Volume 1: Student Life Chapter 203: Starting from the Digital Pyramid
After saying goodbye to the instructors, the teachers and students who participated in the training returned to school together by bus.
In the car, when the team leader asked Chen Zhou and others what was their most profound impression of this military training.
Chen Zhou unconsciously thought of the digital pyramid.
This is also his biggest gain.
Now that the military training is over, he can finally pour out all the thoughts in his mind.
As soon as he got off the car, Chen Zhou went straight to the dormitory. He was a little impatient.
So much so that when the team leader was still counting the number of people and preparing to say something, Chen Zhou's figure had disappeared.
Of course, Chen Zhou didn't just turn around and leave. He told Li Li that if he counted the number of people, he would help him.
On the way, Chen Zhou sent another message to Yang Yiyi, telling her that he would wait for her in the library in the afternoon.
However, Chen Zhou returned to the dormitory , packed his things, and went directly to the library.
He is chasing the rotation of the earth.
“First, list the digital pyramid.”
Library, old location.
Chen Zhou took out a piece of draft paper and began to draw triangles. He then graded them and listed the numbers for each level, thus completing the construction of the digital pyramid model.
"What follows is a calculation based on the hailstone conjecture."
Thinking this way, Chen Zhou continued writing on the draft paper with the pen in his hand.
[After the odd number a is subjected to m hailstone conjecture operations, the m+1th odd number obtained is recorded as a(m). Obviously, a(m)=3^m/2^(b1+b2+···+bm)·a+3^(m-1)/2^(b1+b2+···+bm)+3^(m-2)/2^(b1+b2+···+bm)+···+3/2^(bm-1+bm)+1/2^bm]
[Here we can list the m+1 odd numbers obtained after a is calculated m times using the hailstone conjecture: a, a(1), a(2), ..., a(m)]
Having written this, Chen Zhou looked back at the model of the digital pyramid.
The two can be connected in series.
In the digital pyramid, Chen Zhou marked the positions of the m+1 odd numbers obtained after a was calculated m times using the hailstone conjecture.
Then use a one-way arrow from a(n) to a(n+1) to connect these points one by one.
After doing all this, Chen Zhou looked at the digital pyramid and wrote next to the curve with a pen:
[Odd number, the route Lm calculated by hail conjecture]
"Well, regarding the route, we must be clear..."
Chen Zhou habitually tapped the draft paper with a pen, thinking about the route issue.
After verifying the route in his mind, Chen Zhou began to write:
[If ab can be divided by 2 in m hailstone conjecture operations, b1, b2, b3, ..., bm times in turn, and ac can be divided by 2 in m hailstone conjecture operations, c1, c2, c3, ..., cm times in turn...]
[…When the condition br=cr (r=1, 2, 3, …, m) holds, ab and ac can be said to have the same “operational path”. When the condition br=cr does not necessarily hold, but r=1→m∑br=r=1→m∑cr holds, ab and ac can be said to have similar “operational paths”. ]
At this point, Chen Zhou completed the preliminary preparations.
Looking at the contents that filled up a whole sheet of draft paper, he couldn't help but smile: "This idea has a future..."
Putting down the pen, Chen Zhou stretched lazily, feeling a little tired.
He was the only one who came to the library to solve the hail conjecture right after military training.
"Little brother, what were you laughing at just now?"
Hearing this familiar voice, Chen Zhou tilted his head slightly and saw Yang Yiyi standing next to him with a backpack and a lunch box.
Well, Yiyi also got tanned. Could it be that the sunscreen is fake?
Chen Zhou thought about it and before the military training, he even did some searching and prepared a military training strategy for Yang Yiyi.
But looking at Yang Yiyi's face, Chen Zhou suddenly felt that this strategy was unreliable. It was a pity that hundreds of people liked it.
"Hey, you come here without eating lunch, do you want to practice Taoism?" Yang Yiyi handed the lunch box in her hand to Chen Zhou with a hint of blame in her tone.
Chen Zhou took the lunch box and promised in a low voice: "I won't do that next time."
After a pause, he added: "Who made me have such a caring girlfriend?"
After saying that, without paying any attention to Yang Yiyi's reproachful look, he got up with his lunch box and walked away.
After all, it is not convenient to eat directly in the study area of the library.
He was afraid of being beaten.
But what Chen Zhou didn't know was that as he left, the cough that was about to happen was retracted.
The dishes that Yang Yiyi bought for Chen Zhou were all his favorites.
Chen Zhou finished this box lunch full of happiness with satisfaction.
After cleaning up the garbage, Chen Zhou slowly walked back to the study area.
Looking at Yang Yiyi's back, Chen Zhou thought for a while and went out again.
When he came back, he was holding an ice cream bowl.
Patting Yang Yiyi's back gently, Chen Zhou handed over the bowl of ice cream when Yang Yiyi turned around: "Here."
Yang Yiyi took it happily and ate the ice cream while reading the book in her hand.
Chen Zhou looked at Yang Yiyi and rubbed her head lovingly .
Yang Yiyi pouted and turned to look at Chen Zhou: "It's delicious~"
Chen Zhou smiled softly.
Putting the previously filled draft paper aside, Chen Zhou took out a new draft paper and continued his research on the hail hypothesis.
From the digital pyramid, we can also obtain some characteristics of the nth-level odd numbers when performing hailstone conjecture operations.
[Feature 1: If the hailstone conjecture is performed on each of the 2^(n-2) odd numbers at the nth level of the digital pyramid, there will be 2^(n-3) odd numbers that can only be divided by 2 once when the hailstone conjecture is performed; by analogy, there is one odd number that can be divided by 2 n+d(n) times when the hailstone conjecture is performed, where d(n) is equal to 1 (when n is an odd number) or -1 (when n is an even number)]
This is what Chen Zhou thought about during military training.
But property 1 needs to be proven.
After a brief thought, Chen Zhou began to prove it.
The proof is not difficult and requires some number theory.
Chen Zhou first wrote down the number theory content he would need, and then listed the 2^(n-2) odd numbers in the nth level of the number pyramid in order.
Then set them equal to a1, a2, a3, …, a2^(n-2) respectively.
This is an arithmetic progression with a common difference of 2.
Using this feature, the series can be converted again.
That is, the conversion is performed in the form of a2=a1+2.
After the conversion was completed, Chen Zhou tapped the tip of his pen, thought for a moment, and then wrote:
[When performing the first hailstone conjecture calculation on each term in the above formula, first multiply each term by 3, then add 1, and you get...]
Chen Zhou kept writing without stopping, and calculated each item according to his thoughts.
Then make , considering 3·2 as a and 3a1+1 as an arbitrary integer b.
At this point, we can deduce based on the number theory lemma next to it.
[…The term with the serial number 2^(n-4)+2^(n-6)+…+2+1, multiplied by 3 and then added with 1, is…]
【Therefore, this formula can be divided by 2 n+1 times. From this, we can know that the description in characteristic 1 is correct. 】
