Catalan's conjecture is Mihăilescu's theorem
Martin Klazar
TL;DR
Catalan's conjecture asks for the unique nontrivial solution to x^m−y^n=1 with m,n≥2; the text outlines Mihăilescu's proof that there are no such solutions for distinct odd primes p>q>2, ultimately validating Catalan's conjecture. It synthesizes elementary case analyses (Pell equations, UFDs, and elementary resolutions of x^2−y^3=1 and x^2−y^q=1) with deep algebraic number theory, including Cassels' divisibility relations, obstruction groups, and Stickelberger ideals in cyclotomic fields. The work presents a structured path through M0.95–M4 theorems, the obstruction framework, and cyclotomic arithmetic to exclude all potential nontrivial solutions, culminating in Mihăilescu's theorem. The results have profound implications for Diophantine equations and the landscape of exponential equations in number theory, delivering a complete proof of Catalan's conjecture.
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 Mihăilescu'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. Mihăilescu'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.