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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 Stella octangula number ： ウィキペディア英語版 Stella octangula number In mathematics, a stella octangula number is a figurate number based on the stella octangula, of the form ''n''(2''n''2 − 1).〔.〕〔.〕 The sequence of stella octangula numbers begins :0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, ....〔 ==Ljunggren's equation== There are only two positive square stella octangula numbers, 1 and 9653449 = 31072 = (13 × 239)2, corresponding to ''n'' = 1 and ''n'' = 169 respectively.〔〔.〕 The elliptic curve describing the square stella octangula numbers, :$m^2\; =\; n\; (2n^2\; -\; 1)$ may be placed in the equivalent Weierstrass form :$x^2\; =\; y^3\; -\; 2y$ by the change of variables ''x'' = 2''m'', ''y'' = 2''n''. Because the two factors ''n'' and 2''n''2 − 1 of the square number ''m''2 are relatively prime, they must each be squares themselves, and the second change of variables $X=m/\backslash sqrt$ and $Y=\backslash sqrt$ leads to Ljunggren's equation :$X^2\; =\; 2Y^4\; -\; 1$〔 A theorem of Siegel states that every elliptic curve has only finitely many integer solutions, and found a difficult proof that the only integer solutions to his equation were (1,1) and (239,13), corresponding to the two square stella octangula numbers.〔.〕 Louis J. Mordell conjectured that the proof could be simplified, and several later authors published simplifications.〔〔.〕〔.〕 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「Stella octangula number」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.