I am interested in understand the proof of infinitely many primes. It seems like quite an easy proof, ( I know there are many but I am referring to the proof that goes as follows);
" Suppose there are only finitely many primes, lets say n of them, we can denote them as p1,p2,..,pn. Now we construct a new number P=(p1)(p2)..(pn)+1. Clearly P is larger than any of the primes, so it does not equal any one of them. Since p1,p2,,pn constitute all primes, P cannot be a prime. Thus it must be divisible by at least one of our finitely many primes, say pn. But when we divide P by Pn we get a remainder of 1 which is a contradiction. So our original assumption must be false."
I understand a lot of what it is saying, I am just not understanding where the +1 ties in. I see that if we assume P is not prime, and then cannot divide without a remainder there is a contradiction, but what if that +1 was not in the statement question originally?
I hope what I am asking makes sense to you guys,
Thanks a lot.
(Source for proof Hans Riesel, Prime Numbers and Computer Methods for Factorization, Birkhaeuser, 1985, ISBN 0-8176-3291-3.)