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Erdős distinct distances problem : ウィキペディア英語版 | Erdős distinct distances problem In discrete geometry, the Erdős distinct distances problem states that between distinct points on a plane there are at least distinct distances. It was posed by Paul Erdős in 1946. In a 2010 preprint, Larry Guth and Nets Hawk Katz announced a solution.〔. See also (The Guth-Katz bound on the Erdős distance problem ) by Terry Tao and (Guth and Katz’s Solution of Erdős’s Distinct Distances Problem ) by János Pach.〕 ==The conjecture== In what follows let denote the minimal number of distinct distances between points in the plane. In his 1946 paper, Erdős proved the estimates for some constant . The lower bound was given by an easy argument, the upper bound is given by a square grid (as there are numbers below ''n'' which are sums of two squares, see Landau–Ramanujan constant). Erdős conjectured that the upper bound was closer to the true value of ''g''(''n''), specifically, holds for every .
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