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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Fermat–Catalan conjecture ： ウィキペディア英語版 Fermat–Catalan conjecture In number theory, the Fermat–Catalan conjecture combines ideas of Fermat's last theorem and the Catalan conjecture, hence the name. The conjecture states that the equation has only finitely many solutions (''a'',''b'',''c'',''m'',''n'',''k'') with distinct triplets of values (''a''''m'', ''b''''n'', ''c''''k''); here ''a'', ''b'', ''c'' are positive coprime integers and ''m'', ''n'', ''k'' are positive integers satisfying This inequality restriction on the exponents has the effect of precluding consideration of the known infinitude of solutions of (1) in which two of the exponents are 2 (such as Pythagorean triples). As of 2015, the following ten solutions to (1) are known:〔.〕 :$1^m+2^3=3^2\backslash ;$ :$2^5+7^2=3^4\backslash ;$ :$13^2+7^3=2^9\backslash ;$ :$2^7+17^3=71^2\backslash ;$ :$3^5+11^4=122^2\backslash ;$ :$33^8+1549034^2=15613^3\backslash ;$ :$1414^3+2213459^2=65^7\backslash ;$ :$9262^3+15312283^2=113^7\backslash ;$ :$17^7+76271^3=21063928^2\backslash ;$ :$43^8+96222^3=30042907^2\backslash ;$ The first of these (1''m''+23=32) is the only solution where one of ''a'', ''b'' or ''c'' is 1, according to the Catalan conjecture, proven in 2002 by Preda Mihăilescu. While this case leads to infinitely many solutions of (1) (since we can pick any ''m'' for m>5), these solutions only give a single triplet of values (''a''''m'', ''b''''n'', ''c''''k''). It is known by Faltings' theorem that for any fixed choice of positive integers ''m'', ''n'' and ''k'' satisfying (2), only finitely many coprime triples (''a'', ''b'', ''c'') solving (1) exist, but of course the full Fermat–Catalan conjecture is a much stronger statement since it allows for an infinitude of sets of exponents ''m'', ''n'' and ''k''. The abc conjecture implies the Fermat–Catalan conjecture.〔 Beal's conjecture is true if and only if all Fermat-Catalan solutions use 2 as an exponent for some variable. ==See also== *Sums of powers, a list of related conjectures and theorems 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「Fermat–Catalan conjecture」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.