[LegionMammal978]

Then again, even an 'elegant' proof can be surprisingly inflexible. I've recently been working through Apéry's proof that ζ(3) is irrational. It's so simple that even a clueless dabbler like me can understand all the details. Yet no one has been able to make his construction work directly for anything else (that hasn't already been proven irrational). C'est la vie, I suppose.

[rocqua]

There was a post yesterday of a quanta article: https://news.ycombinator.com/item?id=42644896.

The article explains that two mathematicians were able to place Apery's proof that ζ(3) is irrational into a much wider (and hence more powerful) framework. I doubt that framework is as easy to understand as the original proof. But in the end something with wider applicability did come out of the proof.

[LegionMammal978]

Yeah, many of the fancy analytic methods are beyond my level of dabbling. I've been trying to learn more about them, so I can solve the myriad exercises left to the reader in all the Diophantine approximation papers.

Still, the newer methods publicized in the Quanta article definitely get more involved, and at least from my perspective they don't establish things as elegantly as Apéry's ζ(2) and ζ(3) arguments do. Hopefully they turn out to be powerful in practice, to make up for it.