Omnipotent Data

Chapter 445

"This is because, among the p1p2 positive integers from 1 to p1p2, the p2 positive integers p1, 2p1, p2p1 have the same prime factor p1 with p1p2; the p1 positive integers p2, 2p2, p1p2 have the common prime factor p1p2 prime factor p2; the rest are all coprime with p1p2."

"Thus, it can be obtained that φ(p1p2) is p1p2-p2-p1, and the above reasoning can be repeated infinitely, which further shows that there are infinitely many prime numbers."

In less than four or five minutes, Cheng Nuo non-stop said three proofs of using the new direction, which opened the eyes of the two teammates.

If these three methods of proof were just variants of the Euriage's method of proof, the two of them would at most think that Cheng Nuo had studied the Euriage's method of proof deeply, and they would not have any admiration for it.

However, the three proof methods are all different from Euricide's proof of multiplication of integers and then doing some addition and subtraction. develop in a completely different direction.

The three proofs Cheng Nuo gave were not too complicated, and could even be said to be too simple.

But the simpler it was, the more it surprised the two of them.

For the proof process of a proposition, no matter what mathematician it is, it is of course hoped that the simpler the better.

Don't look at the proving process of many high-level mathematical theorems is extremely complicated, but the group of mathematicians are not willing to do this!

It is not because there is no simpler proof method.

The simpler it is, the easier it is for people to understand. But the requirements for mathematicians are higher.

For the same theorem, a mathematician who can prove it in a one-page paper has at least twice the academic level of a mathematician who can prove it in a five-page paper.

Because of this, the two looked at Cheng Nuo now as if they were looking at a monster.

This guy... is really just a graduate student?

I thought Cheng Nuo's strength was just on par with the two of them. Now, I feel that with Cheng Nuo's current strength, he is qualified to be an associate professor in their school!

"Is there any water? I'm a little thirsty." While the two were still thinking about it, Cheng Nuo asked hoarsely.

"oh oh,

I have water here. "One person hastily handed over a bottle of mineral water in his backpack.

"Thanks."

Cheng Nuo gulped down half a bottle, waited for the discomfort in his throat to pass, and said, "Where did I talk before, oh, I've finished the third proof, now let's talk about the fourth."

Cheng Nuo forgot to glance at his teammate who was holding a pen to record, and said, "If you're tired, you can ask him to help you."

After finishing speaking, Cheng Nuo continued to speak.

"The fourth one is to use the proof of analytic number theory. This method is similar to the proof method I used above using algebraic number theory. As you all know, the Euler product formula is: Σnn-s=Πp(1-p-s)-1( s\u003e1), the left side can be transformed into a very important function in analytic number theory: the Riemann ζ function ζ(s) after analytical extension on the left side.

"For s=1, the left-hand side of the Euler product formula is a diverging series called a harmonic series..."

Cheng Nuo cleared his throat and continued, "The above are all related to number theory. Next, I will talk about some proof methods in other fields."

With the two of them dumbfounded, Cheng Nuo said, "Fifth, you can use the combination proof method. The idea of ​​the proof is this: any positive integer N can be written in the form of N=2, where r cannot be written by any A positive integer divisible by a square number greater than 1, and s2 is the product of all square number factors. If there are only n prime numbers, then in the prime number decomposition of r..."

"Uh, Cheng Nuo, can you explain it again." The student in charge of recording scratched his head and said with a little embarrassment, "I was so distracted by patronizing just now that I forgot to record it."

Cheng Nuo shrugged helplessly, "Okay, let me say it again, this time you have to listen carefully."

The light from the bonfire reflected on Cheng Nuo's side face, looking extremely radiant.

The two doctoral students under Cheng Nuo's seat nodded in unison like good babies, showing the attitude of students humbly accepting teaching.

"...Sixth, use the method of topology to prove."

The two suddenly became suspicious.

Cheng Nuo noticed their puzzled eyes and smiled, "I understand the doubts in your hearts. Topology and number theory seem to be two fields that you don't want to do. Why do I say that. After I finish speaking, you will Clear."

"We can define a topology on the set of integers, whose open set consists of and only consists of the empty set and the union of the arithmetic sequence a+b (a≠0 and b are both integers). It is not difficult to prove that the open set thus defined satisfies the definition of topology, which is:  …”

"...From this, we know that there are infinitely many prime numbers. Do you understand now?"

Both of them nodded like chickens pecking at rice, thinking about Cheng Nuo's words in their minds.

But Cheng Nuo didn't leave much time for the two of them to reflect.

After briefly going through the ideas in his mind, Cheng Nuo explained the next method of proof.

Now almost half of the half hour has passed, if you don't hurry up, it is really possible that you can't finish talking.

"The seventh one is to use the application of prime numbers in information, coding and other fields to prove. The process is very simple. A positive integer N can be decomposed into the continuous product of prime numbers: N=p1m1·p2m2"

"...Eighth, use the direction of the function to prove that f(N) is the number of different prime numbers that can divide N. If there are only a finite number of prime numbers, and their continuous product is P, then obviously for all N there are f (N)=f(N+P)..."

"... the ninth one, I call it the one-line proof of prime numbers, the one-line expression is: 0\u003c∏sin(π/p)=∏sin(π(1+2∏p')/p), assuming prime numbers There are only a finite number. If there are only a finite number of prime numbers, the left side of the expression "0,..."

"Hoo-hoo-!"

After finishing the ninth proof method, Cheng Nuo felt his mouth dry and gulped down the remaining half bottle of mineral water.

One of them tactfully handed Cheng Nuo another bottle of mineral water.

Seeing that Cheng Nuo hadn't moved for a long time, the classmate in charge of recording flipped through the four-page formula he had written, swallowed his saliva, and asked cautiously, "Is there any more?"

Cheng Nuo waved his hand and said with a wry smile, "These are the only nine proof methods I can think of in the new direction. Alas, it's still far from the more than 500 proof methods of the Pythagorean Theorem!"

Cheng Nuo smiled bitterly, and they smiled bitterly too.

There are more than 500 proof methods of the Pythagorean theorem, but they were formed after thousands of years of history and the development of dozens of generations of mathematicians.

Cheng Nuo was able to come up with nine proofs of infinite prime numbers in less than half an hour, which was beyond the scope of the two of them.

But from Cheng Nuo's tone, he seemed quite dissatisfied.

This……

What else can they say!