Chapter 445: 9 directions

Chapter 445
"This is because, among the p1p2 positive integers from 1 to p1p2, p1, 2p1, and p2p1 have a common prime factor p1 with p1p2; p2, 2p2, and p1p2 have the same prime factor p1 as p1p2. Prime factor p2; the rest are all relatively prime to p1p2."
"From this, we can get φ(p1p2) as p1p2-p2-p1, and the above reasoning can be repeated infinitely, which shows that there are infinitely many prime numbers."
In less than four or five minutes, Cheng Nuo has been constantly saying three proofs for using new directions, which opened the eyes of the two teammates.
If these three methods of proof are all just variants of the method of Proof by Euricid, the two would at most think that Cheng Nuo has studied the method of Euricid's proof quite deeply, and cannot promote any worship.
But the three proofs are all different from Eurich’s proof of multiplying integers and then adding and subtracting. Instead, they take a different approach, using "coprime sequence", "prime number distribution", and "algebraic number theory". Expand in a completely different direction.
The three proofs that Cheng Nuo said are not too complicated, and even too simple and too simple.
But the simpler, the more surprised the two of them.
Regarding the proof process of a proposition, no matter which mathematician is, it is hoped that the simpler the better.
Regardless of the fact that the proving process of many tall mathematics theorems is extremely complicated, but that group of mathematicians are not willing to do this!
It's not because there is no simpler way to prove it.
The simpler, the easier it is for people to understand. But the higher the requirements for mathematicians.
For the same theorem, a mathematician who can prove it with a one-page essay is at least twice the academic level of a mathematician who can prove it with a five-page essay.
Therefore, the two of them now look at Cheng Nuo's eyes as if they were looking at a monster.
This guy... is really just a graduate student?
I thought that Cheng Nuo's strength was just between them. Now I feel that Cheng Nuo is qualified to serve as an associate professor in their school with the strength that Cheng Nuo has shown now!
"Is there any water? I'm a little thirsty." Cheng Nuo asked with a dumb voice while the two were still thinking.
"Oh, I have water here." One person hurriedly handed over a bottle of mineral water in his backpack.
"Thanks."
Cheng Nuo drank half a bottle, waited for the discomfort in his throat to pass, and said, "Where did you go before, oh, I've finished the third proof, let's talk about the fourth."
Cheng Nuo forgot to glance at his teammate who was holding a pen and preparing to record, "If you are tired, let him help you."
After speaking, Cheng Nuo continued from above.
"The fourth is to use the proof of analytic number theory. This method is similar to the proof method I used in algebraic number theory above. As you all know, the Euler product formula is: Σnn-s=Πp(1-ps)-1( s>1), after analytic continuation on the left, it can be transformed into an extremely important function in analytic number theory: Riemann ζ function ζ(s)."
"For s=1, the left side of Euler's product formula is a divergent series called a harmonic series..."
Cheng Nuo cleared his throat and continued, "These above are all related to number theory, and I will talk about a few proof methods in other fields."
Under the dumbfounding of the two, Cheng Nuo said, "The fifth one, you can use the method of combined proof. The idea of ​​proof is this: any positive integer N can be written in the form of N=2, where r cannot be any A positive integer divisible by a square number greater than 1, and s2 is the product of all square factors. If there are only n prime numbers, then in the prime number decomposition of r..."
"Uh, Cheng Nuo, can you say it again." The student in charge of the recording scratched his head and said with a slight embarrassment, "I was stunned just now and forgot to record it."
Cheng Nuo shrugged helplessly, "Well, I'll say it again, you have to listen carefully this time."
The light of the bonfire reflected on Cheng Nuo's side face, which looked extremely radiant.
The two doctoral students under Cheng Nuo nodded together like beloved babies, showing a humbly teaching attitude.
"...Sixth, use the topological method to prove."
The two suddenly became suspicious.
Cheng Nuo noticed the doubts in their little eyes, and laughed, "I understand the doubts in your hearts. Topology and number theory seem to be two areas that you don't want to do. Why do I say that? When I finish, you will Clear."
"We can define a topology on a set of integers. The 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 Meet the definition of topology, namely:..."
"...From this, we know that there are infinitely many prime numbers. Do you understand now?"
The two nodded like chickens pecking at rice, and they continued to recall Cheng Nuo's words in their minds.
But Cheng Nuo didn't leave them too much time for aftertaste.
After passing through the thoughts briefly in his mind, Cheng Nuo will tell the next proof method.
Almost half of the half hour has now passed. If you don't hurry up, you may not be able to finish it.
"Seventh, use the application of prime numbers in information, coding and other fields to prove. The process is very simple, positive integer N can be decomposed into the product of prime numbers: N=p1m1·p2m2"
"...The eighth, use the direction of the function to prove that, let f(N) be the number of different prime numbers that can divide N. If there are only a finite number of prime numbers, and the continuous product is P, then it is obvious that there is f for all N (N)=f(N+P)……

"...The ninth one, I call it a one-line proof of prime numbers, the one-line expression is: 0＜∏sin(π/p)=∏sin(π(1+2∏p＇)/p), assuming prime numbers Only a finite number. If there are only a finite number of prime numbers, the left side of the expression "0,..."
"Huh-!"
After talking about the ninth method of proof, Cheng Nuo felt dry and dry, so he poured the remaining half bottle of mineral water Gudong Gudong down ~EbookFREE.me~ One person was very witty and handed Cheng Nuo a bottle of mineral water. water.
Seeing that Cheng Nuo hadn't moved for a long time, the student in charge of recording turned over and wrote a four-page formula, swallowed, and carefully asked, "Anything else?"
Cheng Nuo waved his hand and smiled bitterly, "I can think of only these nine proof methods for the new direction. Alas, it's still too far away from the more than 500 proof methods of the Pythagorean theorem!"
Cheng Nuo smiled bitterly, and they were also smiling bitterly.
The five hundred kinds of proofs of the Pythagorean theorem were formed after thousands of years of history and the development of several decades of mathematicians.
Cheng Nuo was able to come up with nine proofs of prime infinity in less than half an hour, which was beyond the understanding of the two.
Hearing Cheng Nuo's tone, he seemed quite dissatisfied.
This……
What else can they say!