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.

单词	EuclidsProofOfTheInfinitudeOfPrimes
释义	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.
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