Technology invades the modern world

Chapter 216 Who is Randolph? (6K)

"Thus, the weak form of Goldbach's conjecture has been perfectly solved."

The formulas on the blackboard were written, erased, and written again.

If printed out as formulas, it would be a full 70 pages.

(Helfgert's proof of the weak form of Goldbach's conjecture, published on Arxiv, went through three revisions, with the final version being 79 pages long.)
After Lin Ran finished speaking, he went to the blackboard and bowed to the audience.

Thunderous applause erupted from the audience.

Whoever achieved this result deserves such applause.

What's more, they witnessed a miracle once again.

According to Harvey Cohen and Lin Ran, Harvey Cohen only notified the other party five days ago, meaning that the other party only took five days to provide a complete proof of the weak form of Goldbach's Conjecture.

This would seem unbelievable to anyone else.

But when it comes to Lin Ran, the Göttingen miracle happened at the beginning of the year, and the Goldbach conjecture happened at the end of the year. Considering Lin Ran's status as a professor at Columbia University, it doesn't seem so incredible that they both have the surname "Ge".

Lin Ran was surrounded by the crowd.

"Randolph, I'll have someone compile the paper and publish it in an academic journal as quickly as possible. Goldbach's conjecture is a major achievement, even if it's just the weak form, it's still a big deal for the mathematics community," Fox said.

Fox comforted himself, thinking that no matter how Göttingen's miracle happened in Göttingen, the author's affiliation on the published paper would always be Columbia University.

Siegel said, "Randolph, I agree with Fox. Also, when can you make the strong form of Goldbach's conjecture?"
I believe everyone here is very curious about this.

If you find inspiration in Göttingen, Göttingen welcomes you anytime.

Harvey Cohen said, "Professor, look, I pushed you, and you solved the weak form of Goldbach's Conjecture."

If we push you any further, wouldn't that just be a direct result of forcing the issue?

How about we make an agreement here today that at the Mathematicians' Congress a year from now, you will solve the strong form of Goldbach's Conjecture?

This is also a way to have some fun outside of your NASA work.

Solving the strong form of Goldbach's conjecture is fun, and that applies to Lin Ran as well.

Lin Ran himself found it absurd, saying that Goldbach's Conjecture was like a Sudoku puzzle, something you could solve in your spare time by picking up a booklet and doing it.

"Let's forget about it, I feel it's too difficult to force a change."

My intuition tells me that there are obstacles in our path that require other tools to overcome.

For example, I need to use algebraic geometry to find algebraic varieties to complete the geometric modeling.

This is also thanks to the consistent efforts of mathematicians such as Grothendieck, Pierre, and Andrew over the past two decades to integrate number theory and algebraic geometry.

That's how I came up with this method.

Therefore, I suspect that relying solely on traditional number theory tools such as the sieve method and the circle method is definitely not realistic in order to solve the strong form of Goldbach's conjecture.

Chen's work, in terms of technique itself, has already reached perfection. No matter how much you try to improve it, it will be difficult to approach the strong form of Goldbach's conjecture.

I think we either need to wait for tools from other fields to provide a framework breakthrough for cross-disciplinary solutions, or we need to wait for new tools to emerge in the field of number theory.

In short, it is virtually impossible to solve strong forms using existing tools.

Lin Ran patted Chen Jingrun on the shoulder: "Chen, maybe you can complete the strong form proof of Goldbach's Conjecture before I do."

He looked at Chen Jingrun, who was much more vibrant than in his original memories, and felt a deep sense of emotion.

In the original timeline, even though Chen Jingrun made world-class achievements and gained national fame through Xu Chi's reportage "Goldbach's Conjecture," he still achieved a certain academic and political standing.

However, his personal life was clearly not very good, mainly due to his health. Early tuberculosis and later Parkinson's disease greatly limited this mathematical genius's peak achievements to "1+2".

Chen Jingrun smiled shyly and said softly, "Professor, I will definitely try my best."

At this pivotal moment in his life, he should have been the center of attention, but because of Lin Ran's appearance, the spotlight was firmly drawn to him. If it were someone else, like some mathematicians who care about fame and fortune, they might not say it, but they would definitely feel resentful inside.

In the original timeline, he lived in a small 6-square-meter room in the single dormitory building No. 88 in Zhongguancun. Later, the Institute of Mathematics arranged a 16-square-meter south-facing room for him.

His first reaction was: "My current housing is already very good. Everyone's housing is very tight. I live alone, and this is good enough!"

Chen Jingrun wouldn't. He was by nature indifferent to fame and fortune, and he had already gained more than enough from the changes Lin Ran had brought to his life.

He was deeply grateful for the heavy responsibility he bore, and similarly, from his perspective, Lin Ran, who was active in the White House, was shouldering a far heavier responsibility than he was.

Just thinking about it made Chen Jingrun feel immense pressure; if it were him, he would have succumbed to the pressure long ago.

Professor Chen Jingrun, dressed in his pajamas and sitting in the moonlight in his office at the City University of New York's Department of Mathematics, still often recalls that scene in the New York twilight.

"Professor, our advertising business has suffered a huge loss this time," IBM CEO Thomas Watson said with a wry smile after Lin Ran finished exchanging pleasantries with the mathematicians.

In addition to mathematicians and the television broadcast team, this mathematicians' conference also included Thomas Watson and IBM executives.

Everyone originally thought they could ride on Lin Ran's popularity.

What a sensation that an IBM-sponsored professor would demonstrate Goldbach's Conjecture on the spot!

Compared to the twin prime conjecture, Goldbach's conjecture has a longer history, is more legendary, and is more widely known because it is easier to understand than the twin prime conjecture.

Therefore, faced with Harvey Cohen's astronomical figure of five million dollars, IBM gritted its teeth and agreed.

The Blue Giants have the confidence to back it up.

They thought it would take five days or even a week to get it featured on two of the three major television networks in the United States for a week.

Five million isn't a loss.

Unexpectedly, the lecture was finished in just one afternoon.

The five million wasn't entirely wasted, but it certainly didn't serve any purpose.

Thomas Watson naturally dared not blame Lin Ran; he could only complain to Lin Ran in a teasing manner.

Lin Ran asked, "What's wrong?"

Thomas Watson, dejected, told Lin Ran the details of what had happened.

The goal was to leave the other party with the impression that IBM had suffered a loss because of them, so that they would remember IBM if there were any good things like Deep Blue in the future.

After listening, Lin Ran laughed and said, "In that case, I will design the New York Mathematicians' Dinner next year. IBM will be responsible for contacting the TV station to broadcast the whole event live. It will definitely be a much better large-scale marketing event than this Goldbach Conjecture event."

Not only Thomas Watson, but other mathematicians also became interested.

“Professor, can I give you a heads-up?” Thomas Watson asked.

Lin Ran nodded and said, "Of course, this is what I'm thinking: we can do a live broadcast during next year's Christmas party."

I'm playing chess against the top eight chess masters in the US all at once, and my goal is to beat them all.

Then I'll play another game of chess with Deep Blue.

That way, my name will be at the very top of Deep Blue's tech Ark kill list.

"If Deep Blue can beat me, then it will mean it has beaten humanity."

The reason for eight is that Lin Ran plans to set up a Bagua formation on the ground.

After Lin Ran finished speaking, a gasp of surprise rang out from the crowd.

Because everyone thinks it's too difficult to do; just hearing about it makes it seem difficult.

There is still a difference between mathematicians and professional chess players.

The Elo system was developed by Arpad Elo, and the United States Chess Federation (USCF) began using this scoring system in 1960.

In other words, although there is no global ranking of chess masters, there is a ranking of chess masters in the United States.

Thomas Watson's eyes lit up: "Professor, it's a deal."

He could clearly tell how much hype and advertising value this was.

Lin Ran was well aware of this, and he said, "Watson, I'm basically advertising for IBM."

Let's make a small bet: if I can remain undefeated in every game, it'll be a ten million dollar advertising fee.

Including Deep Blue, there are a total of 9 games. If I lose even one game, I won't charge you a single penny.

What do you think?

Lin Ran doubled it for him. Ten million US dollars in this era, even if calculated simply according to the consumer price index, is equivalent to 8200 million US dollars in later times.

If calculated based on gold prices, this would be equivalent to $5 million in 2020.

The former number is too low, and the latter is too high.

In short, $1000 million is a huge sum of money.

If Thomas Watson were just an ordinary manager, he wouldn't even be able to make the decision on this amount of money.

Thomas Watson is the son of Thomas Sr., the founder of IBM.

But he hesitated for almost half a second: "No problem, Professor, it's a deal. I look forward to your performance at next year's Christmas party."

There are plenty of people who want to give money to Lin Ran but can't find a way. IBM has this opportunity, so why don't they hurry up and seize it?

These top American billionaires are just as adept at cultivating connections as Chinese businessmen.

Even if we take a step back and consider the ten million as Deep Blue's technology fee, it wouldn't be a loss.

Moreover, in everyone's opinion, Lin Ran could at least serve as NASA's director for at least twenty years.

How long he works depends entirely on his personal mood.

During the dinner that evening, Jenny asked Lin Ran in casual conversation, "Professor, how confident are you in winning that ten million dollars from Thomas?"

After thinking for a moment, Lin Ran said, "Saying 100% is a bit of an exaggeration, but it's at least 95%."

During the daytime academic presentation, Professor Cohen's suggestion that I treat Goldbach's Conjecture as a Sudoku game to play in my spare time seemed like a huge exaggeration.

But playing chess against yourself in your mind during your free time can indeed be considered a form of entertainment.” Jenny rolled her eyes at Lin Ran: “Professor, what you’re saying is equally exaggerated. You can beat the top eight chess players in the US with just this kind of entertainment.”

When I was a child, I dreamed of becoming the first female chess grandmaster in the United States to win the U.S. Chess Championship, just like Nona Gaprindashvili.

America started organizing chess tournaments for women in 1938.

But Nona Gaprindashvili, a female chess player from the Soviet Union, won a series of victories in England, defeating many English master players, and winning the Hastings International Chess Tournament.

I don't know why the Soviet female chess player went to England to play chess.

In short, Nona became the first female chess grandmaster since the title of chess grandmaster was established.

Lin Ran explained, "Jenny, you know, people are different."

Jenny raised her glass to toast, downed the red wine in one gulp, and then sighed, "Professor, you're right."

I'm curious, in your eyes, are we all similar to apes, even though those sitting here are among the smartest people in America, or even the world?

Lin Ran also downed his drink in one gulp. He was used to the feeling of drinking to stimulate his brain nerves, and sometimes he even felt that such a feeling helped him think: "No, no, no, people have different talents, just like I can never guess what you are thinking."

I know some men are very good at interpreting women's thoughts and feelings.

The virus pandemic is spreading globally.

Although Australia only has sporadic cases of the virus, schools have already encouraged everyone to stay home and take classes remotely, without having to go to school.

This is good news for Terence Tao, meaning he has more time to browse MathOverflow and Arxiv.

MathOverflow is a professional forum in the field of mathematics, where Terence Tao has been active for many years using his real name.

After GPT was released, he used it to do a lot of interesting work, which he published on MathOverflow.

These interesting works are somewhat removed from professional mathematics journals, which can hardly fully express what they want to say. However, they are much more valuable than general popular science, and the average person would find it difficult to understand the connotation of their work.

Therefore, he published these AI-related mathematical works on MathOverflow.

As usual, he stayed at home, enjoying this long, special, global holiday.

In his study, the bookshelves were densely packed with mathematics books, from the classic "Introduction to Number Theory" to his latest research monographs, each one a testament to his academic career.

A whiteboard hung on the wall, covered with formulas and sketches. The light from the computer screen illuminated his face.

Outside the window, the streets were deserted, and even dog walkers were a rare sight.

For foreigners, walking their dogs is as natural as eating.

Terence Tao has long been accustomed to browsing Arxiv daily to find the latest academic developments. For him, it's instinctive.

But that doesn't mean he reads every single article.

After all, a massive number of papers are uploaded to Arxiv every day, but for him, he can judge whether most articles are worth paying attention to just by glancing at the title.

He skips titles that don't appeal to him.

Occasionally, some headlines would make him stop and read the abstract.

But the number of papers he actually read in depth was very small, probably less than one in a thousand.

His selection criteria are exceptionally strict: the title must be novel enough, and the abstract must be sufficiently in-depth; otherwise, it will be rejected outright.

The rigor is comparable to that of the novels recommended to readers by Qidian.com.

As usual, he opened Arxiv and scrolled through the pages. The titles on the screen flowed by like water, most of which he ruthlessly ignored.

Suddenly, a title caught my eye: "Application of Algebraic Geometry Methods in the Proof of the Ternary Goldbach Conjecture".

This title made him stop typing.

He was very familiar with the weak Goldbach conjecture.

In 2013, Helfgett proved this conjecture using the circle method and the sieve of large numbers method, namely that every odd number greater than 5 can be expressed as the sum of three prime numbers.

Helfgott’s work combined classical number theory techniques with modern computing power, and Terence Tao vividly remembers its proof.

But this new paper claims to have used methods of algebraic geometry to improve Helfgett's proof, which surprised him greatly.

Although algebraic geometry and number theory are both important branches of mathematics, their research objects and methods have only begun to overlap slightly in the last forty years.

However, they are mostly not very relevant, especially in the realm of prime numbers.

Algebraic geometry focuses on geometric objects defined by polynomial equations, while the prime number subfield of number theory focuses on the properties of integers.

How to apply algebraic geometry to additive number theory problems such as Goldbach's conjecture is a perplexing question.

A question flashed through Terence Tao's mind: Is this possible?

But it must be said that the title alone was enough to grab his attention.

He glanced at the author again, Randolph Lin, was he Chinese? he wondered.

It's normal that there's only one author's name.

Stony Brook University, State University of New York, is not a university known for its differential geometry. When did they start working on a combination of number theory and algebraic geometry?

Terence Tao had even greater doubts. As a well-known internet surfer in the mathematics community, he was extremely social, and many mathematics professors at the State University of New York were his friends.

He had never heard of any professor attempting to conduct research in this direction.

Filled with curiosity and a touch of skepticism, he clicked on the paper link and began reading the abstract.

The abstract states that the authors construct a specific algebraic variety on which rational points correspond to the representation of odd numbers as the sum of three primes. By studying the properties of this algebraic variety, the weak Goldbach conjecture can be proved.

Terence Tao frowned. The idea sounded very novel, but was it really feasible?
He decided to delve into the introduction of the paper.

In the introduction, the authors describe in detail how they constructed this algebraic variety and used tools from algebraic geometry to analyze its structure.

The authors claim that this method not only simplifies Helfgott's proof but also provides a new perspective for understanding the distribution of prime numbers.

Terence Tao's eyes lit up; this line of thought reminded him of unexpected connections he had encountered between different mathematical fields during his research.
These connections often lead to unexpected results.

He guessed that perhaps this paper was such an example.

He leaned back in his chair, staring at the computer screen.

If this method holds true, it will be a major breakthrough with profound implications not only for number theory but also for the entire mathematical community.

He recalled encountering similar situations in his research, such as introducing analytical methods into combinatorics or using probability theory to solve number theory problems.

These interdisciplinary attempts often open new doors to research.

He decided to download the full text and study it carefully later.

But just then, his wife walked into the study and asked, "Terry, lunch is ready. Would you like something to eat?"

Terence Tao looked up, smiled, and replied, "Oh, okay, I'll be there in a bit."

His thoughts were still on that paper.

He spent the entire day going through the pages, checking the derivation of each theorem.

The mathematical language used in the proof is complex and elegant, interweaving the distribution of prime numbers in number theory and the cluster theory in algebraic geometry.

He would stop from time to time to consult relevant literature and make sure he understood each step.

Late at night, Terence Tao closed his notebook and rubbed his temples. Although he roughly understood the framework of the paper, some technical details still puzzled him.

The next morning, Terence Tao organized a video conference, inviting several colleagues and graduate students to share the paper.

He opened his screen on Zoom, showing the abstract of the paper, his tone slightly excited: "This paper claims to prove the weak Goldbach conjecture using algebraic geometry. What do you think?"

The discussion quickly became lively.

A colleague questioned, "Can algebraic geometry handle the additive properties of prime numbers? That sounds a bit far-fetched."

Another graduate student, specializing in algebraic geometry, brightened up: "If they really construct a suitable algebraic variety, it's theoretically possible. I think this idea is very novel!"

He further explained the geometric meaning of points on a cluster, helping Terence Tao to understand the core ideas of the paper more clearly.

However, another professor raised concerns: "Helfgate's proof is already quite complete. What substantial improvement can this new method bring? Or is it just a different form?"

Terence Tao nodded slightly and noted down these questions.

He knew that academic breakthroughs often lay hidden in controversy.

He decided to continue his research and personally verify every derivation in the paper.

On the third day, Terence Tao arrived at his study early, brewed a new cup of coffee, and reopened his thesis.

This time, he jumped straight to the core of the proof, focusing on how the author connected odd numbers with algebraic varieties.

The paper introduces a construction based on elliptic curves. By analyzing the rational points of the curve, the authors establish a representation of the sum of prime numbers.

He stared at the screen, when a sudden inspiration flashed through his mind.

"I see!" he murmured to himself with a smile.

The author utilized the special properties of elliptic curves to transform the problem of prime number sums into a geometric problem, and then solved it using tools of algebraic geometry.

This method is not only elegant, but it is also likely to provide new perspectives on other number theory problems.

Tao Zhexuan leaned back in his chair, closed his eyes, and countless formulas and geometric figures floated through his mind.

He felt a long-lost surge of excitement.

This feeling was like the racing heartbeat he experienced years ago when he tackled a difficult problem.

He knew that if the proof were true, it would not only be an improvement on the weak Goldbach conjecture, but could also be a revolution in the intersection of number theory and algebraic geometry.

Terence Tao thought to himself that he had to find this author and discuss the idea with him in person.

But who is Randolph Lin? How come I've never heard of a mathematician of that caliber?
　Here's a 10,000-word update for you. Asking for a monthly vote isn't too much to ask, is it?
　　Crow's writing was quite enjoyable.
　　
　
(End of this chapter)