
Simple proofs of great theorems - ColinWright
http://mathscholar.org/2018/09/simple-proofs-of-great-theorems/
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aoki
It seems to me that most commenters are missing the point, which is that
results like these usually remain unproven until the end of 2-3 course
sequences that are mainly taken by math majors. What the authors are doing is
to reduce each result to something that can be verified by a high school
calculus student (even if it does not provide very much insight). These are
not elegantly concise proofs; they are maximally elementary proofs that you
can give to a curious kid. I think there’s value in that, even if that’s not
what mathematicians do with their proofs.

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hansbo
I think the simplicity of the proofs were somewhat overstated.

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goldenkey
I agree. The integral function required to prove pi as irrational is quite
ugly. I suspect there are much simpler proofs that convey a better intuition
rather than appearing contrived.

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ColinWright
It would be interesting if you could find one. This is certainly the simplest
I've ever seen.

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goldenkey
I dont know the specifics of it but one could use the zeta function and the
periodicity requirement of infinite series that equal rationals. It is clear
that the zeta(2) is not periodic but one would have to prove that and then use
the existing theorems about infinite series, periodicity, and rationals.

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ColinWright
Doesn't feel like it's simpler - there's a lot more going on in this. A _lot_
more. The proof in the original article is completely self-contained.

Still, I suppose simplicity is in the eye of the beholder.

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goldenkey
I wouldn't consider an obfuscated 1 line C program simple. Thats how I look at
unintuitive proofs that simply evaluate to true but appear to lack
premeditation. Might as well just replace the proof with a brute force
computer assisted proof.. I get that people want simple... simple and small is
nice..its a challenge for any mathematician. But the beauty is really in
something other than the proof - its origin and creation story. Every
"challenge proof" should come with an addendum. Kinda like Obfuscated C or
JS2k contest entries usually come with a write-up.

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ColinWright
These aren't intended to be "beautiful" proofs, they are _simple_ proofs. They
can be followed without having to grok vast amounts of external material. They
are self-contained, without external dependencies. They are short, and while
you may feel that they fail to impart insight, I'm not so sure. Having lived
with these proofs for some years ( I knew them before I saw this series of
blogs posts) I find that they are becoming simpler over time as I understand
more about what's going on around them.

Fermat's Last Theorem is trivial to prove because we just quote the Taniyama-
Shimura-Weil Theorem, observe a particular elliptic curve, and there you are.
It's "simple," once you have T-S-W. But that's means it's not self-contained,
and while it is probably more enlightening as to the underlying _why_ all this
works, it's _not_ simple.

So your comment about using the zeta function and the periodicity requirement
of infinite series exactly shows why your suggestion might be enlightening,
but is not _simple._

At least, not in any sense I would use.

