Chapter 268: Hail Conjecture

Translator: Henyee Translations  Editor: Henyee Translations

The MRS conference was one of the regular academic activities of the American Society for Materials Research and was the most influential conference in the field of materials science.

It covered almost all research directions in the field of materials science, and its status was probably equivalent to the “International Conference of Mathematicians”, but in the field of materials science. Almost all of the material science scholars would attend the conference.

However, unlike the “International Conference of Mathematicians” which was held once every four years, the MRS conference was held twice a year, once in spring and once in autumn. Spring one was generally in Phoenix, Arizona, while the autumn one was usually in Boston, Massachusetts.

The main purpose of the conference was to show off the technology to the industry. Laboratories could connect with rich companies for funding. It also gave a chance for people to catfight with their peers.

Yes, catfight.

It would be strange for someone to throw a show on stage. If the conference was quiet, and everyone calmly exchanged ideas, praised each other’s technology… Then the industry people would have doubts.

The nuttier people were, the more they would try to catfight others.

This type of situation would not be seen at mathematics conferences.

In some sense, the style of mathematics was different than other disciplines.

As a mathematics professor, Lu Zhou was not interested in catfighting.

However, this conference was still an opportunity for him.

Also, since MRS sent him an invitation, there must have been a lot of people interested in his research.

Of course, Lu Zhou did not forget who he was.

He was a mathematics professor.

No matter what, he was still a mathematician. He could not let his mathematics level fall behind because that level determined the upper limit level of his other subjects.

On the last day of August, Lu Zhou sat in his office at the Institute of Advanced Studies. He was testing his two other students.

10 questions, two-hour limit.

After handing them the test, Lu Zhou sat in his chair and picked up a book.

Time slowly passed by…

When Lu Zhou’s phone rang, he closed the book and looked at the two people who were struggling with the test.

“Time’s up, let me see the results of your studies for the past six weeks.”

Hardy put down his pen reluctantly. Qin Yue did the same. They were both nervous.

“Professor, the time frame you gave was way too short,” said Hardy. He got up and handed Lu Zhou the paper as he said, “I can definitely solve another question in 10 minutes.”

“The time frame isn’t important. I’m not asking you guys to solve every question. I want to test what you know.”

Lu Zhou took the two test papers and looked at the questions.

For him, these were all very simple questions. He could ballpark the answer in his head.

Qin Yue was up to question six, and he was halfway through question seven. His thought process was correct.

In general, not bad. This was what Lu Zhou expected.

Hardy did five. He had barely completed the requirement. This was somewhat unexpected.

Lu Zhou thought that there would at least be one person failing the test and it would most likely be Hardy because he was the most impetuous student out of the three.

However, it seemed that all three of them were qualified to participate in his research project.

Lu Zhou placed the test papers aside. He then cleared his throat and said, “First of all, congratulations on joining my research project.”

When Hardy heard this, his eyes widened in surprise. Qin Yue also had a strange expression.

Lu Zhou said in a relaxed tone, “My passing requirement is five questions. If you could complete five questions, that means you followed my task and didn’t waste the past month and a half… ”

“… As for the details of our research project, I’ll explain it shortly.”

Lu Zhou took a sip of his coffee before he stood up. He then walked over to his whiteboard and picked up a marker.

Vera was sitting in the corner of the office, quietly reading documents. She stopped and as the other students, she looked at the whiteboard.

“Six weeks ago, I told you guys that the research project is related to hail.”

“If you know your additive number theory, then you guys have probably already guessed what the research project is.”

Qin Yue and Hardy nodded.

As per what Lu Zhou said, they already guessed what the research project was.

As for Vera, she obviously knew about it since she joined the research project two weeks ago.

Lu Zhou paused for a second before he continued, “The so-called Hail conjecture, also known as the Collatz conjecture, or 3n+1 problem, describes that for any positive integer N, after continuous iteration of fokn(n) = 1, it would fall into the trap of {4,2,1}…”

“… Simply speaking, start with any positive integer n. Then each term is obtained from the previous term as follows: If the previous term is even, the next term is one half the previous term. If the previous term is odd, the next term is 3 times the previous term plus 1. The conjecture is that no matter what value of n, the sequence will always reach 1.”

Lu Zhou paused for a second. He then smiled and added, “It’s like a black hole.”

Hail’s conjecture was no doubt more popular than Goldbach’s conjecture.

In the 1970s, almost all of the America Universities were delving into this magical “number game”. This phenomenon was even reported in the “Washington Post”.

Of course, for most people, this was just a game of numbers, but for mathematicians, this was something deeper.

“This is a number theory problem, and one of the classics in additive number theory. But, the essence is actually a complex analysis problem!”

“… The Collatz conjecture will be your mission for the next three years. I’m not asking you guys to fully prove this conjecture, but you should all at least complete one thesis worthy of publication…”

Lu Zhou picked up the pen and wrote down an equation on the whiteboard.

[h(z^3)=h(z^6)+{h(z^2)+λh(λz^2)+λ^2h(λ^2z^2)}/3z] (where λ=e^ {2πi/3}]

When Qin Yue saw this line of equations, he took out his notebook. Even Hardy also started to pay attention.

As for Vera, she was as focused as ever.

“The community is pessimistic about this problem. In fact, the number theory community has made no progress on this problem.”

“In 1994, Professor L. Berg and G. Meindardus proved that the conjecture is equivalent to the function h(z^3), which is what I wrote on the whiteboard… ”

“… This equation placed down the first brick to solving this problem…”

Some things could not be described in words.

Lu Zhou turned around and continued to write on the whiteboard.

[g(z)=z/2+(1−cosπz)(z+1/2)/2+1/π(1/2−cosπz)sinπz+h(z)sin2πz satisfies: N⊂Φ(g)]

[…]

Vera looked at the lines of equations and her eyes lit up.

Hardy and Qin Yue also have a thoughtful expression.

Lu Zhou finally stopped writing and placed the marker on the table. He smiled at his three students.

“This step is crucial… ”

“… If you can prove that there is an integer function h(z), for each g(z) above, each branch of Φ(g) containing a positive integer has z0∈D, so that [gok(z0)] converges. To 1…”

Lu Zhou paused for a second and looked at the three faces of anticipation. He then smiled and said in a positive tone, “Therefore, we can prove that…”

“3n+1 is true!”

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