visit "" Does there exist any simpler or different proof of Catalans conjecture?
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$\begingroup$I'm not aware of any proof of Mihăilescu's theorem other than that by Mihăilescu himself. Searching turns up this article of Henri Cohen, but I'm not sure if it's a modification of Mihăilescu's proof or an expository account of it. (It's also in French.)
There is a nice book by René Schoof expositing the proof of Mihăilescu. It is mostly self-contained; the preface states that
To read the first few chapters requires little more than a basic mathematical background and some knowledge of elementary number theory. The other chapters involve Galois theory, some more algebraic number theory, and a little bit of commutative algebra. [...] Our exposition is self-contained with one small exception. This regards chapter 14. Here we explain an argument of Mihăilescu’s that is based on Francisco Thaine’s famous theorem. Our proof of Thaine’s theorem involves an application of Chebotarev’s density theorem to the Hilbert class field of a cyclotomic field. While we do provide a proof of Chebotarev’s theorem, we do not prove the existence and the basic properties of the Hilbert class field. A proof would involve a good deal of class field theory, and this is not included in these notes.
(It is quite reasonable to not want to exposit class field theory for this purpose!)
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