The characteristics of the nth odd number in the hail conjecture have been demonstrated.

But Chen Zhou's pen did not stop.

I took out a new piece of scratch paper, and the tip of the pen made close contact with the paper.

He plans to continue the study of the hail conjecture in one go.

At least, all sorts of thinking during military training.

He needs to be fully unleashed.

[Feature 2: If the first hail conjecture is performed on the nth level of the number pyramid, the 2^ odd terms that can only be divisible by 2 once will continue to perform the second hail conjecture operation.] 】

[There will be 2^ terms that are only divisible by 2, 2^ terms that are only divisible by 2 times, and 2^ terms that are only divisible by 2 and 3 times,...... , 2 terms are only divisible by 2 n-4 times, only one is divisible by 2 n-3 times, and the other is divisible by 2 n-2 times or more. 】

[If you continue to perform the hail conjecture operation on the nth level of the number pyramid, the 2^ odd number that can only be divisible by 2 once will continue to perform the third hail conjecture operation......]

Chen Zhou wrote down the characteristics of the nth odd number obtained from the number pyramid when performing hail conjecture calculations2.

Writing fills an entire sheet of A4 scratch paper.

Remember in a second https://

These contents are the contents of Chen Zhou's thinking.

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

Regarding the proof of characteristic 2, Chen Zhou also started the proof from the first hail conjecture operation.

Here Chen Zhou took a trick.

He linked feature 2 to the feature.

The same is done by means of a series of numbers.

In this case, the proof will have:

【...... For the first time in the nth level, the terms that are only divisible by 2 are a2, a4, a6,...... ，a2r，...... ，a2^。 】

[In this sequence, the distance between them is 2 terms, and the tolerance is 2^2, so you can write the sequence as a2, a2+2^2, a2+2·2^2,...... ，a2+r·2^2，...... ,a2+-)·2^2 form ......】

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

Looking at the results obtained, Chen Zhou thought about it for a while and converted it.

[Treat 3^2·2 as a, 3a2+ as any integer b......]

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

He glanced at the two number theory conclusions written to prove the feature, which were also needed in the process of proving feature 2.

Using these two number theory conclusions, Chen Zhou easily deduced that "in the above equation, there is one of any adjacent 2^r terms that is divisible by 2^".

Thus, Chen Zhou completed the first step of the feature 2 proof.

This is also the most important step.

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

The train of thought is constant, steady as an old dog.

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

It's a very hearty feeling.

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

At this point, Chen Zhou has completed all the preparations to prove the hail conjecture in the early stage.

And these conclusions are all derived from the use of digital pyramids.

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 the proof of the two characteristics, which look simple and the thinking is also smooth, actually took me so much time......"

Muttering to himself, Chen Zhou didn't think about it anymore, gathered his thoughts, sorted out the previous scratch paper, and took it in his hand and brushed it over.

This is Chen Zhou's attempt to make his thinking 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 account.

Therefore, it is necessary to rationalize the way of thinking.

Moreover, in the face of world-class problems, Chen Zhou feels that it is not too cautious to be more cautious.

This is also the reason why he is praised for being extremely rigorous in calculations.

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

Chen Zhou once again entered the world of proof of the hail conjecture.

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

That is, the proof of the hail conjecture is transformed into a narrative form that is more in line with the way he now proves it.

The transformation of the narrative form also transforms the proof form of the hail conjecture.

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

Therefore, Chen Zhou needs to first prove that "all odd numbers of the nth level in the digital pyramid can become an odd number smaller than itself after a finite number of hail conjectures".

Formulating conclusions is a necessary process of proof.

[Let the odd number a be in the form of a=3^/2^a+3^/2^+3^/2^+......+3/2^+/2^b]

[When the power exponent of the denominator in the first coefficients of 3^/2^ in the above equation appears for the first time b+b2+b3+......+b≥2......]

【...... Therefore, it can be determined that the odd number a is able to become an odd number smaller than itself through several hail conjecture operations, referred to as a, and meets the condition "a>a". 】

Once the formula is complete, it is a proof of the conclusion.

This step is not so brain-intensive.

With the foreshadowing in the early stage, Chen Zhou was much easier in terms of thinking and calculation when he verified the "calculation method of the odd number of 'eligible a>a' in the nth level".

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

Combined with the content of the number pyramid, Chen Zhou sorted out a table about "the odd number of 'eligible a>a' obtained each time when the odd number in the nth level is continuously hail conjecture".

The first operation, the second operation, and the odd number of coefficients until the first operation are calculated.

In the column of the second hail conjecture operation, the rules are obtained by using the similarity of the calculation routes.

After the proof of this part of the content was completed, it was already dark outside.

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

glanced at Yang Yiyi beside him, and was looking at the textbook.

Yang Yiyi felt something in her heart and turned her head to look at Chen Zhou.

She smiled at Chen Zhou and said softly, "Let's go, finish eating, and come back?"

Chen Zhou nodded: "Have you waited for me for a long time?" Why don't you call me?"

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