Pythagoras used equidistant small stones to form equilateral triangles or squares, or pentagons, hexagons and other shapes. He called the numbers of small stones used triangle numbers, square numbers, and pentagons respectively.
number.
Triangular numbers: 1, 3, 6, 10... are the sum of the first n natural numbers;
Square numbers: 1, 4, 9, 16... are square numbers;
Then there are pentagonal numbers, hexagonal numbers, etc.
Don’t think this is very simple and not very difficult, there are more mysteries of shapes and numbers than you can imagine!
To talk about a simple thing, the Pythagorean theorem we studied is actually a special case of square numbers. It is equivalent to, under what circumstances can two small squares be placed into a large square.
If the Pythagorean theorem is expanded to powers, a^n+b^n=, it is Fermat's conjecture, and of course it is now Fermat's last theorem;
If the number of terms is expanded, there is a four-sum-of-squares theorem: any integer is expressed as a^2+b^2+c^2+d^2... Does this form require at most four terms?
This is completely in the field of formal numbers, and was finally proved by Euler and Lagrange.
But if we continue to expand, we will reach the Hualin problem. Four terms are needed for a square number, and how many terms are needed for a cubic number? What about the 5th power? What about the 6th power? This is a big pitfall that has not yet been solved.
Not only that, Fermat also dug another hole in the field of morphological numbers, called the polygonal number conjecture.
The conjecture was won by Gauss, the little prince of mathematics, and Cauchy completed the final proof, which took more than two hundred years.
Although it has been proved, if we continue to expand, we will reach the perfect cube problem. This is another big pitfall that cannot be proven or disproven so far...
So even though Gan Dadi just raised his head, Ye Han already felt something was not good.
It's not that he can't answer the question, of course there is a possibility that he can't answer it, but even if he can answer it, if he throws his answer to the other party, the probability that the other party can understand it is close to zero.
Sure enough, Gan Dadi first asked two relatively simple questions to ask questions. If you know that the sum of adjacent triangular numbers is a square number, or that the nth cubic number is the square of the nth triangular number, you can easily give the answer.
Give the answer.
Then he showed up!
A few examples are given first, 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 I asked Ye Han, can all numbers be represented by up to three triangular numbers?
Yes.
Triangle numbers can be represented by three numbers, square numbers can be represented by four numbers, and the number of polygons can be represented by as many numbers as there are. This is the polygon number conjecture. One of Fermat's famous conjectures that "the space is too small to fit"
.
The above is only the case of n=3.
But even if n=3, it is not so easy to prove. I think when the little prince of mathematics came out and proved it, he was so excited that he shouted Eureka. Ye Han didn't think that if he copied out the proof, the guy above would be able to understand it.
After thinking about it for a while, he said: "Not only do I know that all positive integers can be represented by three triangular numbers, I also know that they can be represented by four square numbers, or five pentagonal numbers, or six hexagonal numbers...
It’s just that the proof process is too complicated and I won’t be able to explain it for a while.”
Although his emotional intelligence is not high, it is not difficult to copy Fermat's show-off routine.
Gan Dadi was once again stunned on the spot.
Why, because this was his follow-up question, and Ye Han answered it before he could say it.
And since the other party came up with a conclusion without even thinking about it, even though there was no proof process, it seems that they have really studied this issue quite deeply. Should this... continue?
Gan Dadi was in a dilemma for a while.
If you say he is thick-skinned, he is definitely thick-skinned.
But thickness also has its limits. The key is that since the contact, Ye Han's understanding of the art of numerology has far exceeded his imagination. He has been hit hard one after another on the most important issues. Even if he is Gan Dadi, he is a little unable to hold on.
.
Ye Han felt that his knowledge was as vast as the ocean, and that he could never reach the bottom of it.
While Gan Dadi was in a daze, a note written by his grandson was handed into Ye Han's hands by one of his suicide squad members.
Before receiving the note, Ye Han had a vague love for Gan Dadi.
Imagine that a person is staying on this cliff with no way to reach the sky or the ground, relying only on the gravel at hand to calculate. One moment, Euler's natural numbers and results are laid out, and the next moment, he explores the field of formal numbers in depth...
You must know that all of this is self-study and there is no reference material. If there is information and guidance, wouldn't it be a rising star of mathematics?
【……】
However, after reading dozens of lines at a glance and reading the contents of the note written by Cheap Sun Tzu, his love for talent... became even stronger.
Emotionally, this is a Qin Jiushao and Gavaro type talent.
Qin Jiushao, a master of mathematics in the Southern Song Dynasty, made world-class contributions to the Chinese Remainder Theorem, triclinic quadrature, and Qin Jiushao's algorithm. In the BBC documentary on the history of mathematics, other Chinese mathematicians mentioned very little, just a few sentences.
, only for Qin Jiushao, it can be called rich and colorful.
But what do you say about this guy? He is greedy, cruel, cliquey and selfish... all the words to describe a corrupt official are not exaggerated when applied to him.
Almost all of his mathematical achievements were made during the break between Ding You and his dismissal from office... As soon as he got an official position, this guy immediately stopped doing his job and started doing evil.
As for Gavarro, he was indeed a genius and not a greedy man like Qin Jiushao. However, due to family reasons, he became a radical movement. During the turbulent period of the French Revolution, going in and out of prison became commonplace, although he died.
He was only 21 years old.
Many people say that if he had not died so early, with his talent in group theory at the age of 21, he would have created at least another Gauss or Euler!
But Ye Han doesn't think so.
Because this guy is not someone like Gauss or Euler who will devote his life to mathematics. If he keeps committing crimes and is imprisoned, his achievements may be higher than those of Euler or Gauss. But if he is free and becomes a member of the establishment, his achievements will be higher.
It's hard to say how.
Even if he hadn't been imprisoned many times, he might not have been able to deduce the group theory so smoothly.
After understanding the cause and effect, Ye Han gradually made a decision in his heart.
Originally, he planned to leave here after cutting off a section of the Seven Shrunk Buttons to meet his friends, but now he wants to stay for a while longer.
Energy absorption and cooling. Although low temperature will not affect the magnetism of Maotianqi Shrink Button, and may even enhance it, it will reduce the toughness of Maotian Qi Shrink Button, making it brittle and brittle. As long as it is brittle to a certain extent, it is purely magnetic.
It is still difficult to tie down a person with nearly two tons of strength.
When it reached a certain level, Ye Han decisively slashed it down with his sword.
With a crisp sound, Maotian Qi's buckle snapped, and he ejected, finally regaining his freedom!
At the same time, the seven-button buttons about two meters long surrounding the waist also came loose, which should be enough for research.