Rampage

Ou Ye entered the defense room and projected her Doctoral Thesis onto the screen.

"Professor Flamont, Professor Numanberg, Professor Hanks, good afternoon," Ou Ye said politely, glancing at Shen Qi and Lyndon Strauss in the audience.

Professor Flamont, the chief examiner, had a poker face. He said sternly, "Oh, this is your fourth semester of doctoral studies."

Ou Ye nodded. "Yes."

Professor Flamont was a strict man, and Shen Qi was worried for Ou Ye.

However, Ou Ye played steadily after entering the game and did not falter, which is a good sign.

Professor Flamont: "Oh, we three examiners have reviewed your Doctoral Thesis, 'The Proof of the Yesmanovich Conjecture.' You will now give a 3-5 minute presentation, after which we will ask questions."

Ou Ye: "Okay."

A 3- to 5-minute presentation? Shen Qi was somewhat surprised, as the opening presentation for doctoral students is normally between 15 and 20 minutes.

Lyndon Strauss turned his head and smiled, his eyes telling Shen Qi: We are tolerant, it varies from person to person.

Ou Ye held a page turner and switched between the PowerPoint slides of her Doctoral Thesis.

Ou Ye cut to page 3: "This, the Lucas sequence."

Without pausing on page 4, Ou Ye skipped directly to page 5: "This, Lucas pair, is equivalent."

The PPT slides show 101 pages, Ou Ye updates one page every 5 seconds on average.

The three examiners did not raise any objections and simply watched as Ou Ye rapidly scrolled through the PowerPoint presentation.

PowerPoint, this is the real PPT... Shen Qi had never seen such a concise PPT presentation, and this is precisely the essence of PPT: a strong viewpoint.

The key to creating a PowerPoint presentation is to highlight the main points of each slide. The presenter must use the most concise language to express the strongest viewpoint within the limited time.

Ou Ye's PPT presentation was extremely concise; she finished the 101-page presentation in just 5 minutes, maintaining her usual style of speaking, getting straight to the point without rambling.

"OK, thank you for your presentation. Now, let's move on to the Q&A session." Professor Flamont began by asking, "You just mentioned the Lucas sequence and defined it in your paper as un = un(α,β) = α^n - β^n / α - β, where n is a positive integer. This definition is correct; it's the premise. So, my question is, based on this definition, how do you reverse the process to find the primitive divisors of un(α,β)?"

Professor Flamont's question is a trap... Shen Qi has already gone through Ou Ye's printed paper. Finding the primitive divisor of un(α,β) in reverse is a logical trap because un(α,β) does not have a primitive divisor.

Ou Ye, clear-headed and quick-witted, replied, "It cannot be found."

Professor Flamont pressed, "Why?"

Ou Ye switched the PPT to page 13, and the laser of the page-turning pen shone on un(α1,β1)=±un(α2,β2), while simultaneously explaining, "It does not possess the property of element division."

"Really? Are you sure?" Professor Flamont pressed.

"I'm sure," Ou Ye with unwavering certainty.

"Now, Professor Numanberg and Professor Hanks will ask questions." Professor Flamont stopped asking questions and began writing and drawing on the defense record sheet.

Professor Numanberg, with a round face, bald head, and a smiling face resembling a white version of Maitreya Buddha, asked, "Oh, regarding Lemma 1, I don't quite understand your basis for choosing 5≤n≤30 and n≠6?"

"Hmm." Ou Ye was prepared. She switched the PPT to page 39, where the most eye-catching element was equation (11): (2k+1)^x±(2k(k+1)))^y√-2k(k+1)=±(1±√-2k(k+1))^z

"Given a positive integer k, there are no positive integer solutions with z ≥ 3," Ou Ye said.

"OK, I have no questions for now." Professor Numanberg lowered his head to take notes, presumably grading Ou Ye.

The second question and answer session lasted less than a minute, but Shen Qi, who was listening in, knew that the question was far more complex than it appeared.

If (x, y, z) is a positive integer solution to equation (11), then according to the definition, 1+√-2k(k+1) and 1-√-2k(k+1) form Lucas pair.

From equation (11), we can obtain a new equation, namely equation (12) in Ou Ye's paper, which can verify that uz(1+√-2k(k+1),1-√-2k(k+1)) has no essential factor.

By BHV Theorem, there is no positive integer solution (x, y, z) with z≥3. Returning to the premise definition, if un(α,β) does not have a primitive divisor, then n must be 5≤n≤30 and n≠6.

Logically, it's quite convoluted. Ou Ye's answer, "Given a positive integer k, there is no positive integer solution with z≥3," is a conclusive summary. She understands this logic, which is why she can summarize the core conclusion derived from it in one sentence.

If you asked Ou Ye to give a long explanation of her entire reasoning, she would have to talk for a whole day.

Fortunately, this was Princeton, and the three examiners had studied Ou Ye's paper beforehand. They were all renowned mathematics professors, and they could see the big picture from a single sentence. A few key points from the candidate's defense were enough for the three examiners to give a score.

At this point, Professor Hanks spoke up, "Let me say a few words. Oh, you proved that there is no z≥3, that is, z is either 1 or 2, and your final conclusion is z=2. But based on Ryan Principle, I calculated that z can take the values 1 or 2, so I think your proof of the Yesmanovich Conjecture is invalid."

Upon hearing this question, Ou Ye was stunned. "..."

Shen Qi was stunned. What the hell was the Ryan Principle?

Professor Lyndon Strauss was stunned: z must be 2, z can only be 2 and cannot be 1! I have confirmed Ou Ye's conclusion; it cannot be wrong!

Only when the condition z=2 is satisfied, by substituting it into the previous formula, can we prove that the equation a^x+b^y=c^z has only integer solutions (x, y, z)=(2,2,2), that is, the complete proof of the Yesmanovich Conjecture is true.

Professor Hanks calculated z = 2 or 1 based on Ryan Principle. If this conclusion holds true, it will overturn Ou Ye's Doctoral Thesis. The Yesmanovich Conjecture still has not been fully proven. Ou Ye's current work is not fundamentally different from Jesmanovich's proof work decades ago.

"Don't let the results of my two years of hard work be overturned!" Ou Ye panicked, her face turning pale and red. She clenched her fists and argued loudly. "Professor Hanks, please look at pages 92 to 101 of my paper. For any (x, y, z) in S, there exists a unique rational number l that satisfies the algebraic integer ring! Modulo 2(n+1) on both sides of equation (22) we get 2|x, then modulo 2n(n+1)+1 we get 4|x, and so on. We can certainly rule out the case of z=1, so z can only be 2!"

Ou Ye suddenly erupted, startling the three examiners, and Professor Hanks accidentally dropped his pen.

"This... Little Yezi who's gone berserk?" Shen Qi was also shocked. He had never seen Ou Ye so agitated. This was probably the longest sentence Ou Ye had uttered in one breath since she fell ill. It was reasonable, well-founded, and based on the truth. It was quite impressive.