# 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, 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 EuclidsProofOfTheInfinitudeOfPrimes 2013-03-22 12:44:07 2013-03-22 12:44:07 mathwizard (128) mathwizard (128) 9 mathwizard (128) Proof msc 11A41