Proof: By Euclid
(related to Proposition: Prop. 9.20: Infinite Number of Primes)
- For let the least number measured by $A$, $B$, $C$ have been taken, and let it be $DE$ [Prop. 7.36].
- And let the unit $DF$ have been added to $DE$.
- So $EF$ is either prime, or not.
- Let it, first of all, be prime.
- Thus, the (set of) prime numbers $A$, $B$, $C$, $EF$, (which is) more numerous than $A$, $B$, $C$, has been found.
- And so let $EF$ not be prime.
- Thus, it is measured by some prime number [Prop. 7.31].
- Let it be measured by the prime (number) $G$.
- I say that $G$ is not the same as any of $A$, $B$, $C$.
- For, if possible, let it be (the same).
- And $A$, $B$, $C$ (all) measure $DE$.
- Thus, $G$ will also measure $DE$.
- And it also measures $EF$.
- (So) $G$ will also measure the remainder, unit $DF$, (despite) being a number [Prop. 7.28].
- The very thing (is) absurd.
- Thus, $G$ is not the same as one of $A$, $B$, $C$.
- And it was assumed (to be) prime.
- Thus, the (set of) prime numbers $A$, $B$, $C$, $G$, (which is) more numerous than the assigned multitude (of prime numbers), $A$, $B$, $C$, has been found.
- (Which is) the very thing it was required to show.
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"