Chapter 444

The most recognized method of proving that there are infinitely many primes is the process of proof listed by the mathematician Eurijde in the 20th proposition of Book 9 of the Geometry Primitives.

Therefore, this proposition is also known as "Euclid's theorem".

Eurich's method of proofing was simple and mundane, and he was able to enter the classroom of elementary mathematics.

He first assumes that prime numbers are finite, that there are only finite number of primes, and that 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 prime, then, q is divisible by the numbers in 2, 3、...、p.

And q is divisible by any of these 2 and 3、...、p will be left with 1, which contradicts it. So, prime numbers are infinite.

This ancient and simple method of proof, even after more than 2,000 years, cannot deny its power.

............

"I think since it's a comparison of numbers, then we'd better make a variant on the basis of Eurigit's proof, so that the time wasted is estimated to be a little less. ”

"Well, I think so too, after all, we only have half an hour, and the three of us at least each have to come up with a variant to have any hope of winning. ”

"No, no, no, three is definitely not enough, and the other schools are not all incompetent, I think it is safer to compete for the top three, at least five! We will spend a maximum of twenty minutes each coming 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 having a heated discussion. After reaching an agreement, they all turned their heads to look at Cheng Nuo.

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

"Uh...... There is a sentence that I don't know whether to say or not. Cheng Nuo scratched his head and said.

The two were stunned for a moment and replied, "But it doesn't hurt to say it." ”

"Why do we have to ponder the variants of Ori's proof method, instead of looking for a new direction to prove it?" asked Cheng Nuo.

Cheng Nuo's words left the two speechless.

They don't want to look for another new direction to prove the proposition of prime infinity.

But it's a competition, not a research.

And the measure is quantity, not quality.

Variants on the basis of Euric's proof method are like standing on the shoulders of giants, and both the difficulty of research and the research time will be greatly reduced.

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

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

The teammate smiled bitterly, "It's not that we don't want to, but we 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 proposition of prime infinity. ”

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

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

One of them asked suspiciously, "Classmate Cheng Nuo, can you just give us a few chestnuts?"

Cheng Nuo moved to the center of the bonfire, changed to a comfortable sitting position, and said slowly, "Of course it's no problem." ”

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

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

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

"What kind of sequence is both an infinite sequence and an interprime sequence?" one couldn't help but ask.

Cheng Nuo snapped his fingers and said with a smile, "Actually, you should have all heard of this sequence, the mathematician Goldbach mentioned in a letter to the mathematician Euler, a number that is completely composed of Fermat: Fn = 2^2^n + 1 (n = 0, 1,...) The concept of the composition of the sequence, by Fn - 2 = F0F1 · The formula Fn-1 can prove that the Fermat numbers are mutually symmetric. ”

Above, using the sequence composed of Fermat numbers, we can easily obtain a proof of prime infinity. Cheng Nuo's tone paused and said, "I'll talk about the second one." ”

"Wait a minute!" a teammate shouted to stop Cheng Nuo, hurriedly took out a stack of scratch paper from the schoolbag behind him, and wrote down the first proof proposed by Cheng Nuo before he said to Cheng Nuo embarrassedly, "You go on." ”

He was so loud that he naturally attracted the attention of many schools next to him.

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

But time was pressing, and everyone's eyes only stayed on the Cambridge University team for a few seconds, and then hurriedly resumed their own calculations.

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

"In the prime theorem proved by the French mathematician Adama and the Belgian mathematician Valle Pusen in 1896, it was pointed out that the asymptotic distribution of the number of primes within N π(N) is π(N)~ N/ln(N), and N/ln(N) tends to be infinitely ...... 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 n < p < 2n. Cheng Nuo said, and the teammate on the side wrote it down on the paper, and his eyes were full of excitement that could not be concealed.

I thought that Cheng Nuo could propose a proof method for a new direction, which was already really rare, but he never expected that Cheng Nuo directly proposed two in one go.

But Cheng Nuo's surprise for the two continued.

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

"And?" the teammate exclaimed.

"Of course. Cheng Nuo said with a smile, looking at his teammates who were rubbing their wrists, "Where is this!"

"The third is to use the knowledge proof of algebraic number theory. 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 Eulerian φ function. ”

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

"Then, for two different primes, p1 and p2, φ(p1p2)=(p1 - 1)(p2 - 1), this is because ......"