Volume 1: Student Life Chapter 204: One Step Ahead
The property 1 of n-th-order odd numbers in the hailstone conjecture calculation has been proved.
But Chen Zhou's pen did not stop.
Take out a new piece of draft paper, and the pen tip begins to come into close contact with the paper.
He planned to continue his research on the hail hypothesis.
At least, these are the various thoughts during military training.
He needs to be completely released.
[Characteristic 2: If the first hailstone conjecture operation is performed on the nth level of the digital pyramid, the 2^(n-3) odd numbers that can only be divided by 2 once will continue to perform the second hailstone conjecture operation.]
[Among them, 2^(n-4) items are only divisible by 2 once, 2^(n-5) items are only divisible by 2 twice, 2^(n-6) items are only divisible by 2 three times, ..., 2 items are only divisible by 2 n-4 times, only one item is divisible by 2 n-3 times, and another item is divisible by 2 n-2 times or more. ]
[If we continue to perform the hailstone conjecture calculations on the nth level of the digital pyramid, the 2^(n-4) odd numbers that can only be divided by 2 once in the first two times will continue to perform the hailstone conjecture calculations for the third time...]
Chen Zhou quickly wrote down what he had learned from the digital pyramid: the properties of n-th-level odd numbers when performing hailstone conjecture operations 2.
The handwriting filled an entire A4 draft paper.
These are what Chen Zhou is thinking about.
The characteristic 2 of the nth-level odd number in the hailstone conjecture operation is extended step by step to the general form.
Regarding the proof of property 2, Chen Zhou also started from the first hailstone conjecture operation.
Chen Zhou took a shortcut here.
He linked Feature 2 to Feature 1.
The proof is also made using the number series method.
In this case, the proof will have:
[…When the hailstone conjecture is first performed at level n, the terms that can only be divided by 2 once are: a2, a4, a6, …, a2r, …, a2^(n-2).]
[In this sequence, the interval is 2 terms and the common difference is 2^2, so the sequence can be written as a2, a2+2^2, a2+2·2^2, ..., a2+r·2^2, ..., a2+(2^(n-3)-1)·2^2...]
Following this idea, Chen Zhou performed the first hailstone conjecture operation on the new form of number series, and then performed the second hailstone conjecture operation.
Looking at the calculation results, Chen Zhou thought for a moment and converted them.
[Regard 3^2·2 as a, and 3a2(1)+1 as an arbitrary integer b...]
After the conversion, Chen Zhou's thoughts became clearer.
He glanced at the two number theory conclusions he had written down to prove Property 1, which would also be needed in the process of proving Property 2.
Using these two number theory conclusions, Chen Zhou easily deduced that "in the above formula, among any adjacent 2^r (where 0≤r≤2^(n-3)) terms, there is one term that is divisible by 2^(r+1)."
Thus, Chen Zhou completed the first step in proving Property 2.
This is also the most important step.
With the first step laid, it will be much easier to prove it to the general form step by step.
The ideas are endless and steady as an old dog.
The pen in my hand constantly writes on the draft paper, turning the thoughts in my mind into reality one by one.
It's an extremely satisfying feeling.
[…From this, we can infer that the general is correct.]
At this point, Chen Zhou has completed all the preparatory work for proving the hail conjecture.
And all these conclusions were obtained by using the digital pyramid.
Chen Zhou put down his pen and looked at the time. It was already 3 o'clock in the afternoon.
"I didn't expect that it would take me so much time to prove these two properties which seemed simple and logical..."
Muttering to himself, Chen Zhou stopped thinking about it, gathered his thoughts, sorted out the previous draft paper, and smoothed it over in his hand.
This is what Chen Zhou did to make his thoughts clearer.
Because the proof idea triggered by the digital pyramid occurred during military training, there may be some details that Chen Zhou did not take into consideration.
Therefore, it is necessary to sort out your thoughts.
Moreover, facing world-class challenges, Chen Zhou felt that it would not be excessive to be more cautious.
This is also the reason why he is praised for his extremely rigorous calculations.
Put down the draft paper and take out a new piece of draft paper.
Chen Zhou once again entered the world of proving the hail conjecture.
First, Chen Zhou needs to make a formulaic conversion.
That is, the proof of the hailstone conjecture, converted into a narrative form that is more consistent with his current proof method.
The change of narrative form also changes the form of proof of the hail conjecture.
Of course, this form of proof relies on Chen Zhou’s previous preparations.
Therefore, Chen Zhou first needs to prove the conclusion that "all odd numbers at the nth level in the digital pyramid can become an odd number smaller than itself (n is any positive integer, n＞56)" after a finite number of hailstone conjecture operations.
Formulating the conclusion is a necessary process in proof.
[Suppose an odd number a (>56) is calculated through the hailstone conjecture m times, and its form is a(m)=3^m/2^(b1+b2+b3+…+bm)a+3^(m-1)/2^(b1+b2+b3+…+bm)+3^(m-2)/2^(b2+b3+…+bm)+…+3/2^(bm-1+bm)+1/2^bm]
[When the exponent of the denominator in the leading coefficient 3^m/2^(b1＋b2＋b3＋…＋bm) in the above formula first appears b1＋b2＋b3＋…＋bm≥2m…]
[…Therefore, it can be determined that the odd number a can become an odd number smaller than itself through several hailstone conjecture operations, referred to as a, and meets the condition "a＞a（m）". ]
Once the formulation is completed, it is the proof of the conclusion.
This step doesn't require that much brain cells.
With the preparation in the early stage, Chen Zhou found it much easier both in terms of thinking and calculation when he was trying to verify the "calculation method of odd numbers that meet the condition a＞a(m) in the nth level".
In particular, Chen Zhou's use of Characteristics 1 and 2 can be said to have supported the entire verification process.
Combining the contents of the digital pyramid, Chen Zhou compiled a table about "the number of odd numbers that meet the condition a＞a(m)" obtained each time when the hailstone conjecture operation is performed continuously on odd numbers in the nth level.
The first operation, the second operation, and the number of odd numbers after the first coefficient of the mth operation are listed in detail .
In the column of the mth hail hypothesis operation, the rules are obtained by utilizing the similarity of the operation routes.
By the time I finished proving this part, it was already dark outside.
When Chen Zhou put down his pen again and prepared to stretch, he realized that it was already seven o'clock in the evening.
He glanced at Yang Yiyi beside him, who was concentrating on reading the textbook.
Yang Yiyi felt something and turned to look at Chen Zhou.
She smiled at Chen Zhou and said softly, "Let's go. Come back after dinner?"
Chen Zhou nodded: "Have you been waiting for me for a long time? Why didn't you call me?"
Yang Yiyi smiled and said, "Seeing you are so focused on your work, how can I bear to interrupt you?"
