We can start by assuming that there are finitely many prime numbers. The set of these primes is

where each p is a prime number. Then we will define an integer m such that

i.e., m is one more than the product of all the prime numbers. Now, m itself cannot be a prime number since it is greater than all of the prime numbers listed. All integers can be uniquely factored into primes, so it must be the case that for at least one of the prime numbers we can take m divided by that prime and get another integer

which can be rewritten as

We know that is an integer, since we chose the prime number in the denominator to be a factor of m, and we know that is an integer, since the prime number in the denominator is contained in the list of primes in the numerator. Finally, we know that cannot be an integer, since every prime number is greater than one. The set of integers is closed under addition, and so we have run into an absurd claim. Clearly, then our assumption that there is a finite list of prime numbers is a false assumption. Therefore, there are infinitely many prime numbers.