Chapter 49: Absurd News

"Sorry, I know that ever since I arrived in Hong Kong, everyone has been looking forward to me hosting a more professionally oriented academic lecture here.

I have always been prepared. The reason I scheduled this academic lecture for today is entirely because I wanted the young people from Hong Kong in my seminar to be able to understand the content, find directions of interest from the topics I introduce, and produce valuable work."

Inside the tiered lecture hall of the University of Hong Kong, compared to the sparse single-digit attendance of the previous month, it was now packed to capacity.

In addition to local mathematicians from Hong Kong, there were also mathematicians from all over Asia, with the largest contingents coming from Japan and India.

Japan's presence was due to their rapid post-war economic recovery, and Kunihiko Kodaira winning the Fields Medal in 1954, which fostered a strong mathematical atmosphere in the country.

Kunihiko Kodaira's main research direction was complex algebraic geometry, which shared a massive amount of overlap with the Langlands program.

Consequently, led by Kunihiko Kodaira, a group of mathematicians from the University of Tokyo, Kyoto University, and Osaka University came to the University of Hong Kong, hoping to communicate directly with Lin Ran.

If you won't come to Japan, then we will come to Hong Kong.

India's presence was due to the legacy of Srinivasa Ramanujan; Indian mathematical research was primarily concentrated in the fields of number theory and statistics, and Fermat's Last Theorem happened to be the crown jewel of number theory.

They, too, eagerly desired direct communication with Lin Ran.

The tiered lecture hall at the University of Hong Kong was completely packed, and Hong Kong reporters who caught wind of the news stood at the back taking photos, having already prepared their headlines: *The Pride of the Chinese People's Inaugural Lecture in Hong Kong Draws Mathematicians from Across Asia to Make a Pilgrimage.*

From the perspective of the Hong Kong media, even though Kunihiko Kodaira had won a Fields Medal, his status was certainly inferior to Lin Ran, who had solved the Fermat Conjecture.

"I trust that those of you who have traveled from afar to attend my lecture already have some understanding of the Fermat Conjecture and its proof.

I would like to follow the thread of the Fermat Conjecture to introduce my new conjecture.

I'd like to start first with Fermat's problem concerning Diophantine equations."

Lin Ran was truly milking Fermat for all he was worth.

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

Fermat had found a positive integer solution set {1, 3, 8, 120} and posed the question: could a fifth integer be added to this set such that the new set also satisfies the Diophantine condition?

"Fermat's Diophantine conjecture—I only need a single sheet of paper to complete its proof."

An uproar broke out among the mathematicians present. Although Fermat's Diophantine conjecture was not as famous as Fermat's Last Theorem, it had similarly troubled the mathematical community up to the present day without being solved.

And now you're saying you only need a single sheet of paper? That is simply too exaggerated.

***

"The general workflow is like this: first, establish the Diophantine equation, then convert it into a Pell's equation, and finally utilize the theory of linear forms in logarithms to rule out all other solutions."

The Indian scholars in the audience could no longer contain themselves. They raised their hands one after another and questioned, "Professor Lin, what is this theory of linear forms in logarithms?

Why have I never heard of this theory before?"

"I've never heard of it either."

Discussions broke out across the audience. Chen Jingrun had already realized what Lin Ran was about to talk about.

"That's right. Next, I will continue by discussing the theory of linear forms in logarithms.

Let us be given algebraic numbers $alpha_1, alpha_2 dots$"

"This theory expands upon the theories of Gelfond and Schneider regarding transcendental numbers, extending the theoretical scope to linear combinations of multiple logarithms.

In addition, it improves upon classical techniques in Diophantine approximation, allowing everyone to use this method to estimate the lower bounds of linear forms."

Warm applause erupted in the room. Everyone present was a mathematician, and they all knew how useful this stuff was.

To put it simply, as long as there are no loopholes in the method constructed by Lin Ran, this so-called theory of linear forms in logarithms will become a powerful tool in modern number theory, capable of helping solve a whole host of problems in the fields of Diophantine analysis and transcendental numbers.

"This method can transform abstract number theory problems into actionable computations, connecting several branches of the Langlands program."

Because the proof of Fermat's Last Theorem utilized the Taniyama-Shimura conjecture, Goro Shimura, who had leaped to become an associate professor in the Department of Mathematics at the University of Tokyo, also accompanied the main Japanese contingent to Hong Kong.

Sitting in the tiered lecture hall, Goro Shimura felt that Lin-kun was practically a god. The light from the window behind cast its rays upon him, making him look as though he were bathed in divine radiance.

He thought to himself: "Lin-kun is no longer content with merely making conjectures; he is already creating tools, completing and connecting the mathematical map he proposed?

He truly lives up to his reputation as the man known as the Gauss of our time."

Not only did Goro Shimura admire him with absolute reverence, but not a single number theorist present failed to feel the same awe.

Lin Ran continued, "Next is the most important content for today. Aside from using the aforementioned theory of linear forms in logarithms to prove Fermat's Diophantine conjecture, its more significant meaning is to support this conjecture of mine.

I have named it the ABC conjecture."

This was precisely the ABC conjecture that Shinichi Mochizuki would later claim to have proven.

After Lin Ran finished introducing the ABC conjecture in detail, the audience broke into thunderous applause. The content of this academic lecture was simply too rich.

***

First, Fermat's Diophantine conjecture was proven; then, Lin Ran proposed a new number theory method; and finally, he pulled out a conjecture that looked excruciatingly difficult at first glance.

"The reason I conceived of the ABC conjecture is because it contains a more concise proof path for Fermat's Last Theorem. My intuition tells me it is even harder than Fermat's Last Theorem.

Furthermore, it connects multiple branches including prime distribution, Diophantine equations, and modular forms, serving as the key to understanding the essence of integers."

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

Not a single one of these Asian mathematicians could escape the fleet-footedness of the Hong Kong reporters.

"Lin-kun is the greatest mathematician in Asia; his achievements in the field of mathematics far exceed my own." These were Kunihiko Kodaira's original words.

When they appeared in the Hong Kong newspapers, they transformed into: *Japanese Mathematical Emperor Claims He Wanted to Kneel Before Professor Lin.*

When Kunihiko Kodaira saw that, he must have been dumbfounded. When did I become the Japanese Mathematical Emperor, and when did I ever want to kneel?

The act of kneeling was even accompanied by a photograph in the report, which greatly increased its credibility.

Because after the lecture ended, the Japanese side strongly invited Lin Ran to dinner, with Kunihiko Kodaira taking the lead in bowing at a ninety-degree angle.

The Japanese capability for superficial pageantry was truly formidable.

And it happened to be captured on camera by a Hong Kong reporter.

This led to the rumor being widely believed on the simplified Chinese internet in the future, even transforming in unofficial mathematical folklore into: *Lin Ran's mighty aura shook them to the core, and Kunihiko Kodaira capitulated on sight.*

Lin Ran had found an excuse in advance, stating that he needed to treat his own students to dinner, and declined the Japanese contingent on the grounds of a scheduling conflict.

During the banquet, regarding the requests for recommendation letters from the local Hong Kong students in attendance, Lin Ran accepted every single one without exception.

It was only Chen Jingrun's desire to go to Columbia University to pursue his PhD in mathematics under his guidance that genuinely surprised him somewhat.

"Dehui, you are very talented. From the second half of the year onward, I'm afraid I won't have much time to teach you. How about this—I will recommend you to pursue your PhD under Harvey Cohn. What do you think of that?"