Chapter 388: Proof Of The Collatz Conjecture

Translator: Henyee Translations  Editor: Henyee Translations

When Vera heard Lu Zhou’s praise, she smiled brightly.

This was undoubtedly the best compliment she had ever been given.

As she stood next to Lu Zhou, she said in a low voice, “Your guess is correct, Collatz conjecture is a number theory problem. It’s also a complex analysis problem…”

In as early as 1994, L.Berg and G.Meinardus proved that the 3n+1 conjecture was equivalent to the function equation h(z3) = h(z^6)+{h(z2)+λh(λz2)+λ2h(λ2z2)}/3z (where λ=e^(2πi/3)). This could be expressed through the unit disc {z:|z|<1} as h(z)=h0+h1z/(1−z) (where h0 and h2 are complex constants).

In 1998, D.Schliecher used this foundation to prove that any integral function in the form of h(z) results in g(z) = z/2 + (1-cos(πz)(z+1/2)/2+1/π(1/2-cos(πz)sin(πz)+h(z)sin^2(πz)).

Vera used these two conclusions to construct a wonderful transcendental integral function. She proved that both g(z) and Φ(g) contained a positive integer, such that for branch D, there was z0∈D, by which {g^ok(z0)}∞/k=1 converged to 1.

The Collatz conjecture was established by inference!

“Excellent proof…” Lu Zhou smiled happily as he said from the bottom of his heart, “I’m amazed.”

He met Vera in the summer of 2016; it was now the end of 2017.

Lu Zhou was happy to see his student’s progress.

He was also happy to see that the Group Structure Method he used to solve the Goldbach’s conjecture was applied by his own students.

He now knew exactly what kind of “joy” the high tech system was talking about.

“This is all thanks to your guidance,” Vera said humbly. Her eyes were full of gratitude as she looked at her professor.

Although the proof process was completed by her, the proof idea was provided by Lu Zhou.

Ever since her in-progress report at Berkeley at the beginning of the year, she had been working with Hardy and Qin Yue on the final proof. All of this work was based on Lu Zhou’s idea.

Vera was the one that completed most of the work. Therefore, she knew better than anyone how important Lu Zhou’s proof idea was.

Lu Zhou smiled. “You don’t have to be humble. I only suggested a direction, but you’re the one that ran to the finish line.”

He paused for a second before continuing, “I suggest you submit the thesis to Annual Mathematics, but the editors are on vacation these days. So, you can post it on arXiv first… Maybe the editors of Annual Mathematics will notice your thesis before the end of their holidays.”

Posting the thesis on arXiv could prevent someone else from posting a similar thesis first. However, this didn’t matter too much. The Collatz conjecture was popular in the ’80s and ’90s, but these days, it wasn’t a trending topic at all. The chance of a similar thesis was almost zero.

Of course, Lu Zhou made this suggestion with his own benefits in mind.

Using the inferences he made when he solved the Goldbach’s conjecture, the system’s deciding factor of when the mission was completed was when the thesis was released to the public.

If the thesis was posted on arXiv, then his mission would be completed.

When Vera heard Lu Zhou’s advice, she nodded her head seriously.

“I understand, I’ll do that right now.”

Lu Zhou smiled and said, “Okay then, I’ll help you erase the blackboards… Thank you for the gift.”

The moment Vera heard Lu Zhou’s praise, she smiled.

After a while, she suddenly lowered her head and blushed.

“Professor.”

Lu Zhou: “What?”

She gave herself some courage and looked up at Lu Zhou.

“Can… I ask you for a wish?”

“I’m not Santa, I can’t put your gifts in a sock next to your bedside table,” Lu Zhou joked. He then paused for a second before adding, “But, as long as it’s not a violation of any principles, I’ll do anything to help you.”

“I’m…”

Vera opened her small mouth.

However, she suddenly realized that her wish might cause trouble for him.

The words were at the tip of her tongue, but she couldn’t say it out loud.

“I… want to do a PhD under you.”

Lu Zhou smiled. “I welcome you! I’ll be offended if you choose another supervisor.”

Vera felt a little more comfortable and a gentle smile appeared on her face.

She was disappointed at herself for not having enough courage.

However, she had gained a lot of courage over the past two years…

But, it still wasn’t enough.

…

Lu Zhou spent the day hanging around the Princeton Institute for Advanced Study. He then returned to his home while carrying all of the gifts from his colleagues and students.

The gifts weren’t extravagant; they were less than US$30. However, they were meaningful.

As for his gifts to his students, he gave them the pens that he had collected at conferences from all over the world.

The gifts weren’t expensive, but they were memorable.

Lu Zhou put the gifts away and sat on the sofa by the fireplace. He then closed his eyes and went into the system space.

As he walked in front of the translucent holographic panel, he suddenly saw two lines of text floating in front of him.

[A. “Research on the even coherent state of a q-distorted harmonic oscillator in a finite-dimensional Hilbert space.”

Student: Wei Wen.

Student engagement: 25%.

Type of experience: Mathematics, Physics. ]

[B. Collatz conjecture.

Student: Vera, Hardy, Qin Yue.

Student engagement: 70%.

Type of experience: Mathematics.]

Before Lu Zhou got home, Vera already uploaded the thesis to arXiv and satisfied the system’s mission conditions. Therefore, the Collatz conjecture was posted on the mission panel.

Lu Zhou didn’t know how the system determined whether a person was his student, but the system was quite accurate.

Not just that, but the system could even calculate the proportion of the student’s work participation.

“Is this a multiple-choice question?”

Lu Zhou looked at the two options on the screen and crossed his arms.

According to the system mission description, he could choose one thesis as a mission submission.

The total amount of subject experience reward was based on the academic value of the thesis multiplied by the coefficient of the student’s participation in the thesis.

It was obvious that the system wanted him to nurture genius students who could do their own research. Therefore, the system required him to avoid direct participation in the thesis.

Actually, Lu Zhou had been trying to be more “hands-off”.

However, even with his hands-off attempts, the system rated his contribution as 30%.

Because in the beginning, he developed a clear research direction for them and developed a framework for the entire research project.

His students followed his research direction and completed the proof.

Lu Zhou thought that the system would allow this, but now it seemed that he was too optimistic.

“It really doesn’t want me to help at all.”

Lu Zhou looked at the translucent screen and shook his head.

When he reached out and selected “B”, a pop-up came out. He then selected “Confirm”.

After that, a wave of blue light swept through the information screen, and a line of text appeared.

[Congratulations, User, for mission completion!]

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