There are infinitely many prime numbers.
This is a proof by contradiction.
Suppose that there are finitely many prime numbers, p1, p2, p3, ... , pn, where n is a positive integer.
Consider the product of the prime numbers. Call it P. So we have P = p1 * p2 * ... * pn.
Now consider one more than this number, P + 1.
By the distributive property, if one of the primes, say px, divides P + 1, then since px also divides P, px must divide 1. But the only integer that divides 1 is 1, and 1 is not prime, so there is no such px. Therefore, P + 1 is not devisible by any lower prime numbers, which implies that P + 1 is prime, which contradicts the assumption that p1, p2, ... , pn was the full list.