Number Theory 1 - Infinitely Many Prime Numbers

Prove that there are infinitely many prime numbers. This is one of the classic proofs in number theory, and in math. It was first proved by Euclid. This version is slightly different from Euclid's proof.

[wmichaelbooth]

I guess he should have said it at the end, but the whole point is that N+1 is always prime.

[Iamatheist]

ya if N+1 is a prime, than we have already prove that this finite set cannot be all the set of primes since it does not contain N+1. But when it is not a prime, than what he said.

[cipherunity]

Reply to Stefan Divic Comment:
You are right N+1 could be prime. But in this case nothing has to be proved. N+1 is prime and is not included in finite set of primes in our assumption.
Now if N+1 is composite then it can be written as a product of primes and as per our supposition all prime factors of N+1 must be included in our set of primes.
Mathxpress considered that one of these factors is Pi.
Now Pi is a factor of N+1. Let us write N+1 = XPi for some X. Similarly N = YPi for some Y
N+1-N=XPi-YPi =Pi(X-Y)
Hence 1 = Pi(X-Y) which means Pi is a factor of 1. But Pi is prime and can not be a factor of 1. Hence our assumption that there are finite primes is not correct.
Mathxpress has provided a very good and simple proof.

[quiveirojason]

Wow that was great. Short and sweet proof. Thanks for sharing!

[Eididfarah]

Starting from 1:07, you contradict your own assumption saying "Pi is somehwere here.." and you put P1<Pi<Pr.. Which is WRONG because, Pi is greater than N, it should surely be greater than Pr..