Simple City

In a paper with the unassuming title of Inter-universal Teichmuller theory IV: log-volume computations and set-theoretic foundations [pdf], Shinichi Mochizuki has released a purported proof of the ABC conjecture. This would be huge news if correct, as this single conjecture is known to imply all sorts of exciting facts about the world of numbers. Proposed by Joseph Oesterlé and David Masser in 1985, its most famous consequence is Fermat’s Last Theorem… but of course that has already succumbed to other methods. So, to give a flavour of its power, I’ll discuss another: Pillai’s conjecture.

This starts with the observation that the numbers 8 & 9 are rather unusual. They are neighbours which are both powers of other positive whole numbers: 8=2³ and 9=3². In 1844, Eugène Catalan conjectured that this is the only instance of two powers sitting next to each other, a delightful and surprising fact which was eventually proved by Preda Mihăilescu in 2002.

However, a host of related questions remain unanswered. What about powers which are two apart, as 25 (=5²) and 27 (=3³) are? Or three apart, as happens for 125 (=5³) and 128 (=2⁷)?

In 1936 Subbayya Pillai conjectured that for every whole number k there are only ever finitely many pairs of powers exactly k apart. But so far the only case this is known is for k=1, i.e. Catalan’s original 8 & 9.

A proof of the ABC conjecture would confirm Pillai’s conjecture for all the remaining values of k at a stroke… and a great deal else besides. So watch closely as the world’s number theorists now descend on Mochizuki’s paper!