The contrapositive proof is more direct than the direct proof in this case...
I came to MHF through google, looking for a *direct* proof that for all integers x it holds that if is even, then is even. There's a post on this here: proof that if a square is even then the root is too., but it doesn't give a direct proof, unfortunately. Anyway, MHF seemed like a nice and helpful site, so I registered, and here I am.
Yes, it's quite neat, that's precisely the point: I'm looking at this as an example of a case where proving the contrapositive is a lot easier than giving a direct proof. Of course, I need the direct proof to justify this claim. (I'm talking about deriving that there exists an s.t. from the given that for some integer .
Well, of course if is odd, then is odd, and if is even, then is even. But without wanting to sound pedantic, I'd say that's a contrapositive proof in disguise. I'd like to start reasoning about and end at .
I just discussed it with a colleague, and we came up with this.
By the fundamental theorem of arithmetic, has a unique prime factorization, so , for certain primes , and positive whole numbers . (Assume these primes are listed in ascending order.) Because is even, and because it is a square, we have that for all , . (This shows why, if is even, .) So , which means is even. QED.
OK, thank you. What I meant was "direct" in the sense that I wanted to reason from being even to being even. What I meant by "contrapositive in disguise" is that just one of those cases (the one of x being odd) already provides the contrapositive proof.
I'm sorry, I'm new here; still learning the proper way of expressing myself on the forum. Thanks.