Catalan's conjecture is Mihăilescu's theorem

2026/01/21 by Martin Klazar · 1 voice
Mathematics · #math.HO #math.NT

paper · pdf

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: x²-y³=1; 2. V. Lebesgue's theorem: x^m-y²=1; 3. Chao Ko's theorem: x²-y^q=1 with q≥5; 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: p² | y and q² | x; 8. Theorem M4: p=3,5 or q=3,5; A Results from mathematical anlysis; and B Results from algebra.

Citations

• Theorematum quorundam arithmeticorum demonstrationes
• Primary cyclotomic units and a proof of Catalans conjecture
• Catalan’s Conjecture: Another old Diophantine problem solved
• A Note on the Equation x 2 = y q + 1
• The Axioms for Euclidean Domains
• On the Equation a x - b y = 1
• INTERSECTION THEOREMS FOR SYSTEMS OF FINITE SETS

Discussions

• Martin Klazar: Catalan's conjecture is Mih\u{a}ilescu's theorem https://arxiv.org/abs/2601.14900 https://arxiv.org/pdf/2601.14900 https://arxiv.org/html/2601.14900 [bsky, 0 points, 0 comments]

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