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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 Euler's sum of powers conjecture ： ウィキペディア英語版 Euler's sum of powers conjecture Euler's conjecture is a disproved conjecture in mathematics related to Fermat's last theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers ''n'' and ''k'' greater than 1, if the sum of ''n'' ''k''th powers of non-zero integers is itself a ''k''th power, then ''n'' is greater than or equal to ''k''. In symbols, the conjecture falsely states that if $$ \sum_^ a_i^k = b^k where $n>1$ and $a\_1,\; a\_2,\; \backslash dots,\; a\_n,\; b$ are non-zero integers, then $n\backslash geq\; k$. The conjecture represents an attempt to generalize Fermat's last theorem, which is the special case ''n'' = 2: if $a\_1^k\; +\; a\_2^k\; =\; b^k$, then $2\; \backslash geq\; k$. Although the conjecture holds for the case ''k'' = 3 (which follows from Fermat's last theorem for the third powers), it was disproved for ''k'' = 4 and ''k'' = 5. It is unknown whether the conjecture fails or holds for any value ''k'' ≥ 6. == Background == Euler had an equality for four fourth powers $59^4\; +\; 158^4\; =\; 133^4\; +\; 134^4;$ this however is not a counterexample because no term is isolated on one side of the equation. He also provided a complete solution to the four cubes problem as in Plato's number $3^3+4^3+5^3=6^3$ or the taxicab number 1729.〔(【引用サイトリンク】title=Euler's Extended Conjecture )〕 The general solution for: :$x\_1^3+x\_2^3=x\_3^3+x\_4^3$ is :$x\_1\; =\; 1-(a-3b)(a^2+3b^2),\; x\_2\; =\; (a+3b)(a^2+3b^2)-1$ :$x\_3\; =\; (a+3b)-(a^2+3b^2)^2,\; x\_4\; =\; (a^2+3b^2)^2-(a-3b)$ where $a$ and $b$ are any integers. 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「Euler's sum of powers 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.