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 Integer points on circles. (Posted on 2011-02-28)
If a circle is centered on the origin and has radius r, it may pass through some points with integer coordinates. In fact, by symmetry, this will be a multiple of 4.

Find the smallest radius that will pass through 4n integer points where n=1,2,3,4,5.

Feel free to go further.

 See The Solution Submitted by Jer Rating: 3.5000 (2 votes)

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 A symmetry argument | Comment 5 of 14 |
Any radius that goes through exactly 4n points where n is odd must have 4 of the points on either the axes or the diagonals (i.e., y= +/-x).  All other points are part of a set of 8 symmetry-siblings.

Thus any radius that goes through exactly 4n points where n is odd must either be an integer or an integral multiple of sqrt(2).

 Posted by Steve Herman on 2011-02-28 18:03:38

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