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Catalan’s equation just before 2000. (English) Zbl 1065.11019
Jutila, Matti (ed.) et al., Number theory. Proceedings of the Turku symposium on number theory in memory of Kustaa Inkeri, Turku, Finland, May 31–June 4, 1999. Berlin: de Gruyter (ISBN 3-11-016481-7/hbk). 247-254 (2001).
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 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.
For the entire collection see [Zbl 0959.00053].

##### MSC:
 11D61 Exponential Diophantine equations