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Catalan’s conjecture. (English) Zbl 1094.11014
Bourbaki seminar. Volume 2002/2003. Exposes 909–923. Paris: Société Mathématique de France (ISBN 2-85629-156-2/pbk). Astérisque 294, 1-26, Exp. No. 909 (2004).
Author’s summary: The subject of the talk is the recent work of P. Mihăilescu [J. Reine Angew. Math. 572, 167–195 (2004; Zbl 1067.11017)], who proved that the equation $$x^p-y^q=1$$ has no solutions in non-zero integers $$x,y$$ and odd primes $$p,q$$. Together with the results of Lebesgue (1850) and Ko Chao (1865) this implies the celebrated conjecture of Catalan (1843): the only solution to $$x^u-y^v =1$$ in integers $$x,y >0$$ and $${u,v>1}$$ is $$3^2-2^3 = 1$$. Before the work of Mihăilescu the most definitive result on Catalan’s problem was due to Tijdeman (1976), who proved that the solutions of Catalan’s equation are bounded by an absolute effective constant.
Comprehensive historical accounts can be found in P. Ribenboim’s book [Catalan’s conjecture. London: Academic Press (1994; Zbl 0824.11010)] and M. Mignotte’s survey [Jutila, Matti (ed.) et al., Number theory. Proceedings of the Turku symposium on number theory in memory of Kustaa Inkeri, Turku, Finland, 1999. Berlin: de Gruyter, 247–254 (2001; Zbl 1065.11019)].
For the entire collection see [Zbl 1052.00010].

##### MSC:
 11D61 Exponential Diophantine equations
##### Keywords:
cyclotomic units; Wieferich pairs