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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 Tree of primitive Pythagorean triples ： ウィキペディア英語版 Tree of primitive Pythagorean triples In mathematics, a tree of primitive Pythagorean triples is a data tree in which each node branches to three subsequent nodes with the infinite set of all nodes giving all (and only) primitive Pythagorean triples without duplication. A Pythagorean triple is a set of three positive integers ''a, b,'' and ''c'' having the property that they can be respectively the two legs and the hypotenuse of a right triangle, thus satisfying the equation $a^2+b^2=c^2$; the triple is said to be primitive if and only if ''a, b,'' and ''c'' share no common divisor. The set of all primitive Pythagorean triples has the structure of a rooted tree, specifically a ternary tree, in a natural way. This was first discovered by B. Berggren in 1934.〔B. Berggren, "Pytagoreiska trianglar" (in Swedish), ''Elementa: Tidskrift för elementär matematik, fysik och kemi'' 17 (1934), 129–139. See page 6 for the rooted tree.〕 F. J. M. Barning showed〔Barning, F. J. M. (1963), "Over pythagorese en bijna-pythagorese driehoeken en een generatieproces met behulp van unimodulaire matrices" (in Dutch), Math. Centrum Amsterdam Afd. Zuivere Wisk. ZW-011: 37, http://oai.cwi.nl/oai/asset/7151/7151A.pdf 〕 that when any of the three matrices : $$ \begin A = \begin 1 & -2 & 2 \\ 2 & -1 & 2 \\ 2 & -2 & 3 \end & B = \begin 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 3 \end & C = \begin -1 & 2 & 2 \\ -2 & 1 & 2 \\ -2 & 2 & 3 \end \end is multiplied on the right by a column vector whose components form a Pythagorean triple, then the result is another column vector whose components are a different Pythagorean triple. If the initial triple is primitive, then so is the one that results. Thus each primitive Pythagorean triple has three "children". All primitive Pythagorean triples are descended in this way from the triple (3 ,4, 5), and no primitive triple appears more than once. The result may be graphically represented as an infinite ternary tree with (3, 4, 5) at the root node (see classic tree at right). This tree also appeared in papers of A. Hall in 1970〔A. Hall, "Genealogy of Pythagorean Triads", ''The Mathematical Gazette'', volume 54, number 390, December, 1970, pages 377–9.〕 and A. R. Kanga in 1990.〔Kanga, A. R., "The family tree of Pythagorean triples," ''Bulletin of the Institute of Mathematics and its Applications'' 26, January/February 1990, 15–17.〕 ==Proofs== 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「Tree of primitive Pythagorean triples」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.