Chapter 444: Proof of Infinite Prime Numbers

Chapter 444
Regarding the proof method of "there are infinitely many prime numbers", the most recognized one is the proof process listed by the mathematician Euclid in the twentieth proposition of the 9th volume of Geometry.
Therefore, this proposition is also called "Euclidean theorem".
Euclid’s method of proof was simple and ordinary, so he was able to enter the elementary mathematics class.
He first assumed 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 number 2, 3,…, p.
However, if q is divisible by any of these 2, 3, ..., p, there will be a remaining 1, which contradicts it. Therefore, prime numbers are infinite.
This ancient and simple method of proof, even after more than two thousand years, cannot deny its power.
…………
"I think since it's more than quantity, we'd better make variants based on Eurich's method of proof, so that the time wasted will probably be less."
"Well, I think so too. After all, we only have half an hour. At least each of the three of us must come up with a variant before we can hope to win."
"No, no, three are definitely not enough, and other schools are not all incompetent. I think if you want to compete for the top three, at least five are more secure! We can each come up with a variant in 20 minutes at most, and then The three of us will work together in the last ten minutes to see if there are any other ideas."
"Well, that's it."
The two teammates were discussing intensely. After reaching a consensus, they all turned to look at Cheng Nuo.
"Cheng Nuo, are you okay?" Although time is tight, the two still want to ask Cheng Nuo's opinion.
"Uh..., there is one sentence, I don't know when to say it improperly." Cheng Nuo scratched his head and said.
The two were taken aback and replied, "But it doesn't hurt to say."
"Why do we have to ponder the variant of Eurich's method of proof instead of looking for a new direction to prove?" Cheng Nuo asked.
Cheng Nuo's words left them speechless.
They don't want to find another new direction to prove the infinity proposition of prime numbers.
But this is a competition, not a research.
The measurement standard is quantity, not quality.
Making variants on the basis of Eurich's method of proof is like standing on the shoulders of giants. Both the research difficulty and the research time will be greatly reduced.
Looking for another proof direction is simple to say, but it is a process from scratch, which is extremely difficult. And the probability of failure is extremely high.
The two did not have the courage, nor the confidence to try to be the pioneer.
The teammate smiled bitterly, "It's not that we don't want to, but that we don't have the confidence to say that we have the ability 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 infinity proposition."
Cheng Nuo shrugged and smiled, "No, I have many new ideas in my mind now."
The two looked at each other silently, both doubting the authenticity of Cheng Nuo's words.
One asked suspiciously, "Student Cheng Nuo, can you 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 held up a finger, "The first one, use the coprime sequence to prove it."
The two were also very curious what Cheng Nuo would say, and listened.
"Think about it. If you can find an infinite sequence in which any two items are relatively prime, that is, the so-called relatively prime sequence, it is equivalent to proving that there are infinitely many prime numbers because the prime factors of each term are different from each other. The number 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?" someone couldn't help but ask.
Cheng Nuo snapped his fingers and said with a smile, "Actually, you have all heard of this sequence. In a letter to the mathematician Euler, the mathematician Goldbach mentioned a number that is entirely determined by Fermat: Fn =2^2^n+1(n=0,1,) The concept of a sequence composed of Fn-2=F0F1···Fn-1 can prove that 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'll talk about the second one below."
"Wait a minute!" A teammate called Cheng Nuo aloud, and hurriedly took out a stack of draft paper from the schoolbag behind him, and after writing down the first proof proposed by Cheng Nuo, he said to Cheng Nuo embarrassedly. "You go ahead."
He was so loud, he naturally attracted the attention of many nearby schools.
So when everyone saw the two talented doctoral students at Cambridge University, they looked like elementary school students at this time, looking up and looking forward to Cheng Nuo's speech over there, they all looked suspicious.
But time is running out, everyone's sights only stayed on the Cambridge University team for a few seconds, and then hurriedly continued to work hard.
"Uh, then let me go on." 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 Valle-Poussen pointed out in the prime number theorem proved in 1896 that the asymptotic distribution of the number of prime numbers π(N) within N is π(N)~N/ln(N ), N/ln(N) tends to infinity with N..."
"...From the above, we can see that for any positive integer n≥2, there is at least one prime number p such that nlt; plt; 2n." Cheng Nuobian said, and the teammate on the side remembered on the paper with his eyes Zhong is full of excitement that can't be concealed.
I thought it was rare for Cheng Nuo to propose a new way to prove it ~EbookFREE.me~, but unexpectedly, Cheng Nuo directly proposed two in one breath.
But Cheng Nuo's surprise for the two continued.
Cheng Nuo caught a glimpse of the teammate who had recorded the record, cleared his throat, and said, "Let's talk about the third one."
"More?" the teammate said in surprise.
"Of course there is." Cheng Nuo said with a smile, looking at his teammate who was rubbing his wrist, "Where is this!"
"The third kind is to use the knowledge of algebraic number theory to prove. One of the starting points for using algebraic number theory to prove that there are infinitely many prime numbers is to use the so-called Euler φ function."
"For any positive integer n, the value of Euler φ function φ(n) is defined as: φ(n): = the number of positive integers not greater than n and relatively prime to n. For any prime number p, φ (p)=p-1, this is because 1, p-1, p-1 positive integers not greater than p, are obviously relatively prime to p."
"Then, for two different prime numbers p1 and p2, φ(p1p2)=(p1-1)(p2-1), this is because..."