Catalan's equation just before 2000. (Q2710129)

The author gives a survey of the recent developments on the Catalan conjecture (the only consecutive perfect powers are 8 and 9), especially the state of the art of upper and lower bounds on the exponents \(m\) and \(n\) for which \(x^{m}-y^{n} = 1\) might be solvable. This depends upon the recent work of \textit{P. Mihailescu} [J. Number Theory 99, No. 2, 225--231 (2003; Zbl 1049.11036)]. The author also gives a very well readable historical account of the problem.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00053].