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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 Beal's conjecture ： ウィキペディア英語版 Beal's conjecture Beal's conjecture is a conjecture in number theory: :If :: $A^x\; +B^y\; =\; C^z,$ :where ''A'', ''B'', ''C'', ''x'', ''y'', and ''z'' are positive integers with ''x'', ''y'', ''z'' > 2, then ''A'', ''B'', and ''C'' have a common prime factor. Equivalently, :There are no solutions to the above equation in positive integers ''A, B, C, x, y, z'' with ''A, B'', and ''C'' being pairwise coprime and all of ''x, y, z'' being greater than 2. Banker Andrew Beal formulated this conjecture in 1993 while investigating generalizations of Fermat's last theorem.〔(【引用サイトリンク】url=http://www.bealconjecture.com/ )〕 It has been claimed that the same conjecture was formulated independently by Robert Tijdeman and Don Zagier. While more commonly known as the "Beal conjecture", it has also been referred to as the Tijdeman–Zagier conjecture.〔 In the 1950s, L. Jesmanowicz and Chao Ko considered a potential class of solutions to the equation, namely those with ''A, B, C'' also forming a Pythagorean triple.〔 Wacław Sierpiński, ''Pythagorean triangles'', Dover, 2003, p. 55 (orig. Graduate School of Science, Yeshiva University, 1962).〕 For a proof or counterexample published in a refereed journal, Beal initially offered a prize of US $5,000 in 1997, raising it to $50,000 over ten years, but has since raised it to US $1,000,000. ==Related examples== To illustrate, the solution $3^3\; +\; 6^3\; =\; 3^5$ has bases with a common factor of 3, the solution $7^3\; +\; 7^4\; =\; 14^3$ has bases with a common factor of 7, and $2^n\; +\; 2^n\; =\; 2^$ has bases with a common factor of 2. Indeed the equation has infinitely many solutions where the bases share a common factor, including generalizations of the above three examples, respectively :$3^+()^=3^;$ $n\; \backslash ge1$ :$(a^-1)^+(a^-1)^=()^;$ $a\; \backslash ge2,\; n\; \backslash ge3$ and :$()^n+()^n=(a^n+b^n)^;$ $a\; \backslash ge1,\; b\; \backslash ge1,\; n\; \backslash ge3$ Furthermore, for each solution (with or without coprime bases), there are infinitely many solutions with the same set of exponents and an increasing set of non-coprime bases. That is, for solution :$A\_1^+B\_1^=C\_1^$ we additionally have :$A\_^+B\_^=C\_^;$ $n\; \backslash ge2$ where :$A\_=\; (A\_^)\; (B\_^)\; (C\_^)$ :$B\_=\; (A\_^)\; (B\_^)\; (C\_^)$ :$C\_=\; (A\_^)\; (B\_^)\; (C\_^)$ Any solutions to the Beal conjecture will necessarily involve three terms all of which are 3-powerful numbers, i.e. numbers where the exponent of every prime factor is at least three. It is known that there are an infinite number of such sums involving coprime 3-powerful numbers; however, such sums are rare. The smallest two examples are: :$\backslash begin$ 271^3 + 2^3 3^5 73^3 = 919^3 &= 776,151,559 \\ 3^4 29^3 89^3 + 7^3 11^3 167^3 = 2^7 5^4 353^3 &= 3,518,958,160,000 \\ \end What distinguishes Beal's conjecture is that it requires each of the three terms to be expressible as a single power. 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「Beal's 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.