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.
One solution for N=5 will have a radius of 65 units, passing through (0,65), (25,60), (60,25), (39,52), (52,39), and 15 other points located symmetrically in the second, third and fourth quadrants.
I got this solution by scaling a 3-4-5 and a 5-12-13 triangle until they both had a hypotenuse of 65.