Chapter 441

Chapter 444

about the proof method of "there are infinitely many primes", the most recognized one is the proof process listed in the 20th proposition of volume 9 of the original geometry by the mathematician euridge.

Therefore, this proposition is also called Euclidean theorem.

Origi De's proof is very simple and ordinary, so he can enter the elementary mathematics class.

He first assumes that prime numbers are finite, that there are only a finite number of primes, and that the largest prime is p.

Then let Q be the product of all primes plus 1, then q = (2 × 3 × 5 ×...) X P) + 1 is not a prime, then q can be 2, 3 Divide the number in P.

And Q is affected by these 2,3 Any integral division of P will result in 1. So, primes are infinite.

This ancient and simple proof method, even after more than 2000 years, can not deny its powerful.

…………

"I think since it's a comparison of quantity, we'd better make a variation on the basis of eurygiede's proof, so that the waste of time is estimated to be a little less."

"Well, I think so. After all, we only have half an hour. At least each of the three of us has to come up with a variant to win."

"No, no, no, three are not enough, and other schools are not all incompetent. I think if we want to compete for the top three, at least five are more secure! We'll spend at most 20 minutes each to come up with a variant, and then the three of us will work together for the last ten minutes to see if there are any other ideas

"Well, that's it."

The two teammates were in a heated discussion. After reaching an agreement, they all turned to look at Cheng Nuo.

"Cheng Nuo, are you ok?" Although time is tight, they still want to ask Cheng Nuo's opinion.

"Er There is a saying that I don't know when to say it or not. " Cheng Nuo scratched his head.

Two people a Leng, return way, "but say no harm."

"Why do we have to think about variants of eurekit's proof, instead of looking for new directions to prove it?" Cheng Nuo asked.

Cheng Nuo's words made them speechless.

They did not want to find another new direction to prove the proposition of infinite prime number.

But it's a competition, not a research.

The standard of measurement is quantity, not quality.

It is just like standing on the shoulders of giants to carry out the variation on the basis of Euclid's proof method, which will greatly reduce the difficulty and time of research.

Looking for another direction of proof is easy to say, but it is a process from scratch, which is extremely difficult. And the possibility of failure is very high.

They didn't have the courage and confidence to try to be the pioneers.

Teammates wryly smile, "it's not that we don't want to, but we don't have the strength to do it. Even if the three of us work together, half an hour may not be able to find a new direction to prove the prime infinite proposition

Cheng Nuo shrugged and said with a smile, "no, I have a lot of new ideas in my mind now."

Both of them looked at each other in silence, doubting the truth of Cheng Nuo's words.

"Cheng Nuo classmate, can you give us some chestnuts

Cheng Nuo moved to the center of the campfire, changed a comfortable sitting posture, and slowly opened his mouth, "of course, no problem."

In this paper, we use the first sequence of "finger up"

Cheng and Nuo are curious about what they say.

"If you think about it, if you can find an infinite sequence in which any two terms are mutually prime, that is to say, the so-called coprime sequence, then it is equivalent to proving that there are infinitely many prime numbers - because each term has different prime factors, the number of terms is infinite, the number of prime factors, and thus the number of prime numbers is naturally infinite."

"What kind of sequence is both infinite sequence and coprime sequence?" One couldn't help asking.

Cheng Nuo snapped his fingers and said with a smile, "in fact, you should have heard of this sequence. In a letter to Euler, mathematician Goldbach mentioned the concept of a sequence completely composed of Fermat numbers: FN = 2 ^ 2 ^ n + 1 (n = 0, 1,...). Through the formula fn-2 = f0f1 ··· fn-1, we can prove that Fermat numbers are mutually prime."

"Above, using the sequence of Fermat numbers, we can easily get a proof of infinite prime numbers." Cheng Nuo tone pauses for a while, open mouth says, "below I say the second."

"Wait a minute!" A teammate called Cheng Nuo to a halt. He quickly took out a pile of scribbles from the bag behind his back and wrote down the first proof proposed by Cheng Nuo. Then he said to Cheng Nuo with embarrassment, "you go on."

He was so loud that he naturally attracted the attention of many schools nearby.

So when people saw the two gifted doctoral students here at Cambridge University, they looked like primary school students, looking up at Cheng Nuo's speech over there. They all looked puzzled.But time is pressing, people's eyes just stay on the team of Cambridge University for a few seconds, and then rush to their own hard work.

"Well, I'll go on." Cheng Nuo continued, "my second idea is to use the distribution of prime numbers to prove it."

In 1896, the French mathematician Adama and the Belgian mathematician Valle Simpson pointed out in the prime theorem that the asymptotic distribution of the number of primes within n is π (n) ~ NLN (n), and NLN (n) tends to infinity with n.... "

“…… From above, we can know that for any positive integer n ≥ 2, there exists at least one prime number P such that N & P & 2n. " Cheng Nuo said, one side of the team-mates will be in the paper Shua Shua remember, eyes full of excitement can not hide the color.

It is really rare that Cheng Nuo could propose a new direction of proof, but unexpectedly, Cheng Nuo directly put forward two.

But Cheng Nuo's surprise continued.

Cheng Nuo caught sight of the record of that teammate has finished, cleared his throat, and said, "let's talk about the third one."

"What else?" The teammates were surprised.

"And of course." Cheng Nuo said with a smile and looked at his teammates rubbing his wrists, "this is where it is!"

"The third is to use the knowledge of algebraic number theory to prove. One of the starting points of proving the infinite number of primes by means of algebraic number theory is to use the so-called Euler φ function

"For any positive integer n, the value of Euler's function φ (n) is defined as: φ (n): = the number of positive integers not greater than N and coprime with n. For any prime P, φ (P) = P-1, this is because 1,..., P-1, P-1, a positive integer not greater than P, is obviously coprime with P

"Then, for two different prime numbers P1 and P2, φ (p1p2) = (P1-1) (P2-1), because..."

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