Chapter 445

"This is because, among the p1p2 positive integers from 1 to p1p2, p1, 2p1,... , p2p1 This p2 positive integer has a common prime factor p1 with p1p2; p2, 2p2,... , p1p2 is a positive integer that has a common prime factor p2 with p1p2, and all the rest are coprime with p1p2. ”

From this, we can get that φ (p1p2) is p1p2 - p2 - p1, and the above reasoning can be repeated infinitely, thus showing that there are infinitely many primes. ”

In less than four or five minutes, Cheng Nuo had already said three proofs of using new directions, which opened the eyes of the two teammates.

If these three proofs are just variants of Ori's proof, the two of them will at most think that Cheng Nuo has a deep study of Oriji's proof, and they can't afford to worship it.

However, all three proofs are different from the proof of Euclidean's multiplication of integers and then adding and subtracting, but they find another way to expand in three completely different directions: "interprime sequence", "prime number distribution" and "algebraic number theory".

The three proofs that Cheng Nuo said are not too complicated, and can even be said to be simple and excessive.

But the simpler it is, the more surprising it is for the two.

For the process of proving a proposition, no matter which mathematician wants it to be as simple as possible.

Don't look at the fact that the process of proving many high-level mathematical theorems is extremely complicated, but those mathematicians don't want to do this either!

It's not because I can't find an easier way to prove it.

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

For the same theorem, a mathematician who can prove it with one page of a paper is at least twice as academic as a mathematician who can prove it with five pages of paper.

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

This guy ...... Really just a graduate student?

thought that Cheng Nuo's strength was just between the two of them. Now I feel that with the strength that Cheng Nuo is showing now, he is qualified to serve as an associate professor in their school!

"Is there any water, I'm a little thirsty. While the two were still thinking, Cheng Nuo asked in a hoarse voice.

"Oh, I have water here. One of them hurriedly handed over a bottle of mineral water in his backpack.

"Thank you. ”

Cheng Nuo drank half a bottle, waited for the discomfort in his throat to pass, and said, "Where did I talk about it before, oh, I've finished talking about 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, you can ask him to help you." ”

After speaking, Cheng Nuo continued to talk above.

"Fourth, the proof using analytic number theory, this method is similar to the proof method I used in algebraic number theory above, as you know, Euler's product formula is: Σnn-s = Πp(1 - p-s)-1 (s > 1), and the left side can be analytically extended to become the most important function in analytic number theory: the Riemann ζ function ζ(s). ”

"For s = 1, to the left of the Euler product formula is the divergent series known as the harmonic series......"

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

Under the dumbfounded eyes of the two, Cheng Nuo said eloquently, "The fifth one can use the method of combinatorial proof. The idea of the proof is that any positive integer N can be written as N = rs2, where r is a positive integer that is not divisible by any square number greater than 1, and s2 is the product of all square factors. If there are only n primes, then in the prime decomposition of r, ......"

"Uh, Cheng Nuo, can you tell me more. The student in charge of recording scratched his head and said with a little embarrassment, "I was so stunned just now that I forgot to record." ”

Cheng Nuo shrugged helplessly, "Okay, I'll say it again, you have to listen carefully this time." ”

The light of the bonfire was reflected on Cheng Nuo's side face, which looked extremely bright.

The two doctoral students under Cheng Nuo's seat nodded in unison like good babies, with the attitude of being taught by students with an open mind.

“...... Sixth, the method of proof using topology. ”

The two of them were suddenly 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." When I'm done, you'll be clear. ”

"We can define a topology on an integer set consisting of an open set consisting of and only the union of the empty set ∅ and the arithmetic sequence aZ+ b (a ≠ 0 and b are both integers). It is not difficult to prove that the open set of such a definition satisfies the definition of topology, i.e., ......"

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

The two of them nodded like chickens pecking rice, and their minds kept recalling Cheng Nuo's words.

But Cheng Nuo didn't leave much time for the two to reminisce.

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

Now that half an hour has passed, if you don't hurry up, you may not be able to finish it.

"Seventh, the application of prime numbers in information, coding and other fields is used to prove. The process is simple, all positive integers N can be decomposed into the product of prime numbers: N = p1m1·p2m2..."

“...... Eighth, using the direction proof of the function, let f(N) be the number of different prime numbers divisible by N, if there are only a finite number of primes and their product is P, then it is obvious that for all N there is f(N) = f(N + P)......"

“...... The ninth, which I call a one-line proof of primes, has a one-line expression of 0<∏sin(π/p)=∏sin(π(1+2∏p')/p), assuming that there are only a finite number of primes. If there are only a finite number of primes, then the left side of the expression is "0,......"

"Whew!"

After saying the ninth proof method, Cheng Nuo felt dry and poured all the remaining half bottles of mineral water.

One person was very interested and handed Cheng Nuo a bottle of mineral water.

Seeing that Cheng Promise had not moved for a long time, the classmate in charge of recording flipped through the formula he had written for more than four pages, swallowed his saliva, and asked cautiously, "Is there anything else?"

Cheng Nuo waved his hand and said with a wry smile, "There are only these nine proof methods I can think of for the new direction, alas, it is still too far from more than 500 proof methods of the Pythagorean theorem!"

Cheng Nuo smiled bitterly, and they were also smiling bitterly.

There are more than 500 methods of proving the Pythagorean theorem, but it has gone through 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 infinity in less than half an hour, which was beyond the scope of the two of them.

But listening to Cheng Nuo's tone, he seemed to be quite dissatisfied.

This......

What else can they say!