Xueba starts with change

Chapter 204

The characteristic 1 of the nth level odd number in the conjecture operation of hail has been proved.

But Chen Zhou's writing didn't stop.

Taking out a new draft paper, the nib of the pen and the paper began to come into intimate contact.

He intends to push forward the research on the hail conjecture in one go.

At least, all kinds of thinking during military training.

He needs to be completely released.

[Characteristic 2, if the nth level in the digital pyramid is used for the first hailstorm conjecture operation, the 2^(n-2) odd numbers that can only be divisible by 3 once, continue to perform the second hailstorm conjecture operation. 】

【其中将有2^(n-4)项仅能被2整除1次，有2^(n-5)项仅能被2整除2次，有2^(n-6)项仅能被2整除3次，……，有2项仅能被2整除n-4次，只有一项能被2整除n-3次，另一项能被2整除n-2次或n-2次以上。】

[If you continue to perform the hailstorm guessing operation on the nth level of the digital pyramid, the 2^(n-2) odd numbers that can only be divisible by 4 once in the first two hailstorm guessing operations, continue to perform the third hailstorm guessing calculation...]

Chen Zhou succinctly wrote the characteristic 2 obtained from the number pyramid, the nth level odd number when performing hail conjecture calculations.

The handwriting filled an entire sheet of A4 draft paper.

These contents are what Chen Zhou thinks about.

The characteristic 2 of the nth level odd number in the conjecture operation of hail is extended to the general form step by step.

Regarding the proof of characteristic 2, Chen Zhou also started to prove it from the first hail conjecture calculation.

Chen Zhou played a trick here.

He connected feature 2 with feature 1.

Proof is also done using numbers.

In this case, the proof would have:
【... When the hail conjecture operation is performed for the first time in the nth level, the items that can only be divisible by 2 once are: a2, a4, a6, ..., a2r, ..., a2^(n-2). 】

【在这个数列中，其间隔距离为2项，公差为2^2，也就可以把数列写为a2，a2+2^2，a2+2·2^2，……，a2+r·2^2，……，a2+(2^(n-3)-1)·2^2的形式……】

According to this idea, Chen Zhou performed the first hail conjecture operation on the new form of sequence, and then performed the second hail conjecture operation.

Looking at the calculation results obtained, Chen Zhou thought for a while and converted them.

【把3^2·2看作是a，3a2(1)+1看作是任意整数b……】

After the conversion, Chen Zhou's thinking became clearer.

He glanced at the two number-theoretic results written down to prove Property 1, which are also needed in the process of proving Property 2.

运用这两个数论结论，陈舟很容易的就推知了，“在上式中，任意相邻2^r（这里0≤r≤2^(n-3)）项中都有一项能被2^(r+1)整除”这一结论。

As a result, Chen Zhou completed the first step in the proof of characteristic 2.

This is also the most important step.

With the foreshadowing of the first step, it will be much easier to prove to the general form step by step later.

Keep thinking, steady like an old dog.

The pen in my hand is constantly on the draft paper, turning the thoughts in my mind into reality one by one.

This is a very comfortable feeling.

【...From this, it can be deduced that the general form of characteristic 2 is correct. 】

At this point, Chen Zhou has completed all the preparatory work for proving the hail conjecture.

And these conclusions are all 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 the two features that look simple and smooth in thinking..."

After muttering to himself, Chen Zhou stopped thinking about it, restrained his thoughts, sorted out the previous draft paper, and stroked it in his hand.

This is for Chen Zhou to clear his mind.

Because the idea of ​​proof triggered by the digital pyramid occurred during military training, there may be some details in it that Chen Zhou didn't consider.

Therefore, it is necessary to rationalize the thinking.

Moreover, in the face of world-class problems, Chen Zhou felt that it would not be too much to be cautious.

This is why he is praised for his extremely rigorous calculations.

Put down the scratch paper and take out a new scratch paper.

Chen Zhou once again entered the world of proving the hailstone conjecture.

First of all, Chen Zhou needs to make a formulaic transformation.

That is, the proof of the Hail conjecture is converted into a narrative form that is more in line with his current proof method.

The transformation of the narrative form also changes the proof form of the Hailstone Conjecture.

Of course, this form of proof is based on Chen Zhou's previous preparations.

Therefore, Chen Zhou needs to prove first that "all the 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 limited number of hail conjecture operations ", this conclusion.

Formulating the conclusion is a necessary process of proof.

【设奇数a（＞56）经过m次的冰雹猜想运算后，其形式为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】

【当上式中首项系数3^m/2^（b1+b2+b3+……+bm）中分母的幂指数第一次出现b1+b2+b3+……+bm≥2m时……】

【...Therefore, it can be determined that the odd number a can be converted into an odd number smaller than itself through several times of hail conjecture operations, abbreviated as a, which meets the condition "a>a(m)". 】

After the formulation is completed, it is the proof of the conclusion.

This step is not so brain-intensive.

With the foreshadowing in the early stage, when Chen Zhou was proving the "calculation method of the odd number 'meeting the condition a>a(m)' in the nth level", it was much easier in terms of thinking and calculation.

In particular, Chen Zhou's use of characteristics 1 and 2 can be said to support the entire process of verification.

Combined with the content of the digital pyramid, Chen Zhou sorted out a table about "the number of odd numbers 'meeting the condition a>a(m)' obtained each time when the odd numbers in the nth level are consecutively used for hail conjecture operations."

The first operation, the second operation, and the odd number of the first coefficient of the m-th operation are listed in detail.

In the column of the conjecture operation of the m-th hailstorm, the rules are obtained by using similar calculation routes.

After completing the proof of this part of the content, the sky outside has darkened.

When Chen Zhou put down his pen again and was about to stretch, he realized that it was seven o'clock in the evening before he knew it.

Glancing at Yang Yiyi beside her, she was immersed in the textbook.

Yang Yiyi felt something in her heart, 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: "Did you wait for me for a long time? Why didn't you call me?"

Yang Yiyi said with a smile: "Seeing you are so focused on doing things, how can I have the heart to interrupt you?"

(End of this chapter)