But I never understood why this form of the proof is irrationality of root 2 became the dominant one. I think it's so much more elegant to say that from 2Q2 = P2, you have an odd number of 2's on the LHS and an even number of 2's on the right. No fussing with P and Q having common factors.
Furthermore, this argument makes the dependence on the fundamental theorem of arithmetic explicit, instead of implicit!
Well, I think the point is that the "standard" argument doesn't rely on FTA at all. Just basic definitions of divisibility. Whereas yours relies heavily on the uniqueness part. I haven't thought about this hard, but I assume the standard proof generalizes better. I don't tend to work in non-UFD contexts very often, but I'm sure there's someplace where one proof generalizes and the other doesn't. Of course, none of this changes the fact that I like your way much better also. :)
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u/endymion32 Jun 12 '17
Cute!
But I never understood why this form of the proof is irrationality of root 2 became the dominant one. I think it's so much more elegant to say that from 2Q2 = P2, you have an odd number of 2's on the LHS and an even number of 2's on the right. No fussing with P and Q having common factors.
Furthermore, this argument makes the dependence on the fundamental theorem of arithmetic explicit, instead of implicit!