Chapter 444

Regarding the proof method of "there are infinite numbers" most recognized at present is the proof process listed by mathematician Oulijit in the 20th proposition in Volume 9 of "Original Geometry".

Therefore, this proposition is therefore called the "Euclidean theorem".

The proof of Oliji is very simple and ordinary, so he can enter the class of elementary mathematics.

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

Then let q be the product of all prime numbers plus 1, then q=( 2x3x5x…xp )+1 is not a prime number, then q can be divided by the numbers in 2, 3,…, p.

And q is divided by any of these 2, 3,..., p, and will be divided by 1, which is inconsistent with it. Therefore, prime numbers are infinite.

This ancient and simple proof method cannot be denied even after more than two thousand years.

…………

"I think since it's a comparison of quantity, we'd better use the proof method of Olikid, so the waste of time is probably a little less."

"Well, I feel the same way, after all, we only have half an hour. At least each of us three have to come up with a variant to have hope of winning."

"No, no, no, three are definitely not enough, and other schools are not all incompetent. I think it is safer to compete for the top three! We will take at least twenty minutes to come up with a variant each, and then the three of us will work together in the last ten minutes to see if there are any other ideas."

"Okay, that's it."

The two teammates were discussing fiercely. After reaching an agreement, they turned their heads and looked at Cheng Nuo together.

"Cheng Nuo, are you okay?" Although time was tight, the two of them still wanted to ask Cheng Nuo's opinions.

"Uh..., there is a saying that I don't know whether to say it or not." Cheng Nuo scratched his head and said.

The two were stunned and replied, "But it's okay."

"Why do we have to think about the variants of the Oliji proof method instead of looking for new directions to prove it?" Cheng Nuo asked.

Cheng Nuo's words made the two of them speechless.

Why don’t they want to find another new direction to prove the proposition of infinite prime numbers?

But this is a competition, not a research.

The measure is quantity, not quality.

Variations based on the Olijid Proof Method are like standing on the shoulders of giants. Whether it is the difficulty of research or the time of research, it will be greatly reduced.

Finding another direction of proof is simple, but it is a process from scratch, extremely difficult and extremely likely to fail.

The two of them didn't have the courage or the confidence to try to be the pioneer.

The teammates smiled bitterly, "It's not that we don't want to, but we really don't have the confidence to say that we have the strength to do it. Even if the three of us work together, we may not be able to find a new direction to prove the infinite proposition of prime numbers in half an hour."

Cheng Nuo shrugged and smiled, "No, I have many new ideas in my mind now."

The two looked at each other silently, both of them doubting the authenticity of Cheng Nuo's words.

One of them asked suspiciously, "Student Cheng Nuo, can you give us some chestnuts?"

Cheng Nuo moved to the center of the bonfire, changed to a comfortable sitting position, and spoke slowly, "Of course there is no problem."

Cheng Nuo raised a finger and said, "First, use the mutual element sequence to prove it."

The two were also curious about what Cheng Nuo would say and pricked up their ears and listened.

"Think about it, if you can find an infinite sequence, in which any two terms are mutually prime, that is, the so-called mutual prime sequence, it is equivalent to proving that there are infinite numbers of prime numbers - because the prime factors of each term are different from each other, the terms are infinite, the number of prime factors, and the number of prime numbers will naturally be infinite."

"What kind of sequence is both an infinite sequence and a mutual element sequence?" a man couldn't help asking.

Cheng Nuo snapped his fingers and said with a smile, "Actually, you must have heard of this sequence. In a letter to mathematician Goldbach mentioned a concept of a sequence composed entirely of Fermat numbers: fn = 2^2^n + 1 (n = 0, 1,...). Through the formula fn - 2 = f0f1····fn-1, it can be proved that Fermat numbers are mutually exclusive."

"Above, using the sequence composed of Fermatus numbers, you can easily obtain a proof method of infinite prime numbers." Cheng Nuo paused for a moment and said, "I'll talk about the second one below."

"Wait a minute!" A teammate stopped Cheng Nuo loudly, hurriedly took out a stack of draft paper from his schoolbag behind him, and wrote down the first proof method proposed by Cheng Nuo, and then said to Cheng Nuo embarrassedly, "You keep going."

His loud voice naturally attracted the attention of many schools next to him.

So when everyone saw two talented doctoral students at Cambridge University, they were like elementary school students, looking up at Cheng Nuo's speech, all with a look of confusion.

But time was tight, and everyone's eyes were just staying on the team at Cambridge University for a few seconds, and then hurriedly continued to do something hard.

"Uh, then I'll continue." Cheng Nuo continued, "The second way I came up with is to use the distribution of prime numbers to verify."

"French mathematician Adama and Belgian mathematician Valai Pson pointed out in the prime number theorem proved in 1896 that the asymptotic distribution of prime numbers π(n) within n is π(n)~ n/ln(n), and n/ln(n) tends to infinite with n..."

"... From above, we can see that for any positive integer n ≥ 2, there is at least one prime number p, so that n < p < 2n." Cheng Nuo said while the teammate on the side remembered it on the paper, his eyes full of unconcealed excitement.

I thought it was rare for Cheng Nuo to propose a new direction to prove it, but unexpectedly, Cheng Nuo proposed two directly in one breath.

But Cheng Nuo made the two of them surprised.

Cheng Nuo saw that the teammate who was recording had finished recording, cleared his throat and said, "Let's talk about the third one."

"And?" Teammates spoke in surprise.

"Of course there is." Cheng Nuo said with a smile, looking at his teammate who was rubbing his wrists, "This is where he goes!"

"The third type is to use the knowledge proof of algebraic number theory. One of the starting points to prove that there are infinite numbers of prime numbers is to use the so-called Euler φ function."

"For any positive integer n, the value φ(n) of the Euler φ function is defined as: φ(n):= the number of positive integers that are not greater than n and are mutually primed with n. For any prime number p, φ(p)= p - 1, this is because 1,..., p - 1 The p - 1 positive integers that are not greater than p are obviously mutually primed with p."
Chapter completed!