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From the article: The proof just given is conceptually even simpler than the original proof due to Euclid, since it does not use Eudoxus’s method of“reductio a
by shasta 11y ago
From the article:
The proof just given is conceptually even simpler than the
original proof due to Euclid, since it does not use Eudoxus’s method of“reductio ad absurdum,” proof by contradiction. And unlike most other proofs of the theorem, it does not require Proposition 30 of Elements (sometimes called “Euclid’s Lemma”) that states: if p is a prime and p|ab, then either p|a or p|b. Moreover, our proof is constructive, and it gives integers with an arbitrary number of prime factors.
Edit: Actually, even though the article seems to imply that the classic proof ("most proofs") uses prop 30, it doesn't really seem to.
- Chinjut 11y agoThe 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?
- shasta 11y agoYeah, I already edited my post. I agree with you.