universal data

Chapter 441 The Proof of Infinite Prime Numbers

444 chapter

Regarding the proof method of "there are infinitely many prime numbers", the most recognized one at present is the proof process listed by the mathematician Euric in Proposition 9 of Volume 20 of "Elements of Geometry".

Therefore, this proposition is also called "Euclidean theorem".

Euricide's proof is very simple and ordinary, so he can enter the classroom of elementary mathematics.

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

Then let q be the product of all prime numbers plus 1, then, q=(2×3×5×…×p )+1 is not a prime number, then, q can be divisible by the numbers in 2, 3,…, p.

However, if q is divisible by any of these 2, 3, ..., p, the remainder will be 1, which is contradictory.Therefore, the prime numbers are infinite.

This ancient and simple method of proof, even after more than 2000 years, cannot deny its power.

…………

"I think since it's a comparison of quantities, it's best for us to make a variant based on Eurich's proof method. This will probably waste less time."

"Well, I think so too. After all, we only have half an hour. The three of us can only hope to win if each of us has to come up with at least one variant."

"No, no, no, three are definitely not enough, and other schools are not all incompetent people. I think if we want to compete for the top three, at least five are more secure! We will spend at most 10 minutes each to come up with a variant, and then The three of us will work together in the last 10 minutes to see if there are any other ideas."

"Okay, that's it."

The two teammates were having a heated discussion.After reaching a consensus, Qiqi turned to look at Cheng Nuo.

"Cheng Nuo, are you okay?" Although the time was short, the two still wanted to ask Cheng Nuo's opinion.

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

The two were taken aback for a moment, and replied, "But it's okay to say."

"Why do we have to ponder the variants of Euriage's method of proof instead of finding a new direction for proof?" Cheng Nuo asked.

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

Why don't they want to find another new direction to prove the prime number infinite proposition.

But this is competition, not research.

The standard of measurement is quantity, not quality.

Making variants on the basis of Euric's proof method is like standing on the shoulders of giants. Both the research difficulty and the research time will be greatly reduced.

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

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

The teammate 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, half an hour may not be able to find a new direction to prove the prime number infinite proposition."

Cheng Nuo shrugged and smiled, "No, I have a lot of new ideas in my mind right now."

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

One person asked suspiciously, "Student Cheng Nuo, can you just give us some chestnuts?"

Cheng Nuo moved to the center of the campfire, changed to a comfortable sitting position, and said slowly, "Of course it's okay."

Cheng Nuo raised a finger, "The first one, use the coprime sequence to prove it."

The two of them were also curious about what Cheng Nuo would say, so they pricked up their ears to listen.

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

"Then what kind of sequence is both an infinite sequence and a coprime sequence?" One person couldn't help asking.

Cheng Nuo snapped his fingers and said with a smile, "Actually, you should all have heard of this sequence. In a letter from the mathematician Goldbach to the mathematician Euler, he mentioned a Fermat number: Fn = 2^2^n + 1 (n = 0, 1, .) The concept of a sequence, through the formula Fn - 2 = F0F1···Fn-1, it can be proved that the Fermat numbers are mutually prime .”

"Above, using the sequence composed of Fermat numbers, you can easily get a proof of infinite prime numbers." Cheng Nuo paused and said, "I will talk about the second one now."

"Wait a minute!" A teammate yelled to stop Cheng Nuo, hurriedly took out a stack of draft paper from the schoolbag behind his back, and wrote down the first proof proposed by Cheng Nuo, then said to Cheng Nuo in embarrassment , "You go ahead."

He was so loud, which naturally attracted the attention of many nearby schools.

So when everyone saw the two talented doctoral students from Cambridge University, looking up like primary school students, looking up at Cheng Nuo's speech, they all looked puzzled.

But time was running out, and everyone's eyes only stayed on the team of Cambridge University for a few seconds, and then hurriedly followed up with their own calculations.

"Uh, let me continue." Cheng Nuo continued, "The second method I came up with is to use the distribution of prime numbers to prove it."

"The prime number theorem proved by the French mathematician Adama and the Belgian mathematician Valle-Pousse in 1896 pointed out that the asymptotic distribution of the prime number π(N) within N is π(N)~N/ln(N ), N/ln(N) tends to infinity with N..."

"... From the above, it can be known that for any positive integer n ≥ 2, there is at least one prime number p such that n < p < 2n." Cheng Nuo said, while the teammate beside him wrote it down on a piece of paper , eyes filled with unconcealable excitement.

I thought it would be very rare for Cheng Nuo to propose a proof method in a new direction, but unexpectedly, Cheng Nuo directly proposed two methods in one go.

But Cheng Nuo continued to surprise the two of them.

Cheng Nuo caught a glimpse that the teammate who had recorded it had finished writing it down. He cleared his throat and said, "Let's talk about the third one."

"More?" His teammates said in surprise.

"Of course." Cheng Nuo said with a smile, looking at his teammates who were rubbing their wrists, "That's where it is!"

"The third method is to use the knowledge of algebraic number theory to prove. One of the starting points for proving that there are infinitely many prime numbers by means of algebraic number theory is to use the so-called Euler φ function."

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

"Then, for two distinct prime numbers p1 and p2, φ(p1p2)=(p1 - 1)(p2 - 1), because..."

(End of this chapter)