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 $4$