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Aug 5, 2023Liked by Joel David Hamkins

I wrote a paper with Conway on proofs of the irrationality of sqrt(2)


We classified 7 “essentially different” ones, based on how they generalized (each one has a natural set of algebraic numbers its argument can be applied to, and these sets overlap in various ways).

Recently I discovered an 8th proof which generalizes furthest of all, to all algebraic irrationals. Can you guess it?

There is also the ridiculous 9th proof using ultraproducts, which generalizes to all roots of irreducible polynomials that are unsolvable mod p for infinitely many p (which is, in fact, all algebraic irrationals), but is useless for most of them because you need the something like the Chebotarev density theorem to show that condition holds.

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