The Catalan equation. (Q1423727)

This is the German translation (by J. Wolfart) from Gaz. Math., Soc. Math. Fr. 94, 25--29 (2002; Zbl 1116.11305). Catalan conjectured in 1844 that the only solution of the Diophantine equation \(x^m-y^n=1\), in the integer unknowns \(x\geq2\), \(y\geq2\), \(m\geq2\), \(n\geq2\) is \(3^2-2^3=1\). This was confirmed in 2002 by \textit{P. Mihailescu} [J. Reine Angew. Math. 572, 167--195 (2004; Zbl 1067.11017)]. The author gives a valuable historical overview on the most important works, ideas and techniques that have finally led to the resolution of the Catalan equation and presents some detailed proofs. The interested reader should also consult the Bourbaki lecture given by \textit{Yu. Bilu} [Catalan's conjecture (after Mihailescu). Astérisque 294, 1--26 (2004; Zbl 1094.11014)]