Chapter 442

Chapter 445

"this is because, from 1 to p1p2, the positive integers P1, 2p1,..., p2p1 have the same prime factor P1 with p1p2; P2, 2P2,..., p1p2, these P1 positive integers have the same prime factor P2 with p1p2; the rest are all mutually prime with p1p2."

"Thus, it can be concluded that φ (p1p2) is p1p2-p2-p1, and the above reasoning can be infinitely repeated, which indicates that there are infinitely many primes."

In less than four or five minutes, Cheng Nuo has been constantly speaking out three proofs of using the new direction, which has greatly opened the eyes of the two teammates.

If these three proofs were only variations of origi De's proof, the two would at most think that Cheng Nuo's study of origi De's proof was very deep, but could not raise any worship.

However, the three proofs are all different from the proof of Euclid's integer multiplication and point addition and subtraction method. Instead, they develop a new way by using three completely different directions of "mutual prime sequence", "prime distribution" and "algebraic number theory".

Cheng Nuo's three proofs are not too complicated, or even too simple.

But the simpler it is, the more surprising they are.

For the proof process of a proposition, no matter which mathematician, the simpler the better.

Although the proof process of many high mathematical theorems is extremely complicated, the group of mathematicians is not willing to do so!

It's not because we can't find a simpler proof.

The simpler it is, the easier it will be understood. But the more demanding it is for mathematicians.

For the same theorem, a mathematician who can prove it with one page paper has at least twice the academic level of a mathematician who needs five pages to prove it.

Therefore, the two now look at Cheng Nuo's eyes as if they were looking at a monster.

This guy Really just a graduate student?

I thought Cheng Nuo's strength was just between Bozhong and them. Now I feel that Cheng Nuo is qualified to serve as an associate professor in their school!

"Do you have water? I'm a little thirsty." When they are still thinking, Cheng Nuo asked in a hoarse voice.

"Oh, oh, I have water here." A man quickly handed over a bottle of mineral water in his backpack.

"Thank you."

Cheng Nuo Gudong drank half a bottle, and when the discomfort in his voice passed, he said, "what did you say before? Oh, I've finished the third proof, and the fourth is the next."

Cheng Nuo forgot to take a look at the team friend who was holding a pen to record and said, "if you are tired, you can ask him to help you."

With that, Cheng Nuo went on to talk about it.

"Fourth, using analytic number theory to prove, this method is the same as the method I used in algebraic number theory above. As you all know, Euler product formula is: ∑ nn-s = Πp (1-p-s) - 1 (s; 1). After analytic extension on the left side, it can be changed into a very important function in analytic number theory: the Riemann zeta function ζ (s)."

"For S = 1, the left side of the Euler product formula is a divergent series called a harmonic series..."

Cheng Nuo cleared his throat and went on to say, "the above are all related to number theory. Next, I'll talk about several proof methods in other fields."

Under the two people's astonishment, Cheng Nuo said, "fifth, we can use the method of combination proof. The idea of proof is as follows: any positive integer n can be written in the form of n = RS2, where R is a positive integer that cannot be divisible by any square 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.... "

"Well, Cheng Nuo, can you tell me again?" The student in charge of the record scratched his head and said in a slightly embarrassed way, "I just patronized stupidly and forgot to record."

Cheng Nuo shrugged helplessly, "OK, I'll say it again. You should listen carefully this time."

The fire light of the campfire reflected on Cheng Nuo's side face, showing incomparable brilliance.

Cheng Nuo's two doctoral students nodded together as if they were good babies, with an open-minded attitude of being taught.

“…… The sixth is to prove it by topological method

The two men were immediately suspicious.

Cheng Nuo noticed their puzzled eyes and laughed, "I understand your doubts. Topology and number theory seem to be two fields that I don't want to do. Why do I say that. When I'm done, you'll know. "

"We can define a topology on the set of integers whose open set consists of and only the union of the empty set ℤ and the arithmetic sequence a ???? B (a ≠ 0 and B are integers). It is not difficult to prove that the open set defined in this way satisfies the definition of topology, i.e.,.... "

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

Two people together, chicken peck rice nod, brain constantly aftertaste Cheng Nuo's words.

But Cheng Nuo didn't leave them too much time for aftertaste.After a brief thought in his mind, Cheng Nuo tells the next proof.

Now half an hour has almost passed. If you don't grasp the time, you may not be able to finish it.

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

" Eighth, by using the direction 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 the product of them is p, it is obvious that for all n there is f (n) = f (n + P)... "

“…… The ninth, I call it the single line proof of prime numbers. The single line expression is: 0 ＜ Π sin (π P) = Πsin (π (1 + 2 Π p ') P), assuming that there are only a finite number of prime numbers. If there are only a finite number of primes, then all the independent variables π P of sin in the product of left "" and right end of the expression are all between 0 and π, sin (π P); 0,...... "

"Hoo Hoo -!"

After finishing the ninth proof, Cheng Nuo felt thirsty and poured down the remaining half bottle of mineral water.

A person is very witty and handed Cheng Nuo a bottle of mineral water.

Seeing that Cheng Nuo had no action for a long time, the student who was in charge of recording turned over his four page formula, swallowed his saliva and asked carefully, "is there anything else?"

Cheng Nuo waved his hand and said with a wry smile, "the only proof I can think of in the new direction is these nine. Alas, it's too far away from more than 500 proof methods of Pythagorean theorem."

Cheng Nuo smiles bitterly, and they also smile bitterly.

There are more than 500 kinds of proofs of Pythagorean theorem, but they have been formed after thousands of years of history and the development of dozens of generations of mathematicians.

Cheng Nuo can come up with nine proofs of infinite prime in less than half an hour, which is beyond the scope of their understanding.

Can listen to Cheng Nuo's tone, he seems quite dissatisfied.

This

What else can they say!

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