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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 Dickson's lemma ： ウィキペディア英語版 Dickson's lemma In mathematics, Dickson's lemma states that every set of $n$-tuples of natural numbers has finitely many minimal elements. This simple fact from combinatorics has become attributed to the American algebraist L. E. Dickson, who used it to prove a result in number theory about perfect numbers.〔 However, the lemma was certainly known earlier, for example to Paul Gordan in his research on invariant theory.〔.〕 ==Example== Let $K$ be a fixed number, and let $S\; =\; \backslash$ be the set of pairs of numbers whose product is at least $K$. When defined over the positive real numbers, $S$ has infinitely many minimal elements of the form $(x,K/x)$, one for each positive number $x$; this set of points forms one of the branches of a hyperbola. The pairs on this hyperbola are minimal, because it is not possible for a different pair that belongs to $S$ to be less than or equal to $(x,K/x)$ in both of its coordinates. However, Dickson's lemma concerns only tuples of natural numbers, and over the natural numbers there are only finitely many minimal pairs. Every minimal pair $(x,y)$ of natural numbers has $x\backslash le\; K$ and $y\backslash le\; K$, for if ''x'' were greater than ''K'' then (''x'' −1,''y'') would also belong to ''S'', contradicting the minimality of (''x'',''y''), and symmetrically if ''y'' were greater than ''K'' then (''x'',''y'' −1) would also belong to ''S''. Therefore, over the natural numbers, $S$ has at most $K^2$ minimal elements, a finite number.〔With more care, it is possible to show that one of $x$ and $y$ is at most $\backslash sqrt\; K$, and that there is at most one minimal pair for each choice of one of the coordinates, from which it follows that there are at most $2\backslash sqrt\; K$ minimal elements.〕 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「Dickson's lemma」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.