Catalan's conjecture (d'apres Mihailescu) Rene' Schoof The perfect powers (squares, cubes, ...) are 1, 4, 8, 9, 16, 25, 27, 32, 49, 64, 81, ... In 1844 Catalan conjectured that the only _consecutive_ powers are 8 and 9. Only in 2002 this conjecture was proved by Preda Mihailescu. His beautiful proof exploits the arithmetic of cyclotomic fields. The first of two talks on Catalan's conjecture will be aimed at a general audience, followed by further details of Mihailescu's proof on 24 October.