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=2^{3} and 9=3^{2}. In 1844, EugĂ¨ne 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 (=5^{2}) and 27 (=3^{3}) are? Or three apart, as happens for 125 (=5^{3}) and 128 (=2^{7})?

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 kat a stroke… and a great deal else besides. So watch closely as the world’s number theorists now descend on Mochizuki’s paper!

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