Chapter 268: Hail conjecture

The mrs conference is one of the regular academic activities of the American Society for Materials Research, and it is also the most influential in the field of materials science.
Its coverage area 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.
However, unlike the "International Conference of Mathematicians" held every four years, mrs is twice a year, divided into spring and autumn. Spring is generally in Phoenix, Arizona, and autumn is more stable, usually in Boston, Massachusetts.
The purpose of the conference was mainly to exchange technologies, show the results to the industry, matchmaking for rich companies and laboratories lacking money, and at the same time provide a brutal place for peers.
Yes, it is tearing.
If someone throws shoes at the speaker at the meeting, it must not be surprising. On the contrary, no one quarreled at that session. Everyone held the meeting calmly, exchanged opinions, reached a consensus, and praised the technology of their peers ... It will certainly make people in the industry wonder if the sun is coming out west.
The bigger the bull, the more severe it is.
This situation is difficult to see at the International Conference of Mathematicians.
In a sense, the style of painting in mathematics and other disciplines is really different.
As a mathematics professor, Lu Zhou had no interest in tearing.
But for him, the meeting was an opportunity.
In addition, Mrs suddenly sent him an invitation letter, presumably many people were very interested in his research results.
Of course, even so, Lu Zhou did not forget his identity.
He is a professor of mathematics.
Even if he wanted to see the "Future Era" after the tenth grade, he had to take a long-term view, and how can he not let the mathematics that determine the upper limit of the other disciplines.
On the last day of August, Lu Zhou tested the other two students in the office of the Institute of Advanced Education.
The same is a ten-question question, limited to two hours.
After handing over the a4 paper with the title to each of them, Lu Zhou picked the book from his desk and flipped it in his hand.
Time passes by every minute.
With the ringtone of the cell phone, Lu Zhou snapped the book in his hand and looked at the two people who were struggling to write on the paper.
"Time is up, let me see how you have learned this month."
Hardy put down his ballpoint pen with a headache, and Qin Yue, who also stopped writing, was also nervous.
"Professor, the time you gave is too short," Hardy said, getting up and handing the test paper to Lu Zhou's desk. "If you give me another ten minutes, I will definitely write the next question. "
"The length of time is not critical, and I didn't ask you to complete every question so that I can see what you will do."
After receiving the two test papers, Lu Zhou said as he scanned the questions on his eyes.
For him, these are very short-answer things, and at a glance, he probably has the number.
Qin Yue made six questions, and the seventh question was not written, but the idea was not a big problem.
Overall, his condition is pretty good, and this is also expected by Lu Zhou.
Hardy did five things and barely met the eligibility criteria, which was somewhat unexpected by Lu Zhou.
Lu Zhou originally thought that at least one person could not pass his test, and this person was most likely Hardy. Because of these three students, his character is the most impetuous.
But now it seems that the situation is more optimistic, all three have been qualified to participate in this subject.
Putting a4 paper aside, Lu Zhou cleared his throat and said.
"First of all, congratulations for joining me."
Hearing this, Hardy, who was frustrated because he had only done five questions, opened his eyes in surprise. Qin Yue next to him also showed a surprised expression.
As if seeing what they were surprised, Lu Zhou stated in a relaxed tone: "The pass line I set is five questions, and at least five can be done, indicating that you have listened to the tasks I arranged. This one and a half month At least time has not gone to waste. "
"As for the specific content of our subject, I will simply talk about it."
After taking a sip of coffee, Lu Zhou stood up, walked to the whiteboard in the office, and picked up his marker.
Sitting in the corner of the office and silently collecting documents in front of the computer, Willa stopped her work. Like the other two students, she moved her chair to the whiteboard and waited for the boss to speak.
"I told you one and a half months ago that our subject was related to hail."
"If you know something about additive number theory, I believe you have guessed what the subject is."
Hardy and Qin Yue nodded.
As Lu Zhou said, they have already guessed what the subject is.
As for Vera, there was no extra response. After all, she had passed the assessment as early as half a month ago, and even had already participated in the project.
After a pause, Lu Zhou continued to preach.
"The so-called hail conjecture, also known as the Kakutani conjecture, or the 3n + 1 problem. It describes the proposition that, for any given positive integer n, after fokn (n) = 1 has been continuously applied for a limited number of times, they all fall without exception. Enter the digital trap of {4, 2, 1}. "
"In simple terms, 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 finite number of cycles, no matter how its value expands during this period, but eventually Like a hail, it will suddenly fall to the bottom of the valley. "
Having said that, Lu Zhou paused for a moment and continued with a smile.
"It's like a black hole."
Compared to the Goldbach conjecture, the hail conjecture is undoubtedly better in the United States.
In the 1970s, almost all American university campuses saw someone delving into this magical "number game." And this phenomenon has even appeared in the old North American newspaper "Washington Post" and formed a wave in a period of time.
Of course, for ordinary people, this is a number game, but for mathematicians, it contains deeper things.
"This is a number theory problem, and it is a classic problem in additive number theory. But in the final analysis, it is a complex analysis problem!"
"Kakutani conjecture is your task for the next three years. I do not require you to fully prove this proposition, but you must at least complete a paper in this direction that is worthy of being included in the annual mathematics ..."
Lu Zhou thought about it, raised his 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 that he had with him, and Hardy quickly got excited.
As for Vera, she was as attentive as ever.
"Although the outside world's views on solving this problem are generally pessimistic, in fact, the number theory community has not made no progress on this issue ~ EbookFREE.me ~ In the 1990s, it was precisely 1994 .berg) and Professor ginardus proved that the 3n + 1 conjecture is equivalent to the function equation h (z ^ 3), which is the equation I wrote in the previous panel. "
"The emergence of this equation paved the first brick to the summit for subsequent proofs."
Some things cannot be said.
Responding to the three pairs of anticipating eyes, Lu Zhou turned and continued to write on the whiteboard.
[G (z) = z / 2 + (πz) (z + 1/2) / 2 + 1 / π (1 / πz) sinπz + h (z) sin2πz satisfies: nΦ (g). 】
[……]
Seeing these lines of formulas, Vera's eyes gradually brightened.
Qin Yue and Hardy showed thoughtful and incomprehensible expressions, respectively.
After stopping writing, Lu Zhou gently placed the marker on the table next to him and smiled at his three students.
"This step is critical."
"If it can be proved that there is an entire function h (z), for g (z) above, each branch d of Φ (g) containing a positive integer has z0 ∈ d so that [gok (z0)] converge To 1 ... "
After a pause, looking at the three pairs of anticipation, Lu Zhou smiled and said in a positive tone.
"From this, we can prove it."
"3n + 1 is established!"
References: Bergl, ions-connected-with-the-coll, 1994, 25: 1-12
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