Twitter proof: infinite primes


The proof of this post is a very well known proof on the infinitude of primes. For the proof I am going to rephrase an argument used by Euclid more than $2000$ years ago.

Theorem: there are infinitely many primes.

Twitter proof: if $\mathcal{P} = \{p_1, \cdots, p_n\}$ is a finite set of primes, then the number $q = p_1\times\cdots\times p_n + 1$ is such that $q \not\in \mathcal{P}$. Either $q$ is prime or $q$ has a prime factor $q'$ that cannot be in $\mathcal{P}$, otherwise $q'$ would have to divide $1$. Hence, no finite set $\mathcal{P}$ can contain all primes.


  - RGS

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