Chapter 71: Master-Level Figure (Requesting Follow Reads)

Mathematics requires seminars, requires an academic atmosphere, and requires guidance from masters for a very important reason.

Even some cutting-edge papers, even if the authors don’t write “easily proven” or “easily obtained,” and instead lay out the complete proof clearly and elegantly, most mathematicians will still find them baffling when reading.

“Damn, how did he think of this?”

You don’t even need cutting-edge papers—just a slightly difficult high school math problem. If you only read the detailed solution, you’ll still marvel at the thought process behind it.

Especially when it comes to the most advanced theories.

Therefore, the material Lin Ran pulled out was still highly substantive, and everyone’s attention immediately shifted from the earlier gossip to what Lin Ran was now presenting.

As he said, the mathematicians present had all prepared in advance, carefully rereading his recently published paper, and clearly understood that the linear forms in logarithms theory could be applied to many number theory problems.

So everyone was eager to know how Lin Ran had arrived at this theory, as it might help them apply it to solve other number theory problems.

“Everyone knows that besides mathematics, I’m also pursuing a philosophy Ph.D. under Professor Horkheimer, studying his critical theory, including his instrumental critique theory.

He gave me a heavy task: critical theory demands that thought transcend existing social structures, so while thinking about Diophantine problems, I also wondered—if the concept of transcendental numbers exists, could we transcend existing mathematical structures? Could we find a way to break free from the constraints of current algebraic equations?

With this question in mind, I thought of the Gel’fond-Schneider theorem, independently proven by Alexander Gelfond and Theodor Schneider in 1934 as a solution to Hilbert’s seventh problem, a theorem nearly every Göttingen mathematician must know.”

Only Siegel had returned to Göttingen.

If he were sitting in the audience, he’d probably doubt his own sanity: this kid knows so much about the Göttingen school—has he really been to Göttingen? Or am I just forgetting things because I’m old?

Lin Ran erased the linear forms in logarithms theory and began writing the Gel’fond-Schneider theorem:

“As you can see, these two mathematicians used the auxiliary function method in proving this theorem.

They constructed a function with high-order zeros at specific points, analyzed its growth properties, derived a contradiction, and proved that Λ ≠ 0.

However, these results were limited to linear forms involving two logarithms.

So I wondered: could I find a way to extend this method, expanding it from a single form to a broader scope, to handle more general linear combinations of multiple logarithms?

At the time, I only had a vague idea: the core method of the Gel’fond-Schneider theorem could certainly be extended to multiple logarithms.

So I began searching for how to construct an auxiliary function that would have high-order zeros at multiple points related to log αᵢ, while still maintaining controllable growth.

Extending from a single variable to multiple variables inevitably requires more complex tools.

So I thought of multivariate interpolation techniques—in Gelfond and Schneider’s work, the auxiliary function was univariate, but in my work, I needed a more complex tool.

At that point, interpolation theory from multivariate complex analysis and algebraic geometry seemed perfectly suited—and if combined with Siegel’s lemma, it would be perfect!”

The entire seminar was originally scheduled with two topics: the first segment assigned to Lin Ran, the second to Harvey Cohen on his latest discovery.

But Lin Ran used up all the time—the audience discussed the linear forms in logarithms theory for a full half-day, leaving no time at all for Harvey Cohen.

Of course, there was also no time left for Chen Jingrun—he never found a chance to speak privately with Lin Ran.

He only exchanged a couple of casual words with him during dinner that evening.

“Dehui, long time no see,” Lin Ran said.

Chen Jingrun was somewhat stiff: “Professor, Happy New Year.”

Lin Ran said nothing more, but turned to Harvey Cohen: “Professor Cohen, Chen is my student from Xiangjiang. I originally planned to mentor him personally, but you know—I’m very likely to take a position at Bai Gong.

I won’t have much time to teach him, so I’m entrusting him to you.

Chen has excellent talent—I believe his aptitude in number theory is no less than Chen Shenshen’s.”

This was an extremely high evaluation.

Chen Shenshen had completed his most important work fifteen years ago, proving the higher-dimensional Gauss-Bonnet formula.

As for Chen Jingrun, even in China, he was still an unknown figure, let alone in America.

Harvey Cohen didn’t doubt it: “Chen has real talent. During his interview, his understanding and insights into the Goldbach conjecture ran deeper than mine.”

Typically, a Ph.D. interview requires you to explain your research direction, what problems interest you, and your thoughts on the area you wish to pursue.

As a former member of the Chinese Academy of Sciences’ number theory seminar (on the Goldbach conjecture), Chen Jingrun would naturally have chosen the Goldbach conjecture.

“Maybe he really can solve the Goldbach conjecture,” Lin Ran said, half-joking, half-serious.

After the banquet, Harvey Cohen specifically kept Chen Jingrun behind for a chat:

“Chen, how was it? What did you think after listening to Professor Lin’s lecture?”

Chen Jingrun paused, then replied: “Brilliant. It gave me many insights.

Professor Lin demonstrated beautifully the thought process from intuition to systematic theory—this is the most valuable thing for a mathematician.

Starting from the special results of Gelfond, Schneider, and others, by deeply understanding their auxiliary function method, he creatively combined multivariate interpolation, complex analysis, and algebraic tools, gradually extending them to the general case.

He drew inspiration from key problems in transcendental numbers and Diophantine approximation, innovatively constructing auxiliary functions applicable to linear combinations of multiple logarithms.

He derived a lower bound for Λ through growth estimates and contradiction.

I feel Professor Lin possesses master-level mastery in analysis, algebra, and geometry—he skillfully fused these methods together, which is incredibly difficult.”

Harvey Cohen added: “He even embodied Professor Horkheimer’s philosophical ideas.

Lin is a master-level figure—not merely a master in number theory.

So what I want to tell you is this: Lin said you might solve the Goldbach conjecture, that your talent rivals Chen Shenshen’s. I don’t deny you have exceptional talent—but your knowledge has been too narrow, do you understand?

Your theories and methods are too limited. If we remain confined to number theory, even classical number theory, we’ll struggle to produce meaningful results.

Take Lin as an example: if he only knew number theory, could he have realized he needed multivariate complex analysis and algebraic geometry?

So my plan for you is this: you must first catch up—fill in the gaps in other fields. Number theory is far more than just number theory.

Talent is talent—but whether you can realize it is what matters.”