Chapter 269 Hail Conjecture

The MRS conference is one of the regular academic activities of the Materials Research Society of America, and it is also the most influential top conference in the field of materials science.

Its coverage covers almost all research directions in the entire field of materials science, and its status is probably equivalent to the "International Congress of Mathematicians" in the field of materials science. Almost all the big cows in the field of materials science will show their faces at the conference.

However, unlike the "International Congress of Mathematicians" held every four years, MRS is held twice a year, divided into spring and autumn. Spring is generally in Phoenix, Arizona, and fall is more stable, usually in Boston, Massachusetts.

The purpose of the meeting is mainly to exchange technologies, show the results to the industry, matchmaking between rich companies and labs that lack money, and also provide a place for peers to compete.

Yes, it is tearing.

Don't be surprised if someone throws a shoe at a speaker at a meeting. On the contrary, no one quarreled in that session. Everyone finished the meeting calmly, exchanged opinions, reached a consensus, and praised the technology of the peers... That will definitely make people in the industry wonder whether the sun has come out from the west.

The bigger the bull, the harder it will be torn.

This situation is difficult to see at the International Congress of Mathematicians.

In a sense, the painting styles of mathematics and other subjects are really different.

As a mathematics professor, Lu Zhou has no interest in bullying.

But for him, this meeting is an opportunity.

Moreover, MRS suddenly sent him an invitation letter, presumably many people are very interested in his research results.

Of course, even so, Lu Zhou did not forget his identity.

He is a mathematics professor.

Even in order to see the "future era" after the tenth level, he has to take a long-term view. No matter how you say it, he can't leave behind the mathematics that determines the upper limit of other subjects.

On the last day of August, Lu Zhou tested two other students in the office of the Institute for Advanced Study.

There are also ten questions, and the time limit is two hours.

After handing over the A4 paper with the title written on it to the two of them, Lu Zhou picked the book from his desk and read it in his hand.

Time passed by every minute and every second.

As the phone rang, Lu Zhou closed the book in his hand with a snap, and looked at the two people who were writing hard on the paper.

"The time is up, let me see how you have learned this month."

Hardy put down the ballpoint pen in his hand with a headache on his face, and Qin Yue, who also stopped writing, also looked nervous.

"Professor, the time you gave is too short," Hardy got up and handed the test paper to Lu Zhou's desk, with a bitter expression on his face, "If you give me another ten minutes, I will definitely be able to write the next question. "

"The length of time is not the key, and I didn't ask you to solve every question, just let me see what you know."

After receiving the test papers from the two, Lu Zhou glanced at the questions while talking.

For him, these are very simple things to answer, and he probably knows the number in his heart at a glance.

Qin Yue made six questions, and the seventh question was not finished, but there was no major problem with his thinking.

In general, his situation is not bad, and this is also expected by Lu Zhou.

Hardy did five tricks, barely reaching the passing standard, which was somewhat beyond Lu Zhou's expectations.

Lu Zhou originally thought that there was at least one person who could not pass his test, and this person was most likely to be Hardy. Because among these three students, his character is the most impetuous.

But now it seems that the situation is more optimistic, and all three people have obtained the qualifications to participate in this project.

Putting the A4 paper aside, Lu Zhou cleared his throat and said.

"First of all, congratulations on joining my subject."

Hearing this sentence, Hardy, who was a little frustrated because he only solved five questions, opened his eyes wide in surprise. Qin Yue beside him also showed a surprised expression.

As if seeing through what they were surprised about, Lu Zhou explained in a relaxed tone: "The passing line I set is five questions, and at least five questions can be solved, which means that you have listened to the tasks I assigned. This one and a half months At least the time is not wasted."

"As for the specific content of our subject, let me briefly talk about it."

After taking a sip of coffee, Lu Zhou stood up, walked to the whiteboard in the office, and picked up a marker.

Vera, who was sitting in the corner of the office silently collecting documents in front of the computer, also stopped her work. Like the other two students, she moved her chair and sat in front of the whiteboard, waiting for the boss to give a lecture.

"One and a half months ago, I disclosed to you that our subject was related to hailstorms."

"If you have some understanding of additive number theory, I believe you have probably guessed what this topic is."

Hardy and Qin Yue nodded one after another.

As Lu Zhou said, they already guessed what the topic was.

As for Vera, there was no unnecessary reaction. After all, as early as half a month ago, she had already passed the assessment, and even participated in the project long ago.

After a pause, Lu Zhou continued to preach.

"The so-called hail conjecture, also known as the Kakutani conjecture, or the 3n+1 problem. The proposition it describes is that, for any positive integer N, after fokn(n)=1 continuous action for a limited number of times, all fall to fall into the number trap of {4, 2, 1}."

"Plainly speaking, choose an N. If N is an odd number, the next step is 3N+1; if N is an even number, the next step becomes N/2. After a limited number of cycles, no matter how its value expands during this period, but eventually It must plummet like hailstones to the bottom of 1."

Having said that, Lu Zhou paused for a moment, smiled and continued.

"Like a black hole."

Compared with the Goldbach conjecture, the hail conjecture is undoubtedly more famous in the United States.

In the 1970s, almost all American university campuses could see people studying this magical "numbers game". And this phenomenon even appeared in the "Washington Post", an old North American newspaper, and formed a trend for a period of time.

Of course, for ordinary people, this is a numbers game, but for mathematicians, it implies something deeper.

"It's a number theory problem, and a classic one in additive number theory. But it's a complex analysis problem at the end of the day!"

"The Kakutani conjecture is your task for the next three years. I don't ask you to prove this proposition completely, but you must at least complete a paper worthy of being included in the Annals of Mathematics..."

Lu Zhou thought for a while, picked up a pen, and wrote a line of calculations 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})

Seeing this line of calculation, Qin Yue immediately took out the notes he carried with him from his pocket, and Hardy quickly cheered up.

As for Vera, she listened attentively as always.

"Although the outside world is generally pessimistic about solving this problem, in fact, the number theory community has not made no progress on this problem."

"In the 1990s, to be precise, in 1994, Professors L. Berg and G. Meinardus proved that the 3n+1 conjecture is equivalent to the function equation h(z^3) , which is the equation I wrote on the panel above."

"The appearance of this equation paved the first brick to the top of the mountain for subsequent proofs."

Some things cannot be said.

Responding to the three pairs of eyes full of expectation, Lu Zhou turned around and continued writing on the whiteboard.

[g(z)=z/2+(1cosπz)(z+1/2)/2+1/π(1/2cosπz)sinπz+h(z)sin2πz satisfies: NΦ(g). 】

【…】

Seeing these lines of calculations, Vera's eyes gradually brightened.

Qin Yue and Hardy showed thoughtful and half-understanding expressions respectively.

After he stopped writing, Lu Zhou gently placed the marker pen on the table next to him, and smiled slightly at his three students.

"This step is critical."

"If it can be proved that there is an entire function h(z), for the above g(z), every branch D of Φ(g) containing a certain positive integer, there exists z0∈D, so that [gok(z0)] converges to 1..."

Pausing for a moment, looking at the three pairs of expectant eyes, Lu Zhou smiled and said in an affirmative tone.

"Thus, we can prove it."

"3n+1 established!"