An N-cluster is a point lattice configuration in which the distance between every pair of points is an integer, no three points are collinear, and no four points are concyclic. An example is the 6-cluster (0, 0), (132, -720), (546, -272), (960, -720), (1155, 540), (546, 1120).

Call the radius of the smallest circle centered at one of the points of an N-cluster which contains all the points in the N-cluster the extent. Noll and Bell (1989) found 91 nonequivalent prime 6-clusters of extent less than 20937, but found no 7-clusters.


Kreisel and Kurz (2006) subsequently found the 7-cluster given by multiplying the coordinates of the points (0,0), (49595290,0), (26127018,932064), (32142553,411864), (17615968,238464), (7344908,411864), (19079044,-54168) by (1,sqrt(2002))/2227, illustrated above.

See also

Collinear, Concyclic, No-Three-in-a-Line-Problem, Point Lattice, Rational Distance Problem

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Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 187, 1994.Kreisel, T. and Kurz, S. "There are Integral Heptagons, No Three Points on a Line, No Four on a Circle." 7 Nov 2006., L. C. and Bell, D. I. "n-clusters for 1<n<7." Math. Comput. 53, 439-444, 1989.

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Cite this as:

Weisstein, Eric W. "N-Cluster." From MathWorld--A Wolfram Web Resource.

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