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Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu. 23 and 32 are two powers of natural numbers, whose values 8 and 9 respectively are consecutive. The theorem states that this is the ''only'' case of two consecutive powers. That is to say, that the only solution in the natural numbers of :''x''''a'' − ''y''''b'' = 1 for ''a'', ''b'' > 1, ''x'', ''y'' > 0 is ''x'' = 3, ''a'' = 2, ''y'' = 2, ''b'' = 3. ==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). In 1976, Robert Tijdeman applied Baker's method in transcendence theory to establish a 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 exp exp exp exp 730 for the bound. This resolved Catalan's conjecture for all but a finite number of cases. However, the finite calculation required to complete the proof of the theorem was nonetheless too time-consuming to perform. Catalan's conjecture was proven by Preda Mihăilescu in April 2002, so it is now sometimes called ''Mihăilescu's theorem''. The proof was published in the ''Journal für die reine und angewandte Mathematik'', 2004. It makes extensive use of the theory of cyclotomic fields and Galois modules. An exposition of the proof was given by Yuri Bilu in the Séminaire Bourbaki. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Catalan's conjecture」の詳細全文を読む スポンサード リンク
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