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

TL;DR

This paper provides a comprehensive lecture note series detailing Mihăilescu's proof of Catalan's conjecture, covering historical theorems, key lemmas, and algebraic and analytical tools used in the proof.

Contribution

It offers an in-depth, step-by-step exposition of Mihăilescu's proof, including related theorems and new algebraic and analytical results.

Findings

Mihăilescu's theorem proven: the only solution to x^p - y^q = 1 with p,q > 2 is (3,2,2,3)

Development of new algebraic and analytical tools for exponential Diophantine equations

Detailed analysis of related theorems and their role in the proof

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.