Chapter 50

"I apologize; I know since I came to Xiangjiang, everyone has been eagerly anticipating a more specialized academic lecture from me here."

"I’ve always been preparing for this. I chose today for this lecture solely because I wanted the young people in my seminar from Xiangjiang to understand its content and find directions of interest from the topics I present, so they can produce valuable work."

In the lecture hall of Xiangjiang University, compared to the sparse handful of attendees in the past month, this time the room was packed.

Besides local Xiangjiang mathematicians, there were mathematicians from across Asia, with the largest delegations coming from Nihon and India.

Nihon’s rapid postwar economic recovery, coupled with Kunihiko Kodaira’s 1954 Fields Medal, created a strong mathematical atmosphere there.

Kodaira’s research focused primarily on complex algebraic geometry, which overlaps extensively with the Langlands Program.

As a result, led by Kodaira, a group of mathematicians from Tokyo University, Kyoto University, and Osaka University came to Xiangjiang University, hoping to communicate directly with Lin Ran.

You won’t come to Nihon? Then we’ll come to Xiangjiang.

India’s presence stems from Ramanujan; its mathematical research centers on number theory and statistics, and Fermat’s Last Theorem is the crown jewel of number theory.

They too desperately wished to communicate directly with Lin Ran.

The lecture hall was filled with people; Xiangjiang reporters who had rushed to the scene stood at the back taking photos, already drafting headlines: “The Glory of the Chinese People Gives His First Lecture in Xiangjiang, Drawing Mathematicians from Across Asia to Pay Homage.”

To Xiangjiang media, even though Kodaira had won the Fields Medal, his status was certainly inferior to Lin Ran, who had proven Fermat’s Conjecture.

"I believe those of you who traveled far to attend my lecture are already familiar with Fermat’s Conjecture and its proof."

"I intend to build upon Fermat’s Conjecture to present my new conjecture."

"First, I’ll begin with Fermat’s work on Diophantine problems."

Lin Ran was ruthlessly exploiting Fermat.

The Diophantine problem, posed by the ancient Greek mathematician Diophantus, asks: find four rational numbers such that the product of any two, plus one, is the square of a rational number.

Fermat found a positive integer solution {1, 3, 8, 120}, and then asked: can a fifth integer be added to this set so that the new set still satisfies the Diophantine condition?

"I can prove Fermat’s Diophantine conjecture on just one sheet of paper."

The mathematicians in the audience erupted in shock—though Fermat’s Diophantine conjecture was less famous than Fermat’s Last Theorem, it had plagued the mathematical community for centuries without solution.

And now you claim you can prove it on one sheet? That’s absurd.

"The general process is this: first construct the Diophantine equation, then transform it into a Pell equation, and finally apply the theory of linear forms in logarithms to eliminate all other solutions."

The Indian mathematicians in the front row couldn’t hold back anymore and raised their hands in protest: "Professor Lin, what exactly is this theory of linear forms in logarithms?"

"I’ve never heard of this theory before."

"Neither have I."

Murmurs spread through the audience; Chen Jingrun had already realized what Lin Ran was about to present.

"That’s right—I’ll now continue with the theory of linear forms in logarithms."

"We are given algebraic numbers α₁, α₂......"

"This theory extends Gelfond and Schneider’s work on transcendental numbers, broadening the scope to linear combinations of multiple logarithms."

"Additionally, I’ve improved classical techniques in Diophantine approximation, enabling everyone to use this method to estimate lower bounds of linear forms."

The room erupted in applause—every one of them was a mathematician, and they all knew how powerful this was.

One could say that if Lin Ran’s method has no flaws, this so-called theory of linear forms in logarithms will become a powerful tool in modern number theory, capable of resolving countless problems in Diophantine analysis and transcendental number theory.

"This method transforms abstract number-theoretic problems into computable operations, linking parts of the Langlands Program."

Because Fermat’s Last Theorem’s proof relied on the Taniyama-Shimura Conjecture, Shimura Goro, who had just become an associate professor at Tokyo University, had come with the rest of the Tokyo delegation to Xiangjiang.

Sitting in the lecture hall, Shimura Goro felt Lin Jun was nothing short of divine; the light from the window behind him bathed Lin Ran in what seemed like divine radiance.

He thought: “Lin Jun no longer settles for making conjectures—he’s building tools, is he connecting the full map of mathematics he’s laid out?”

“No wonder he’s called the Gauss of our age.”

Not only Shimura Goro was awestruck—every number theorist in the room was equally in awe.

Lin Ran continued: "And now, the most important part of today’s lecture: the theory of linear forms in logarithms, beyond proving Fermat’s Diophantine conjecture, serves a far greater purpose—it supports my new conjecture."

"I name it the ABC Conjecture."

The very ABC Conjecture that Mochizuki Shinichi claimed to have proven.

After Lin Ran’s detailed exposition of the ABC Conjecture, the audience burst into applause—this lecture had been overwhelmingly rich.

First, Fermat’s Diophantine conjecture was proven; then Lin Ran introduced a new method in number theory; finally, he unveiled a conjecture that looked impossibly difficult.

"I conceived the ABC Conjecture because it implies a simpler proof path for Fermat’s Last Theorem; intuitively, I feel it’s even harder than Fermat’s Last Theorem."

"Moreover, it connects multiple branches—including prime distribution, Diophantine equations, and modular forms—and is key to understanding the essence of integers."

The lecture, including the Q&A, lasted three full days.

None of the Asian mathematicians could escape the swift feet of Xiangjiang reporters.

“Lin Jun is the greatest mathematician in Asia; his achievements in mathematics far surpass mine.” — Kunihiko Kodaira’s exact words.

On Xiangjiang newspapers, it became: “Nihon’s Mathematical Emperor Claims He Wanted to Kneel Before Professor Lin.”

Kodaira, if he saw it, would be stunned—when did I become Nihon’s Mathematical Emperor? When did I ever want to kneel?

The rumor was even accompanied by a photo, greatly increasing its credibility.

After the lecture, the Nihon delegation strongly insisted on hosting Lin Ran for dinner, with Kodaira leading the group in a ninety-degree bow.

Nihon’s face-saving rituals were truly impressive.

The Xiangjiang reporters happened to capture it.

As a result, this rumor later gained widespread belief on Chinese-language internet forums, and in mathematical folklore, it evolved into: “Lin Ran shook his mighty frame, and Kodaira surrendered at the sight.”

Lin Ran had already prepared an excuse—he declined the Nihon delegation’s invitation, saying he needed to treat his students to dinner and was unavailable.

During the meal, Lin Ran accepted every request from the local Xiangjiang students for recommendation letters.

Only Chen Jingrun’s request to pursue a Ph.D. in mathematics under him at Columbia University genuinely surprised him.

"Dehui, you have great talent. Starting this fall, I Kongpa won’t have much time to teach you personally. How about I recommend you to study under Harvey Cohn for your Ph.D.?"