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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 Erdős–Anning theorem ： ウィキペディア英語版 Erdős–Anning theorem The Erdős–Anning theorem states that an infinite number of points in the plane can have mutual integer distances only if all the points lie on a straight line. It is named after Paul Erdős and Norman H. Anning, who published a proof of it in 1945.〔.〕 ==Rationality versus integrality== Although there can be no infinite non-collinear set of points with integer distances, there are infinite non-collinear sets of points whose distances are rational numbers. For instance, on the unit circle, let ''S'' be the set of points $(\backslash cos\backslash theta,\backslash sin\backslash theta)$ for which $\backslash tan\backslash frac$ is a rational number. For each such point, both $\backslash sin\backslash frac$ and $\backslash cos\backslash frac$ are themselves both rational, and if $\backslash theta$ and $\backslash phi$ define two points in ''S'', then their distance is the rational number $|2\backslash sin\backslash frac\backslash cos\backslash frac-2\backslash sin\backslash frac\backslash cos\backslash frac|$. More generally, a circle with radius $\backslash rho$ contains a dense set of points at rational distances to each other if and only if $\backslash rho^2$ is rational.〔.〕 For any finite set ''S'' of points at rational distances from each other, it is possible to find a similar set of points at integer distances from each other, by expanding ''S'' by a factor of the least common denominator of the distances in ''S''. Therefore, there exist arbitrarily large finite sets of points with integer distances from each other. However, including more points into ''S'' may cause the expansion factor to increase, so this construction does not allow infinite sets of points at rational distances to be translated to infinite sets of points at integer distances. It remains unknown whether there exists a set of points at rational distances from each other that forms a dense subset of the Euclidean plane.〔 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「Erdős–Anning theorem」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.