If there were only a finite amount of primes then there would be some largest prime . However is not divisible by any number , since is, so cannot be factored by the primes we already know, but every integer greater than one is divisible by at least one prime, so there must be some prime greater than by which is divisible.

Actually Euclid did not use for his proof but stated that if there is a finite list of primes, then the number is not divisible by any of these primes and thus either prime and not in the list or divisible by a prime not in the list.