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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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 n = 4 + full solution (spoiler) | Comment 8 of 14 |
The smallest radius I see for n = 4 is the sqrt(65), which goes through (1,8),(8,1),(4,7), and (7,4).

Putting it all together, my answer (which might be wrong) is:

n  r         r^2
-- --        -----
1  1            1
2  sqrt(5)    5
3  5           25
4  sqrt(65) 65
5  25        625

I am struck by all the 5's.  Probably because of 3-4-5 and 5-12-13 triangles.  Note that 65 = 5*13.

 Posted by Steve Herman on 2011-03-01 03:26:31

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