Intellectual Exchange
A proof typically defines preconditions; if the preconditions are defined incorrectly, then all subsequent work is meaningless.
Shen Qi looked in the direction of the voice and saw that the person making the question was Professor Yan.
Who is Professor Yan? His nameplate reads "Professor Yan Gang of Hunan University".
Shen Qi did not know Professor Yan.
The Chinese mathematics community is small, and the number theory community is even smaller, yet Shen Qi doesn't know every expert and scholar.
In the spirit of friendly exchange, Shen Qi asked in a tone of seeking advice, "Professor Yan, I would like to hear more details."
Professor Yan smiled faintly and began to explain his reasoning, "Shen Qi, in your report, the number of integer points on the curve determined by Equation 2 is determined by the number of prime factors of k. Please return to the previous page."
Shen Qi flipped to the previous slide, waiting for Professor Yan's next explanation.
"I have conducted research on this Diophantine equation for eight years. The general view is that if m equals 0.1, according to the classic theory of William-Junglen, the equation has only positive integer solutions (X, Y)=(1,1), which is somewhat different from the premise you set, Shen Qi." Professor Yan spoke slowly, wearing gold-rimmed glasses, giving him a gentle and refined impression.
"This..." Zhou Yu'an, who was attending the meeting, looked puzzled. He couldn't remember what exactly Jongren's classic theory was about.
Ordinary people sometimes wonder how mathematicians can memorize so many mathematical formulas, theorems, assumptions, and corollaries.
Could mathematicians have made a mistake? Could they have confused Grønland's Theorem with Jack' theorem?
Of course, mathematicians also make mistakes and misremember things. It's just that their memory is stronger than that of ordinary people, and they study mathematics every day, so the probability of them making mistakes is relatively low.
Even the most prodigious mathematicians cannot memorize all mathematical formulas, theorems, assumptions, and corollaries. Therefore, mathematicians usually choose one or two, or at most five, main research directions and specialize in a few areas.
William Junglen was a Norwegian mathematician, not very famous, but his Grønland's Theorem is a relatively obscure theorem in the Diophantine equation branch of number theory.
If one doesn't deal with number theory every day and immerse oneself in the study of Diophantus equations, even a top student in the Mathematics Department of Yenching University might not remember the specific properties of Grønland's Theorem. Take Zhou Yu'an, for example.
It would be extremely embarrassing to bring a math book to look up formulas and theorems during a high-level conference like the Chinese Mathematicians Congress.
Zhou Yu'an had studied the basics of number theory for a semester; all Mathematics Department students had to take this course.
It's normal for Zhou Yu'an, who has chosen Differential Geometry as his Depth, not to remember obscure number theory theorems.
Ou Ye was very familiar with number theory; she remembered Grønland's Theorem, but she was unwell and resting at the hotel, so she did not attend the conference.
Zhou Yu'an was an observer and had no right to speak, but Sun Erxiong, who brought him into the meeting, did have the right to speak.
Sun Erxiong, having worked in the Mathematics Community for so many years, could understand Professor Yan's point of view.
"Is this Professor Yan trying to embarrass Shen Qi?" Sun Erxiong frowned, pondering a solution to help Shen Qi out of the predicament.
However, Sun Erxiong was overthinking it.
After a moment's thought, Shen Qi replied fluently and confidently, "First of all, I completely agree with Professor Yan's point of view; Grønland's Theorem is applicable here. In fact, I used Grønland's Theorem in my first draft of the paper."
"After much deliberation and numerous revisions, incorporating the opinions and suggestions of number theory experts from Yenching University and researchers from Princeton, I finally made up my mind to redefine the Thue equation in the ninth draft of the paper."
"In this definition, if for some k > 0, u2k+1 is a square number, then u1 is also a square number. This does not contradict Professor Yan's viewpoint, and I am grateful for Professor Yan's insightful observation." Shen Qi stated calmly, without arrogance or impatience, adhering to his own viewpoint while not denying Professor Yan's opinion.
At this point, a young or middle-aged expert commented. "Shen Qi's redefinition of the Thue equation as a precondition is perfectly acceptable. I have read Shen Qi's original proof of the Wash Conjecture published in JAMS. The groundwork laid by redefining the Thue equation may seem complicated, but from a holistic perspective, it's like sharpening your axe before chopping wood. All solutions can be given by (u2k+1, v2k+1), which actually improves the accuracy of the entire paper and reduces unnecessary repetitive arguments."
Shen Qi looked at the middle-aged expert, who appeared to be in his mid-thirties, with thick hair, thick eyebrows, big eyes, and a square face. His name tag read "Professor Su Yiwen of Huake University and Technology".
Yes, Professor Su is a wise man, thumbs up. Shen Qi didn't know Professor Su, but Professor Su made a very good first impression on Shen Qi. A kindred spirit, worth getting to know.
Sun Erxiong smiled at Professor Su and said, "Little Su, shouldn't you go back to Yenching University?"
Professor Su exchanged a subtle glance with Sun Erxiong. "Old Sun, you're still so fat. Looks like Peking University is treating you well."
As an observer and learner, Zhou Yu'an didn't quite understand the discussion between Shen Qi, Professor Yan, and Professor Su, but he keenly sensed that there must be a historical connection between Sun Erxiong and Professor Su of Huake University.
Zhou Yu'an has keen and accurate observation skills. Professor Su studied at the Big Mathematics Department from his undergraduate to his doctoral studies and is currently teaching at Huake University.
Professor Su, who has a background at Yenching University, supports Shen Qi with reason and evidence. The work he is doing is similar to annotating Shen Qi's works.
The experts and scholars present had no further objections to this issue. Shen Qi's explanation and Professor Su's annotations were perfect and flawless. Shen Qi's redefined Thue equation is a necessary and irreplaceable prerequisite for proving the Wash Conjecture.
Professor Yan politely said, "I have reservations. Please continue your report, Shen Qi."
Shen Qi smiled slightly and proceeded to the subsequent report segment.
Debates among intellectuals are not like street brawls; whether agreeing with or disagreeing with someone's viewpoint, one must present sufficient evidence and clear logic.
Shen Qi's proof of the Wash Conjecture has been published in four major mathematics journals, and he has just won the Chern Prize in Mathematics, making him a rising star.
Faced with skepticism, Shen Qi could easily retort, "If you're so capable, then you do it! You spent eight years doing research, how come you didn't succeed in proving the Wash Conjecture?"
But Shen Qi couldn't say that. He was a high-level elite, about to become a mathematics master, and needed to possess the demeanor of a master.
What is the demeanor of a master?
I will conquer you with elegance and composure.
Shen Qi continued his report with elegance and composure. He was someone who had passed the test organized by the mathematical giants of Princeton. The number theory textbook for undergraduate students in the Princeton Mathematics Department was revised because of a proof by Shen Qi.
Shen Qi didn't mention this; it would lose its meaning if he brought it up himself.
Once the new edition of Princeton's number theory textbook is published, it will be in black and white, and everyone can study it on their own.
"Here I used an efficient approximation of algebraic numbers. Since you are all experts, I won't go into the proof of the lemma; I'll just get straight to the point." Shen Qi explained all the way to the second half of the report, and most of the experts present nodded frequently, praising him highly.