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# Infinitely many prime numbers

## Euclid's proof

By contradition, let's consider that the set $\{ p_n \}$ of prime numbers has finitely many primes, with some cardinality $N$. Therefore, there is a highest prime $p_N$. Now take the new number

$$p = \left(\prod_{i=1}^N p_i \right) + 1$$

If there are only finitely many primes, any number should be able to get expressed by

$$n = \prod_{i = 1}^N (p_i)^{c(i)}$$