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.

I think the simplicity of the proofs were somewhat overstated.

Perhaps these things are harder to prove than you think, and these really are, in the grand scheme of things, quite simple.

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.

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

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.

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.

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.

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.