Re:A One-Line Proof of the Infinitude of Primes
by yequalsx on Tuesday November 15, 2016 @01:37PM
(#12960848)
Yes, definitely there is more information implicitly wrapped into the sine proof. That's why it's not simpler. However, it is not a worse proof either. It's just a different one that is clever and can help to reinforce concepts from trigonometry.
Re:A One-Line Proof of the Infinitude of Primes
by bonoboTP on Tuesday November 15, 2016 @02:09PM
(#12961164)
There's not much trigonometry in this proof, just the fact that sine has period 2pi is zero at pi. Any function that has these two properties could directly replace sine in this proof.
And actually any periodic function that is zero somewhere could be used in a tiny bit modified version of this proof.
Re:A One-Line Proof of the Infinitude of Primes
by JadeNB on Tuesday November 15, 2016 @03:18PM
(#12961791)
> And actually any periodic function that is zero somewhere could be used in a tiny bit modified version of this proof.
I think that, for the modification to be tiny, you need something like: there is an interval on which the function is positive, and on the boundary of which it vanishes. (Some sign condition is necessary; taking the constant function at 0 doesn't work! Of course, this particular condition is automatically satisfied for any continuous, periodic function that vanishes somewhere, but is not identically 0.)
While your point that every step of a proof requires justification is a good one, this is really, really easy to verify. That doesn't excuse one from actually giving that justification, in the course of a formal proof; but there's at least as much implicit knowledge bound up in the linked proof. (For example, one needs the existence of sine, its 2π-periodicity, the fact that it is positive from 0 to π/2 ….)