting quietly by the window reading a London tabloid, stood up, took a black sun hat from the coat rack: “Let’s go, professor; looks like you have no inspiration.
Or did the big talk you let slip put too much pressure on you?”
The two chatted while walking out of the room; the moment they stepped out, Lin Ran suddenly pulled Jenny, signaling her to look back.
Jenny turned back, seeing only the door and corridor; she was puzzled: “What to look at?”
Lin Ran said: “Originally I had no confidence; now I have full confidence.
Look, what’s the room number?”
Jenny said: “257, what’s wrong?”
Lin Ran said: “This room number is too wonderful; 257 itself is prime, and each of its digits, 2, 5, and 7, are also prime.
Heaven is hinting at me; I will definitely solve the twin prime conjecture on this Göttingen trip.”
Jenny said helplessly: “Professor, I didn’t expect you to be so superstitious.”
Lin Ran explained: “No, this isn’t superstition; sometimes solving some problems needs a little luck; luck can bring a strong psychological hint, and such a psychological hint is the most helpful.”
After arriving at the hotel restaurant, Lin Ran didn’t order; instead, he called the lobby manager.
“Professor, hello; what can I help you with?” The manager in a tuxedo was very polite.
Lin Ran asked: “I’d like to ask if the hotel has a room numbered 523?”
The lobby manager thought for a moment and said: “Yes.”
Lin Ran nodded: “Please arrange it; I want to move to room 523 tomorrow.”
Lin Ran didn’t ask if the room was occupied; the hotel side would handle such small matters.
After arranging, Lin Ran explained in detail to Jenny: “Jenny, I’ll have to trouble you to change rooms with me tomorrow.
A three-digit number, itself prime, and each composing digit also prime; single-digit primes are 2, 3, 5, 7; there are 15 such three-digit numbers, but if we don’t allow repeating composing digits, then there are only 2 such numbers: 257 and 523.
Since the Claridge Hotel has rooms numbered these two primes, my last two days in London will be spent in these two rooms respectively.
In the future, in mathematics history, this will be called the Claridge Hotel prime enlightenment; believe me, all mathematicians working on prime problems coming to London will have to stay one night in these two rooms, because they are about to gain the legendary color I bestow upon them.”
Lin Ran’s gaze was sharp; the whole person was completely different from the previous days of thinking about the twin prime conjecture; Jenny felt the strongest confidence from Lin Ran these days.
This was one of the rare times she directly felt the childlike side of the man before her; she smiled and pinched Lin Ran’s hand: “Professor, after you succeed, I’ll help you properly record this legendary segment in the New York Times.”
