The article makes suggestions here about the classic proof which aren't true. Euclid's proof of the infinitude of the primes was not phrased in terms of an ove

The article makes suggestions here about the classic proof which aren't true. Euclid's proof of the infinitude of the primes was not phrased in terms of an overarching reductio ad absurdum (and even had it counterfactually been, mathematicians would've long ago been able to trivially rephrase it so as not to be, showing "For any finite set of primes, there is some further prime" directly). And the classic proof of the infinitude of the primes does not anywhere use Proposition 30 of the Elements (see for yourself at http://aleph0.clarku.edu/~djoyce/elements/bookIX/propIX20.html http://aleph0.clarku.edu/~djoyce/elements/bookIX/propIX20.ht... ; Proposition 31 (that every composite has some prime factor) is used, but this in turn is argued for without any invocation of Proposition 30). Where in the classic proof would you imagine "if p is a prime and p | ab, then p | a or p | b" would come up?