Rjlipton’s thoughts on the possible ABC Conjecture proof

There is this thing called the abc Conjecture. It’s a big question in number theory, which is the part of mathematics where we learn we don’t understand anything about prime numbers. Nearly a decade ago Shinichi Mochizuki announced a proof. It’s been controversial. Most importantly, it’s not been well-understood.

It’s finally getting published in a proper journal. A lot of mathematics work is passed around as PDFs, usually on arXiv.org, these days. It’s good for sharing fresh thoughts. But journal publication usually means that the paper has been reviewed, critically, and approved by people who could tell whether the reasoning is sound. Mochizuki’s paper is somewhere around 500 to 600 pages (I’ve seen different figures), and by every report hard to understand even for number theory proofs. A proof is, more than mathematicians like to admit, really an argument that convinces other mathematicians that, if we wanted to spend the time, we could find a completely rigorous proof. With very long proofs, and very complicated proofs, the standard of being convincing gets tougher.

In this essay, Not As Easy As ABC, rjlipton discusses some of the conjecture, and the problems of Mochizuki’s paper. Not specifically about whether this proof is right, but about the general problem of how we can trust difficult proofs. So you may find that worth the read.