# Euclid's Theorem/Corollary 1/Proof 2

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## Corollary to Euclid's Theorem

There are infinitely many prime numbers.

## Proof

Assume that there are only finitely many prime numbers.

Let $p$ be the largest of these.

Then from Existence of Prime between Prime and Factorial there exists a prime number $q$ such that:

- $p < q \le p! + 1$

So there cannot be such a $p$.

$\blacksquare$

## Sources

- 1982: Martin Davis:
*Computability and Unsolvability*(2nd ed.) ... (previous) ... (next): Appendix $1$: Some Results from the Elementary Theory of Numbers: Theorem $4$