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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 Pythagorean quadruple ： ウィキペディア英語版 Pythagorean quadruple A Pythagorean quadruple is a tuple of integers ''a'', ''b'', ''c'' and ''d'', such that ''d > 0'' and $a^2\; +\; b^2\; +\; c^2\; =\; d^2$, and is often denoted $(a,b,c,d)$. Geometrically, a Pythagorean quadruple $(a,b,c,d)$ defines a cuboid with integer side lengths |''a''|, |''b''|, and |''c''|, whose space diagonal has integer length ''d''. Pythagorean quadruples are thus also called ''Pythagorean Boxes''.〔R.A. Beauregard and E. R. Suryanarayan, ''Pythagorean boxes'', Math. Magazine 74 (2001), 222–227.〕 == Parametrization of primitive quadruples == The set of all primitive Pythagorean quadruples, i.e., those for which gcd(''a'',''b'',''c'') = 1 where gcd denotes the greatest common divisor, and for which without loss of generality ''a'' is odd, is parametrized by,〔R.D. Carmichael, ''Diophantine Analysis'', New York: John Wiley & Sons, 1915.〕〔L.E. Dickson, ''Some relations between the theory of numbers and other branches of mathematics'', in Villat (Henri), ed., Conférence générale, Comptes rendus du Congrès international des mathématiciens, Strasbourg, Toulouse, 1921, pp. 41–56; reprint Nendeln/Liechtenstein: Kraus Reprint Limited, 1967; Collected Works 2, pp. 579–594.〕〔R. Spira, ''The diophantine equation $x^2\; +\; y^2\; +\; z^2\; =\; m^2$'', Amer. Math. Monthly Vol. 69 (1962), No. 5, 360–365.〕 :$a\; =\; m^2+n^2-p^2-q^2,\backslash ,$ :$b\; =\; 2(mq+np),\backslash ,$ :$c\; =\; 2(nq-mp),\backslash ,$ :$d\; =\; m^2+n^2+p^2+q^2,\backslash ,$ where ''m'', ''n'', ''p'', ''q'' are non-negative integers and gcd(''m'', ''n'', ''p'', ''q'') = 1 and ''m'' + ''n'' + ''p'' + ''q'' ≡ 1 (mod 2). Thus, all primitive Pythagorean quadruples are characterized by the Lebesgue Identity :$(m^2\; +\; n^2\; +\; p^2\; +\; q^2)^2\; =\; (2mq\; +\; 2np)^2\; +\; (2nq\; -\; 2mp)^2\; +\; (m^2\; +\; n^2\; -\; p^2\; -\; q^2)^2.$ 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「Pythagorean quadruple」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.