If there was a finite number of primes, then there would exist some "biggest prime number". Call it .

Therefore, you could create a number that was the product of all the prime numbers. Call it .

So .

Then .

If you were to divide by ANY of the prime numbers, then you would ALWAYS have a remainder of .

That means that either there is a prime number larger than which divides , or else is itself a prime number.

Either way, this contradicts our original statement that is the largest prime.

Therefore, there is an infinite number of prime numbers.