Euclid’s proof of the infinitude of primes

If there were only a finite amount of primes then there would be some largest prime $p$ . However $p!+1$ is not divisible by any number $1<n\leq p$ , since $p!$ is, so $p!+1$ 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 $p$ by which $p!+1$ is divisible.

Actually Euclid did not use $p!$ for his proof but stated that if there were a finite list $p_{1},\ldots,p_{n}$ of primes, then the number $p_{1}\cdots p_{n}+1$ 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.

Title	Euclid’s proof of the infinitude of primes
Canonical name	EuclidsProofOfTheInfinitudeOfPrimes
Date of creation	2013-03-22 12:44:07
Last modified on	2013-03-22 12:44:07
Owner	mathwizard (128)
Last modified by	mathwizard (128)
Numerical id	9
Author	mathwizard (128)
Entry type	Proof
Classification	msc 11A41