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How many prime numbers are there?
So one may now ask, with all the stuff I have told them (they probably asked long ago, but like I said, this isn't ordered too well), 'How many prime numbers are there?' Well the answer is infinitely may. Euclid gave a very nice proof to this effect, so I will repeat it here:
First we assume that there are a finite number of primes. We then take these primes and multiply them together: 2*3*5*7..... = P. If we take P and add 1 to it, it will be an even number with a prime factor, but this prime factor will not be any of the primes that make it up since to divide P by any of them would leave a remainder of 1, so there has to be another prime number that divides into P. Here again:
Theorem:
There are infinitely many prime numbers.
Proof:
Suppose that p1= 2 < p2= 3 <...< pr are all of the primes. Let P = p1p2...pr + 1 and let p be a prime dividing P; then p cannot be any of p1,p2,...pr, otherwise p would divide the difference P-p1p2...pr=1, which is impossible. So this prime p is still another prime, and p1,p1,...,pr would not be all of the primes.
If you have had some expierence with convergence and divergence in calculus than you may want to read on. Although Euclid proved that there are infinitely many primes over 2000 years ago, there is infinity, and then there is a bigger infinity. So how big is the prime number subset? The best place to start, to answer that question is with the inverses of the prime numbers, i.e. 
. Where px is a prime number. The question is what the sum of the reciprocals of prime numbers approach, as the prime numbers approach infinity. Or maybe not so much what they approach (since it diverges) but whether or not it does diverge. If the sum were to diverge, we would be dealing with a large infinity, if they were to converge, we would be dealing with a small infinity, relatively speaking. But as it turns out, the sum does diverge, for a proof of this go to http://www.utm.edu/research/primes/infinity.html.
