Editing Catalan's conjecture (section)

==History==
The history of the problem dates back at least to [[Gersonides]], who proved a special case of the conjecture in 1343 where (''x'', ''y'') was restricted to be (2, 3) or (3, 2).  The first significant progress after Catalan made his conjecture came in 1850 when [[Victor-Amédée Lebesgue]] dealt with the case ''b'' = 2.<ref>{{citation | author=Victor-Amédée Lebesgue | author-link=Victor-Amédée Lebesgue | title=Sur l'impossibilité, en nombres entiers, de l'équation ''x<sup>m</sup>''=''y''<sup>2</sup>+1 | journal=Nouvelles annales de mathématiques | series=1<sup>re</sup> série | volume=9 | year=1850 | pages=178–181 }}</ref>

In 1976, [[Robert Tijdeman]] applied [[Baker's method]] in [[transcendental number theory|transcendence theory]] to establish a [[upper and lower bounds|bound]] on ''a'',''b'' and used existing results bounding ''x'',''y'' in terms of ''a'', ''b'' to give an effective upper bound for ''x'',''y'',''a'',''b''. Michel Langevin computed a value of <math>\exp \exp \exp \exp 730 \approx 10^{10^{10^{10^{317}}}}</math> for the bound,<ref>{{citation | title=13 Lectures on Fermat's Last Theorem | first=Paulo | last=Ribenboim | author-link=Paulo Ribenboim | publisher=[[Springer-Verlag]] | year=1979 | isbn=0-387-90432-8 | zbl=0456.10006 | page=236 }}</ref> resolving Catalan's conjecture for all but a finite number of cases.

Catalan's conjecture was proven by [[Preda Mihăilescu]] in April 2002. The proof was published in the ''[[Journal für die reine und angewandte Mathematik]]'', 2004. It makes extensive use of the theory of [[cyclotomic field]]s and [[Galois module]]s.  An exposition of the proof was given by [[Yuri Bilu]] in the [[Séminaire Bourbaki]].<ref>{{Citation | last1=Bilu | first1=Yuri | title=Séminaire Bourbaki vol. 2003/04 Exposés 909-923 | series=Astérisque | year=2004 | volume=294 | chapter= Catalan's conjecture | pages=1–26| chapter-url=http://www.numdam.org/book-part/SB_2002-2003__45__1_0/ }}</ref>  In 2005, Mihăilescu published a simplified proof.<ref>{{Harvnb|Mihăilescu|2005}}</ref>