Note on Integral Distances
- First Online:
- 62 Downloads
A planar point set S is called an integral set if all the distances between the elements of S are integers. We prove that any integral set contains many collinear points or the minimum distance should be relatively large if |S| is large.
Unable to display preview. Download preview PDF.
- 1.A. Anning and P. Erdös, Integral distances, Bull. Amer. Math. Soc. 51 (1945), 548–560. Google Scholar
- 2.P. Erdös, Integral distances, Bull. Amer. Math. Soc. 51 (1945), 966. Google Scholar
- 3.P. Erdös, On some elementary problems in geometry, Köz. Mat. Lapok 61(7) (1980), 49–54 (in Hungarian). Google Scholar
- 4.H. Harborth, Integral distances in point sets, Charlemagne and His Heritage. 1200 Years of Civilization and Science in Europe, Vol. 2, pp. 213–224, Brepols, Turnhout, 1998. Google Scholar
- 5.H. Harborth and A. Kemnitz, Diameters of integral point sets, Intuitive Geometry (Siófok, 1985), pp. 255–266, Colloq. Math. Soc. János Bolyai, 48, North-Holland, Amsterdam, 1987. Google Scholar
- 7.G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th edition, p. 63, Oxford University Press, London, 1979. Google Scholar
- 8.G. B. Huff, Diophantine problems in geometry and elliptic ternary forms, Duke Math. J. 15 (1948), 443–453. Google Scholar