Alright my brother dug me out of my hollow to set people straight about the proof:
We assume there are finitely many primes p1
. We need only conclude that there exists a prime number NOT in that list, because that would disprove our original assumption that p1
are all the primes (it can irk you all you want but this is how a proof by contradiction works).
So consider a number q = p1
+1 and consider its prime factorization (There is a theorem that states that every number factors UNIQUELY into a product of primes, or is a prime number itself). If q is a prime itself we are done, we found a prime not in our list and therefore we disprove the original assumption.
If q is not a prime it must contain a factor supposedly in our list, call it pi
. Then q/pi
must be an integer, but we clearly see that q/pi
), and since (p1
) is just a product of integers and therefore an integer itself, and (1/pi
) clearly is not an integer, we conclude that pi
must have been a prime not in our list in order to divide q (that is to say, q/pi
cannot be an integer if pi
is part of the list). Therefore q is either a prime itself or contains factors not in our list of "all the primes".
Hope that provides some clarification