What is Shape Number?

Let's start with Pythagoras.

Pythagoras used small stones at equal distances to form equilateral triangles or squares, or pentagons, hexagons, and the number of small stones used were called triangles, squares, and pentagons.

Number of triangles: 1, 3, 6, 10...... is the sum of the first n natural numbers;

Number of squares: 1, 4, 9, 16...... It's the square number;

Then there are pentagonal numbers, hexagonal numbers, and so on.

Don't think it's easy or difficult, there are so many mysteries in the number of shapes that you can't imagine!

To put it simply, the Pythagorean theorem we studied is actually a special case of the number of squares. It is equivalent to two small squares, and under what circumstances can they be placed into one large square.

If the Pythagorean theorem is extended to the power, a^n+b^n=c^n, it is Fermat's conjecture, and of course it is Fermat's great theorem;

If we extend the number of terms, there is a four-square-sum theorem: any integer, expressed as a^2+b^2+c^2+d^2...... Do you need a maximum of four items for such a form?

This is entirely in the realm of shapes, and was finally proved by Euler and Lagrange.

But to continue to expand it is the Hualin problem, the square number needs four terms, and the cubic number needs how many terms? What about the power of 5? What about the power of 6? This is a big hole that has not been solved to this day.

Not only that, but Fermat also dug another hole in the field of shape numbers, called the polygon number conjecture.

The conjecture was won by Gauss, the little prince of mathematics, and Cauchy completed the final proof, which lasted more than 200 years.

Although it has been proved that if we continue to expand, we will reach the perfect cube problem, which is another big hole that has not yet been proved or disproven......

Therefore, although Gan Dadi had only raised his head, Ye Han had already faintly felt bad.

It's not that he can't answer the question, of course, there is a possibility that he can't answer, but even if he can, the probability that his answer will be lost to the other party, and the other party can understand it is close to zero.

Sure enough, Gan Dadi first threw out two relatively simple questions to ask for directions, if you know that the sum of the adjacent triangle numbers is the number of squares, or the nth cubic number is the square of the nth triangle number, you can easily give the answer.

And then he tried to see the dagger!

Let's give a few examples, such as 4=3+1;5=3+1+1;7=6+1;8=6+1+1;9=6+3;14=10+3+1;20=10+10......

Then asked Ye Han, can all numbers be represented by up to three triangle numbers?

Yes.

The number of triangles can be represented by three numbers, the number of squares can be represented by four numbers, and the number of polygons can be represented by as many numbers, which is the polygon number conjecture. One of Fermat's famous conjectures that "places are too small to write".

The above is just a case of n=3.

But even n=3 is not so easy to prove, and I think that the little prince of mathematics was so excited that he shouted Eureka after he proved it. Ye Han didn't think that if he copied out the certificate, the guy on it would definitely be able to understand it.

After a moment's deliberation, he spoke: "I know not only that all positive integers can be represented by three triangular numbers, but also that they can be represented by four square numbers, or five pentagonal numbers, and six hexagonal numbers...... It's just that the proof process is too complicated to explain for a while. ”

Although the emotional intelligence is not high, it is not difficult to copy the routine of Fermat back then.

Once again, Gan Dadi was on the spot. qq

Why, because his follow-up question is this, let Ye Han answer it before he could say it.

And since the other party gave a conclusion without thinking about it, although there is no proof process, I think it is really a deep study of this issue. This...... Do you want to continue?

Gan Dadi was in a dilemma for a while.

If he is thick-skinned, he is definitely thick enough.

But thickness also has a limit. The key is that since the contact, Ye Han's understanding of the way of mathematics has far exceeded his imagination, and he has been critted one after another on the most proud issues, even if he is a sweet land, he is a little unable to hold on.

gave birth to Ye Han's learning like an abyss and a sea, and he couldn't reach the bottom of his water nature.

Gan Dadi was in a daze, and the note written by his cheap grandson was also handed to Ye Han by a daredevil.

Before receiving the note, Ye Han had a faint love for Gan Dadi.

Imagine a person staying on this cliff that does not reach the world and the ground, relying only on the gravel at hand, and for a moment he laid out Euler's natural numbers and results, and for a moment he delved into the field of shape and number......

You know, it's all self-taught, and there are no references. Wouldn't it be appropriate for someone to guide this with information and guidance?

【……】

However, when he read the content of the cheap grandson's note at a glance and dozens of lines, his love for talent ...... It's more abundant.

Feelings: This is a Qin Jiushao and Gavaro-type talent.

Qin Jiushao, a master of mathematics in the Southern Song Dynasty, has made world-class contributions to the Chinese remainder theorem, triclagonal quadrature, and Qin Jiushao's algorithm. The BBC's documentary on the history of mathematics is mentioned very little by other Chinese mathematicians, only a few sentences, but for Qin Jiushao, it can be called a strong color.

But what does this guy say? greedy, brutal, and self-serving...... It is not an exaggeration to say anything to describe corrupt officials.

Almost all of his mathematical achievements were made in the interval between Ding Wei and his dismissal...... Once there is an official to do it, this guy will immediately stop doing his job and start doing evil.

As for Gavaro, this is indeed a genius, and he is not an ink-greedy person like Qin Jiushao. But because of his family, he became a radical movement, and during the turbulent times of the French Revolution, it became a common thing to go in and out of prison, although he was only 21 years old when he died.

Many people say that if he hadn't died so early, with his talent for creating group theory at the age of 21, at least another Gauss or Euler!

But Ye Han felt that it was not necessarily.

Because this guy is not a person like Gauss or Euler who will dedicate his life to mathematics, if he has been guilty of crimes and imprisoned, he may have achieved higher achievements than Euler or Gauss, but if he is free and has become a person in power, it is really hard to say how he will achieve.

Even if he hadn't been imprisoned repeatedly, his group arguments might not have been able to be deduced so smoothly.

Knowing the cause and effect, Ye Han gradually made a decision in his heart.

Originally, he planned to leave here to join his friends after intercepting a section of the Seven Shrinkage Buckles, but now he wanted to stay for a while.

Energy absorption, cooling, although the low temperature will not affect the magnetic properties of the Seven Shrink Buckles, and even increase, but will reduce the toughness of the Seven Shrink Buckles, making them brittle and fragile. As long as it is brittle to a certain extent, it is still difficult for pure magnetism to tie up a person with a strength of nearly two tons.

When it reached a certain level, Ye Han resolutely swung his knife and slashed down.

With a crisp sound, the Seven Shrinkage Buckles of the Sky broke in response, and he ejected out, finally regaining his freedom!

At the same time, the two-meter-long seven-shrink buckle around the waist is also loosened, which should be enough for research.