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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 Hall's conjecture ： ウィキペディア英語版 Hall's conjecture In mathematics, Hall's conjecture is an open question, , on the differences between perfect squares and perfect cubes. It asserts that a perfect square ''y''2 and a perfect cube ''x''3 that are not equal must lie a substantial distance apart. This question arose from consideration of the Mordell equation in the theory of integer points on elliptic curves. The original version of Hall's conjecture, formulated by Marshall Hall, Jr. in 1970, says that there is a positive constant ''C'' such that for any integers ''x'' and ''y'' for which ''y''2 ≠ ''x''3, :$|y^2\; -\; x^3|\; >\; C\backslash sqrt.$ Hall suggested that perhaps ''C'' could be taken as 1/5, which was consistent with all the data known at the time the conjecture was proposed. Danilov showed in 1982 that the exponent 1/2 on the right side (that is, the use of |''x''|1/2) can't be replaced by any higher power: for no δ > 0 is there a constant ''C'' such that |''y''2 - ''x''3| > C|''x''|1/2 + δ whenever ''y''2 ≠ ''x''3. In 1965, Davenport proved an analogue of the above conjecture in the case of polynomials: if ''f''(''t'') and ''g''(''t'') are nonzero polynomials over C such that ''g''(''t'')3 ≠ ''f''(''t'')2 in C(), then :$\backslash deg(g(t)^2\; -\; f(t)^3)\; \backslash geq\; \backslash frac\backslash deg\; f(t)\; +\; 1.$ The ''weak'' form of Hall's conjecture, due to Stark and Trotter around 1980, replaces the square root on the right side of the inequality by any exponent ''less'' than 1/2: for any ''ε'' > 0, there is some constant ''c''(ε) depending on ε such that for any integers ''x'' and ''y'' for which ''y''2 ≠ ''x''3, :$|y^2\; -\; x^3|\; >\; c(\backslash varepsilon)\; x^.$ The original, ''strong'', form of the conjecture with exponent 1/2 has never been disproved, although it is no longer believed to be true and the term ''Hall's conjecture'' now generally means the version with the ε in it. For example, in 1998, Noam Elkies found the example 4478849284284020423079182 - 58538865167812233 = -1641843, for which compatibility with Hall's conjecture would require ''C'' to be less than .0214 ≈ 1/50, so roughly 10 times smaller than the original choice of 1/5 that Hall suggested. The weak form of Hall's conjecture would follow from the ABC conjecture. A generalization to other perfect powers is Pillai's conjecture. ==Notes== 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「Hall'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.