Catalan's conjecture is Mih\u{a}ilescu's theorem

Abstract

This text evolves from the lecture notes for my course on Catalan's conjecture in winter term 2025/26. The ultimate goal is to give full details of Mih\u{a}ilescu's proof. Current chapters: 1. Euler's theorem: $x^2-y^3=1$; 2. V. Lebesgue's theorem: $x^m-y^2=1$; 3. Chao Ko's theorem: $x^2-y^q=1$ with $q\ge5$; 4. Two relations of Cassels: $p\,|\,y$ and $q\,|\,x$; 5. Mih\u{a}ilescu's theorem: $x^p-y^q=1$ with $p>q>2$; 6. An obstruction group; 7. Super-Cassels relations: $p^2\,|\,y$ and $q^2\,|\,x$; 8. Theorem M4: $p=3,5$ or $q=3,5$; A Results from mathematical anlysis; and B Results from algebra.