The article makes suggestions here about the classic proof which aren&#x27;t true.<p>Euclid&#x27;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&#x27;ve long ago been able to trivially rephrase it so as not to be, showing &quot;For any finite set of primes, there is some further prime&quot; directly).<p>And the classic proof of the infinitude of the primes does not anywhere use Proposition 30 of the Elements (see for yourself at <a href="http:&#x2F;&#x2F;aleph0.clarku.edu&#x2F;~djoyce&#x2F;elements&#x2F;bookIX&#x2F;propIX20.html" rel="nofollow">http:&#x2F;&#x2F;aleph0.clarku.edu&#x2F;~djoyce&#x2F;elements&#x2F;bookIX&#x2F;propIX20.ht...</a> ; Proposition 31 (that every composite has some prime factor) is used, but this in turn is argued for without any invocation of Proposition 30).<p>Where in the classic proof would you imagine &quot;if p is a prime and p | ab, then p | a or p | b&quot; would come up?