Chapter 441 (2/2)
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...”