213: Waring conjecture and Goldbach conjecture

"Don't worry! I won't mind such a small matter."

Guo Hao smiled and said to the two people in front of him.

"Alright."

Ma Xin nodded with some worry.

"Let's go!"

Saying that, Guo Hao left the dormitory.

He arrived at the Library.

Shen Luoyan was indeed already sitting there.

"The stuff online..."

Just as Guo Hao sat down, Shen Luoyan had already raised her head, her eyes filled with worry as she looked at Guo Hao.

"You also browse Weibo?"

Looking at Shen Luoyan's expression, Guo Hao asked with a smile.

"No, Zhao Yu told me. Zhao Yu showed me some comments. Are you okay?"

Shen Luoyan looked at Guo Hao hesitantly and asked.

"Don't worry, I'm fine."

Guo Hao smiled and said, looking at Shen Luoyan in front of him.

"It's just some small matter, being attacked by some unknown creatures on the internet. This kind of thing will happen a lot in the future."

"Okay."

Shen Luoyan nodded, her eyes filled with worry as she looked at Guo Hao next to her, clearly not feeling at ease.

However, she generally wouldn't refute Guo Hao.

Looking at Shen Luoyan's expression, Guo Hao felt a slight sense of helplessness.

"Don't worry!"

Guo Hao said to Shen Luoyan with a wry smile.

"Didn't I go out of school the day before yesterday?"

"Yeah."

Shen Luoyan nodded.

"I went to see the Big Leader that time!"

Guo Hao smiled and whispered to Shen Luoyan.

Shen Luoyan's eyes showed a look of surprise as she looked at Guo Hao in front of her.

"Big Leader???"

"Yes!"

Guo Hao smiled and nodded.

"Now you can rest assured, right?"

Hearing Guo Hao's words, Shen Luoyan nodded. Since there was a Big Leader supporting him, Guo Hao would definitely be fine.

Shen Luoyan basically never questioned Guo Hao's words.

"Then you shouldn't look at the stuff online, what they're saying is too unpleasant!"

As she spoke, Shen Luoyan's face showed a look of anger.

Her angry look with puffed-up cheeks was very cute in Guo Hao's eyes.

He gently ruffled Shen Luoyan's hair, a warm smile on his face.

"Don't worry! I won't take the words of those people online to heart. It's not certain who will attack whom!"

"Okay!"

Shen Luoyan nodded.

After looking at Guo Hao seriously for a few moments, she continued to read.

Guo Hao was not in a hurry to read.

He had already passed the beginner stage where he needed to study hard.

In one year, Guo Hao had not only read the hundred books required by the System, but also many papers and related books.

His knowledge base had reached a considerable level.

He quietly watched Shen Luoyan for a while.

A trace of trance flashed in Guo Hao's eyes.

Did he have an influence on Shen Luoyan?

Guo Hao didn't know.

But Shen Luoyan, this girl, was really very hardworking.

Rebirth was the luckiest thing for him, and after rebirth, being with Shen Luoyan was the second luckiest thing.

After watching Shen Luoyan for a while, Guo Hao gradually collected his thoughts.

Without looking at the internet, he continued to calculate the Warings Problem.

Any positive integer can be expressed as the sum of not more than 4 squares of integers, such as 2+1^2, 14 = 3^2+2^2+1^2, etc.; if those less than 4 are added with 0^2, such as 13 = 3^2+2^2+0^2+0^2, then any positive integer can be expressed as the sum of 4 squares of integers.

Also, any positive integer can be expressed as the sum of 9 cubes of natural numbers, the sum of 19 fourth powers of natural numbers, and the sum of 37 fifth powers of natural numbers. Here, natural numbers include 0.

This conjecture can be expressed in a general form: for any positive integer N, there exists a number r(m), such that N can be expressed as the sum of m-th powers of r natural numbers, i.e., N = (x1)^m + ... + (x[r])^m

In 1909, Hilbert proved that the general form is correct, solving the existence problem of r(m). But what is the minimum value of r(m)?

This is the problem that Guo Hao currently needs to solve.

Besides the Warings Problem, up to now, because the value of g(k) heavily depends on the case when the positive integer is small, people have proposed a stronger problem, which is to find for every sufficiently large positive integer, the number of k-th powers into which they can be decomposed, G(k). The progress on this problem is slower, and G(3) is still undetermined to this day.

This problem has a very high correlation with the Waring's problem, and it is also a problem that needs to be solved at the forefront of the Academic Circle.

Guo Hao lowered his head, frowning as he looked at the manuscript paper in front of him.

He slowly wrote out a line of calculation.

Regarding this conjecture, Guo Hao did have some inspiration before, but when he really started to advance this conjecture, Guo Hao felt the heavy resistance.

Also, many top mathematicians have studied the Waring's problem.

Including the elder Mr. Chen Jingrun, many top mathematicsBig Brother have more or less dabbled in this problem.

But many of them have achieved some results.

However, what is the minimum value of r(m)?

To this day, no one knows.

Over the past month, Guo Hao has made some research on this problem, but the progress is still very slow, and he has not touched the core point.

Guo Hao has read the papers of the elder Mr. Chen Jingrun more than once.

Elder Chen used the Hardy–Littlewood circle method to solve this problem.

It's a pity that Elder Chen only proved up to g(5) = 37.

Guo Hao tried to extend and expand from Elder Chen's perspective. From the perspective of the Hardy–Littlewood circle method, calculating this problem up to g(5) = 37 is already the limit, and there is no way to continue calculating downwards.

Is it a problem with the method of solving?

Guo Hao was thoughtful.

Looking at the problem description and the mathematical formulas in front of him.

For some reason, Guo Hao thought of another more famous mathematical conjecture in the field of Number Theory.

Goldbachs Conjecture.

The statement of this problem is that any integer greater than 5 can be written as the sum of three prime numbers. (n > 5: When n is an even number, n = 2 + (n-2), and n-2 is also an even number, which can be decomposed into the sum of two prime numbers; when n is an odd number, n = 3 + (n-3), and n-3 is also an even number, which can be decomposed into the sum of two prime numbers)

The statement of the Waring's problem, to some extent, does have a wonderful similarity with Goldbachs Conjecture, reaching the same destination by different routes.

Elder Mr. Chen improved the Sieve method and applied it to Goldbachs Conjecture, proving "1+2", that is, he proved that any sufficiently large even number can be expressed as the sum of two numbers, one of which is a prime number, and the other is either a prime number or the product of two prime numbers, which is called "Chen's Theorem".

Therefore, he became famous throughout the world.