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.

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.