No rational number squares to 2, after D. Zeilberger

If p/q is a given rational number, the process

write p/q = n + k/q, where 0 ≤ k/q < 1, equiv 0 ≤ k < q.
if k = 0: 
else return q/k, and loop.

is guaranteed to halt, since we must eventually hit k = 0, as the denominator is strictly smaller after each non-terminating cycle.

Assume r is a rational number satisfying r^2 = 2. By inspection, 1 < r < 2. Apply the process above:

r = 1 + (r-1) 
--> 1/(r-1) = (r+1)/(r^2 - 1) = r+1 = 2 + (r-1)
--> 1/(r-1)

Which will never halt, so no such rational number can exist.

I learned this nice proof from Doron Zeilberger’s manuscript Two Motivated Concrete Proofs (much better than the usual one) that the Square-Root of 2 is Irrational .


Joyal: Arithmetisation dans le topos

ON 31st July 1974 André Joyal gave a talk with the title Arithmetisation dans le topos. Anders Kock took notes, and kindly shared them with me a few years back. He gave me permission to host them publicly, so here they are: pdf link.

Dupuy: computations conditional on IUT3 Corollary 3.12

Just a quick note to advertise some slides by Taylor Dupuy recently presented at Rice University

Explicit Computations in IUT, slides for talk in AGNT Seminar, Rice University April 8, 2019 (pdf)

This is joint work with Anton Hilado modelled on his series of YouTube videos presented earlier this year (and there are links to them in the slides, for technical details). Note that all of this is explicitly stated by Dupuy as being conditional on Corollary 3.12 in Mochizuki’s third IUT paper, the written proof of which is not accepted by almost the entire number theory community.

I see the benefit here as at least simplifying what seems like the sound part of Mochizuki’s work (even if dependent on something still regarded as conjectural) to ordinary mathematics; no dismantling alien ring structures or odd metaphors about how school students get confused by logarithms. Not only that, but the statement of Corollary 3.12 is rendered in ordinary mathematics, rather than in the language of Frobenioids and their ilk.

SDT book review done

I’ve been reading Bunge, Gago and San Luis’s book Synthetic Differential Topology to review for MathReviews. I’ve finally sent off the written review, to appear here within short order (requires subscription).


On the whole, the book is ok, pretty much self-contained, but could have done with a bit more copy-editing. I think of this as more a fault of the publisher, but given that it’s in a lecture notes series, they probably didn’t give it as thorough a going-over. I’m very much intrigued by the link between Penon’s intrinsic topology on objects of a topos, which plays a big rôle in SDT, and Mike Shulman’s real cohesion, as hinted at in these slides by David Jaz Meyers.

(My thanks to the AMS for the free reviewers copy of the book).