http://www.mathreference.com/num,inf.html
impressed me with this

concise proof that there are

Infinitely Many Primes

Suppose there is a finite list of primes.

Multiply them together and add 1, giving n.

Now n is not prime - it is larger than all the primes on our list,

which is suppose to be complete.

So n is composite.

Let p be a prime in the unique factorization of n.

Since p is on the list, it divides n-1, as well as n.

Hence it divides 1, which is impossible.

There are an infinite number of prime numbers.