Chapter 442 (1/2)

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.... ”