Chapter 221: Fields Without Suspense
“No, Randolph, I know you are very confident.
At around your age, I solved an important problem in mathematical physics and achieved a breakthrough, and I was just as confident as you were then.
I thought the Nobel Prize in Physics was beckoning to me, and the Fields Medal was nothing more than that, but now I am over sixty, with no Fields Medal and no Nobel.
It’s good for young people to be confident, but the universal gravitation of reality will pull you back to the ground.
Additionally, Randolph, I want to remind you that the most recent International Congress of Mathematicians was in 2022, two years from now.
Half a year, there will be no Fields Medal awarded half a year later.”
Mikhail Lyubich had to admit that he was starting to like this guy a bit.
The other party was completely different from the Chinese people and those of Chinese descent he knew.
New York is the city with the most Chinese people in America, and Stony Brook University is full of those of Chinese descent and Chinese students.
Over these years, Mikhail Lyubich had no shortage of Chinese students, Chinese colleagues, and Chinese peers he met at academic conferences.
Other Chinese people were all very humble.
Even Chinese mathematicians who could give a one-hour academic report (maximum 1 hour) at the main venue of the International Congress of Mathematicians would say, when facing praise from peers, that they still had many areas to improve and that many peers were better than them.
It was the first time he had seen someone as arrogant as Lin Ran.
“Professor Lyubich, what I mean is half a year, just half a year, and the 2022 Fields Medal will lose its suspense.” Lin Ran said indifferently.
Mikhail Lyubich did not dwell on it further; he did not want to strike the young man when he was at his most proud and ambitious, as he had already given all the necessary reminders: “I look forward to that moment arriving.”
Lin Ran said: “Professor Lyubich, you will see it soon, but I still need the school to provide me with a little small help.”
Mikhail Lyubich said: “Of course, this is the school’s obligation.
Tell me, which professor do you want to guide you? To be honest, I’ve been thinking about this these past few days. You know, Stony Brook University is not very strong in number theory.
There are even fewer in the intersection of algebraic geometry and number theory.
Probably only a handful of professors have done some related work, and the relevance is very weak, making it hard to provide effective guidance.”
Mikhail Lyubich had not finished speaking; he originally wanted to say the good news was that Lin Ran would not have to wait too long, as they had poached a professor from England whose research topic overlapped highly with Lin Ran’s and who would soon be on board.
Lin Ran said: “No need, that’s not what I mean.
What I mean is, my current influence is limited. For some of my cutting-edge mathematical work to hold academic reports, I can’t keep troubling Terry. I hope Stony Brook University can help me organize enough influential mathematicians to attend my academic report.
That way, I can spread my reputation throughout the mathematics community at the fastest speed.”
The professor’s brilliance in 2020 would also shine throughout the mathematics community.
Mikhail Lyubich asked directly: “So, Randolph, hasn’t your proof of the weak form of the Goldbach Conjecture already gone through peer review?
Do you have any new achievements?”
Mikhail Lyubich could roughly guess where Lin Ran’s confidence came from; he must have achievements.
With achievements, one would be this confident.
In other words, Lin Ran still had achievements up his sleeve.
Lin Ran said: “Of course, I will release a series of achievements over the next half year.
I hope Stony Brook University can help me organize a Zoom meeting.
My idea is to first use pure mathematical tools to achieve the unconditional bound of 246 for the Twin Prime Conjecture.
Then completely take down the Twin Prime Conjecture with pure mathematical tools.
In this process, my mathematical concepts and ideas need to be disseminated throughout the mathematics community through a large number of mathematical conferences.
This aspect will need to rely on Stony Brook University.”
Mikhail Lyubich was shocked.
Truly shocked.
Because Zhang Yitang proved the bound for the Twin Prime Conjecture in 2013, and then in 2014, global mathematicians used computer-assisted tools to lower that bound to 246.
Among them, without relying on computer assistance, the best was only up to 600; this proof was completed in November 2013.
In other words, from November 2013 to now, over six years, many mathematicians have tried to prove the upper bound of 246 for the Twin Prime Conjecture using pure mathematical tools.
But over six years, no one has succeeded.
(ps: Not to mention over six years, no one has done it even by 2025)
Mikhail Lyubich thought for a moment; the weak form of the Goldbach Conjecture also used cross-field combination methods: “So the method you used on the Goldbach Conjecture can also be applied to the Twin Prime Conjecture?”
“No no no, that method cannot be used here.
The Twin Prime Conjecture is different from the Goldbach Conjecture; proving the upper bound of 246 for the Twin Prime Conjecture with pure mathematical tools is a pure number theory technical method.
It only requires the core ideas of number theory, nothing that complicated.
It’s just that completely solving the Twin Prime Conjecture requires some cross-field content.”
Mikhail Lyubich was half-believing, “When do you plan to start?”
Lin Ran said: “We can start anytime; it depends on when Stony Brook University can arrange it.
Once the time is arranged, I will upload the paper to Arxiv three days in advance, and then we can hold the talk three days later.”
Mikhail Lyubich said: “Then three days from now. Upload it today, and don’t worry about the Zoom meeting; even with time differences, I will arrange it for you.”
Whether it’s a donkey or a horse, it has to be taken out for a trot.
If you can prove yourself, Stony Brook University will allocate more resources to you.
Obviously, using pure mathematical tools to lower the bound for the Twin Prime Conjecture to 246 is also enough for a top four journal.
There are quite a few PhDs who can publish in the top four globally, but when that number becomes 2, very few can do it.
In one month, publishing two top four papers consecutively is even rarer.
Everyone rushes to publish their achievements as soon as they have them; no one holds back achievements.
PhDs who can publish in the top four don’t worry about job placements at all.
They don’t need to worry about up or out or holding onto achievements to publish after joining a university.
“Lin Ran, wait, why did you change your mind again? Weren’t you going to replicate the Apollo Program? How did you suddenly become a great mathematician?”
Li Xiaoman had rushed back to New York from Buffalo University, dusty and tired.
As mentioned earlier, both being part of the State University of New York, Buffalo University and Stony Brook University are very far apart.
Because of the virus outbreak and remote classes, Li Xiaoman had wanted to come back, but hadn’t made up her mind before.
After seeing Lin Ran’s WeChat message, she resolutely returned home, even if it meant facing her annoying uncle and aunt.
“How did I not know you had such talent!” Li Xiaoman was truly shocked.
After seeing Lin Ran’s WeChat, she knew what the Fields Medal was.
Then she searched online.
Not searching means not knowing.
Searching gave her a shock.
Just over a month apart, and Randolph Lin had made such a big name for himself.
He became famous in the mathematics community, hailed as the new star among Chinese mathematicians.
Wait, didn’t we originally plan to fool Bezos? How did you suddenly take off in academia?
No matter how much of a liberal arts student Li Xiaoman was, she had a basic understanding of mathematics.
She knew what Lin Ran’s achievements meant; at least, the path to the top in the mathematics community was now paved.
Lin Ran took a deep breath and said: “In short, a small accident happened.
But the moon landing still has to be done; there was just a little hiccup in between.
Whether you believe it or not, I have always been interested in mathematics.”
Li Xiaoman interrupted: “Stop, you previously said you were interested in the Apollo Moon Landing.
Now it’s mathematics, huh?”
Lin Ran shook his head: “I am equally interested in mathematics.
Recently, while thinking about the Apollo Program, I figured out many mathematical problems that I couldn’t understand before.
My brain works better now; do you understand?
Previously, my level in the field of mathematics could at most be a hobbyist; now, being a mathematician is more than enough.”
Lin Ran was very restrained in front of Li Xiaoman; he said mathematician, not master.
Li Xiaoman sighed: “So, are we still meeting Bezos?
But don’t have too high expectations, because in this current environment, the fundraising dinner will probably go online too.
The Donkey Party senator definitely won’t risk public outrage to hold an offline dinner.
This also means you may not get a chance to chat alone with Bezos.”
Lin Ran nodded: “Sister Xiao Man, if there’s a chance, we’ll meet; if not, forget it.”
“So what exactly is your plan now?”
Lin Ran said lightly: “Let’s get a Fields Medal first.”
(All citation relationship graphs for Twin Prime Conjecture related papers)
“Mathematical Proof on the Twin Prime Conjecture and Prime Gaps Not Exceeding 246”
That night, Randolph Lin’s Arxiv homepage had a new article.
The article title was nothing innovative.
But the article abstract was very innovative.
Because the abstract clearly stated:
“Previous proofs on the Twin Prime Conjecture relied on the following key mathematical tools: Selberg sieve method, GPY sieve method, multidimensional Selberg sieve method. These works optimized its bounds, but the numerical optimization proposals used could not detach from computation-intensive work; computation-intensive method optimization allowable parameters for k-tuples had to involve computers.
The author believes that by utilizing the EH conjecture, with stronger zero control, and then with indirect support from GRH, it can effectively solve”
The full abstract boiled down to one meaning.
We can use pure mathematical methods to lower the bound for the Twin Prime Conjecture to 246.
Terence Tao was very excited after reading it.
Back then, the collaborative effort on the Twin Prime Conjecture was his proposal; he had special feelings for this problem.
Plus, Randolph was a newcomer to the mathematics community that he had excavated.
He called James Maynard, his collaborator in the Twin Prime Conjecture field: “James, have you seen Randolph’s new paper? It’s truly shocking.”
Terence Tao was thrilled by this breakthrough, recalling the efforts of the Polymath project six years ago in his mind.
James Maynard’s exclamation came from the other end of the phone: “Of course, Randolph has been very hot lately.
As soon as his new paper was uploaded, colleagues were shouting at me on WhatsApp to check it out quickly.
I just read the abstract. He claims to have proved with pure mathematical tools that there are infinitely many pairs of primes with gaps not exceeding 246. This achievement is extremely shocking.
Previously, the best I could do was 600; he actually advanced it to 246.
But I haven’t seen how he did it specifically.
Currently, I still have some doubts about this result.”
Precisely because he had worked on such problems, he knew even better how difficult it was.
I could only get to 600, and you got to 246.
Terence Tao explained: “Based on our work in the Polymath project, he introduced some new technologies and further optimized the sieve weights.
From my quick browse of the paper, he seems to have handled the error terms more effectively, possibly using some advanced analytic number theory tools, like improved Fourier analysis.
However, there are still many parts I have doubts about; I’m very curious how Randolph specifically did it.”
Terence Tao’s mind conjured up complex Dirichlet sum estimates; just now, he had been trying to figure out how Randolph optimized the error terms.
James Maynard said excitedly: “That sounds interesting! I need to find time to read his paper carefully.”
But their confusion did not last long.
Because three days later, Stony Brook University held a Zoom meeting where Lin Ran explained his latest achievements to the mathematicians.
In the half year that followed.
Lin Ran’s progress could be described as skyrocketing without exaggeration.
One month later, he released the proof of the EH conjecture.
As mentioned earlier, the EH conjecture was proposed by Elliott and Halberstam in 1968 and published in Symposia Mathematica; originally, it remained unproven until 2025.
To put it this way, proving this conjecture means the distribution error of primes in arithmetic progressions with modulus ≤1 can be effectively controlled, far surpassing the half of standard theorems.
As a conjecture dormant for a full fifty-plus years, the paper caused a sensational effect as soon as it was released.
Because Randolph, with two improvements on others’ work, was no longer a newcomer to the mathematics community.
Holding two top four papers, he could be described as the rising star in the mathematics community without exaggeration.
After he released this paper, many people thought that with the EH conjecture proved, the Twin Prime Conjecture would soon be solved too.
To a certain extent, the two are equivalent.
But before they could react.
Lin Ran iterated a version of the paper on the proof of the Twin Prime Conjecture based on the EH conjecture proof.
“Proof of the Twin Prime Conjecture Based on the EH Conjecture Proof”
Public opinion was in an uproar for a time.
In the first half of 2020, besides the virus raging globally, Randolph was also raging in the mathematics community.
The first two were top four journal achievements, the latter two were Fields Medal achievements.
In half a year, they were published so lightly.
After the Twin Prime Conjecture proof was published, every media outlet covering academic news went crazy asking the big shots for their views.
The most common question among them was whether the 2022 Fields Medal had already lost its suspense ahead of time.
