That&#x27;s not really an accurate representation of the old proof. You&#x27;ve turned it around to make it look like the new one, but that&#x27;s not how it is originally laid out at all.<p>The fundamental problem (for constructivists) with Euclid&#x27;s proof is that it assumes the existence of a set of all primes, and then proves that that set is incorrect (either contains a composite, or doesn&#x27;t contain at least one specific prime).<p>Saidak&#x27;s proof, on the other hand, only assumes the existence of a single integer, and then constructs infinitely many primes from that integer.<p>Your restatement of Euclid&#x27;s proof is basically &quot;look, Euclid and Saidak proved the same thing&quot;, not &quot;look, Euclid&#x27;s proof is constructive too&quot;.