I Just Want to Be a Quiet Top Student

Ouye entered the defense meeting and projected her doctoral thesis on the screen.

"Professor Flamont, Professor Nurmenberg, and Professor Hanks, good afternoon." Ouye said politely, and glanced at Shen Qi and Linden Strauss in the auditorium.

The main respondent, Professor Flamont, had a poker face. He said seriously, "Oh, this is the fourth semester of your doctoral program."

Ouye nodded: "Yes."

Professor Flamont was strict, and Shen Qi was sweating for Ouye.

However, after Ouye entered the field, he played steadily and did not lie, which is a good sign.

Professor Flamont: "Oh, your doctoral dissertation "Proof of the Jesmanovich Conjecture" has been read by the three of us, and you will make a presentation for 3 to 5 minutes, and then we will ask questions. "

Ouye: "Okay."

3 to 5 minute presentation? Shen Qi was a little surprised. Under normal circumstances, the opening presentation time for doctoral students is between 15-20 minutes.

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

Ouye holds a page presenter and switches the PPT of her doctoral dissertation

Ouye cuts to page 3: "This, the Lucas sequence."

Ouye didn't stop on page 4, and cut directly to page 5: "This, Lucas even number, is equivalent."

The PPT page number shows that there are 101 pages, and Ouye passes one page in 5 seconds on average.

The three responding officers did not raise any objections, and just watched Ouye quickly brushing the PPT quietly.

Power-Point, this is the real PPT...Shen Qi has never seen such a concise PPT report, and the essence of PPT is exactly this: a strong point of view.

The point of making a PPT is to highlight the key points of each page. The PPT reporter must express the strongest point of view in the most refined language within a limited time.

Ouye's PPT expression is extremely refined, 101 pages, and she finished the presentation in 5 minutes. The language expression style is similar to usual, only talking about the main points without sacrificing.

"OK, thank you for your statement, Ou, let's move on to the questioning session." Professor Flamont took the lead in asking, and he said: "You just mentioned the Lucas sequence, which is defined in the paper as un=un(α, β)=α^n-β^n/α-β, where n is a positive integer, this definition is no problem, this is the premise. Then what I want to ask is, based on this definition premise, how to reversely find un(α , the primitive divisor of β)?"

Professor Flamont's question is a trap...Shen Qi has already gone through Ouye's printed paper, and it is a logical trap to find the original element divisor of un(α, β) in reverse, because un(α, β) does not have the divisor of this element.

Ouye was conscious and responsive, and she replied: "I can't find out."

Professor Flamont asked: "Why?"

Ouye switched the PPT to page 13, operated the laser light of the page presenter to un(α1, β1)=±un(α2, β2), and explained simultaneously: "It does not have, the original element divides the child."

"Really? Are you sure?" Professor Flamont continued to ask.

"I'm sure." Ouye was extremely firm.

"Professors Numanberg and Hanks will ask questions next." Professor Flamont stopped asking questions, and he lowered his head to write and draw on the defense record paper.

Professor Numanberg has a round face, bald head, and smiles like a white version of Maitreya Buddha. He asked: "Oh, about Lemma 1, I don't really understand that you take 5≤n≤30 and n≠ What is the basis for 6?"

"Hmm." Ouye was well prepared, and she switched the PPT to page 39. The eye-catching focus on this page is equation (11): (2k+1)^x±(2k(k+1)))^y√ -2k(k+1)=±(1±√-2k(k+1))^z

"Given a positive integer k, there is no positive integer solution z ≥ 3." Euler said.

"OK, I have no problem for the time being." Professor Numanberg lowered his head to record, and he should be grading Ouye.

The second question took less than a minute to ask and answer, but Shen Qi, who was listening, knew that this question was by no means as simple as it seemed.

If (x, y, z) is a positive integer solution of equation (11), according to the definition of the premise, we know that 1+√-2k(k+1) and 1-√-2k(k+1) form a Lucasian even number.

A new equation can be obtained from equation (11), that is, equation (12) in Euler's paper, which can verify that uz (1+√-2k(k+1), 1-√-2k(k+1)) does not have this Elemental factor.

From the BHV theorem, it can be obtained that there is no positive integer solution (x, y, z) with z ≥ 3. Going back to the definition of the premise, if un (α, β) does not have an original divisor, then n must be 5 ≤n≤30 and n≠6.

Logically, Ouye's answer "Given a positive integer k, there is no positive integer solution z ≥ 3" belongs to the summary nature of the final word. She understands this logic in her heart, so she can summarize the core derived from this logic in one sentence. in conclusion.

If Ouye were to talk about the whole set of derivation logic at length, she would have to talk all day.

Fortunately, this is Princeton, and the three respondents have studied Ouye's thesis in advance. They are all famous mathematics professors who know each other well. One or two key words of the respondent are enough for the three respondents to give a score.

At this time, Professor Hanks spoke: "Let me say a few words, Ou, you proved that there is no z ≥ 3, that is, z is either 1 or 2, and your final conclusion is that z = 2. And I based on Ryan The principle calculates that z can take either 1 or 2, so I don't think your proof of the Jesmanovich conjecture is valid."

As soon as this question came out, Ouye was stunned: "..."

Shen Qi was stunned, what the hell is Ryan's principle?

Professor Linden-Strauss was stunned, z must be 2, z can only be 2 and not 1! Ouye's conclusion is confirmed by me, it can't be wrong!

Only when the condition of z=2 is satisfied, substituting the previous formula can prove that the equation a^x+b^y=c^z has only integer solutions (x, y, z) = (2, 2, 2), ie The complete proof of Smanovich's conjecture holds.

Professor Hanks calculated z=2 or 1 based on Ryan’s principle. If this conclusion is established, it will overturn Ouye’s doctoral thesis. The Yesmanovich conjecture has not yet been fully proved. Ouye’s current work, and There is no essential difference in the proof work of Jesmanovic himself decades ago.

Don't overthrow the results of my hard work for two years! Ouye was anxious, her face turned white and red, she clenched her fists and argued loudly: "Professor Hanks, please see pages 92 to 101 of my thesis, for any (x, y, z) in S There is a unique rational number l that satisfies the algebraic integer ring! On both sides of the equation (22) modulo 2(n+1), we get 2∣x, and then modulo 2n(n+1)+1, we get 4∣x, and so on, we The case of z=1 must be ruled out, so z can only take 2!"

Ouye broke out suddenly, the three defense officers were startled, and Professor Hanks' pen accidentally dropped to the ground.

"This... the runaway little leaf?" Shen Qi was also frightened. He had never seen Ouye so excited. This was probably the longest sentence Ouye said in one breath after he got sick. Quite 6.