I am trying to find circles on lattice points with exactly $n$ lattice points on its circumference. Schinzel's Theorem only gives one instance for each $n$ and it does not necessarily have the smallest radius. There are two general problem I am trying to solve:
Given $n$, find the circle with smallest radius (arbitrary center).
Given $n$ and $R$, find all circles with radius $r<R$.
The problem can be solved by properties of sum of squares function when the center is on a lattice point (origin). But I found few references when the center is on the axis $(x_0,0)$ or even off the lattice. I also know how to reduce the problem to the same as above, given a center with fraction coordinates. But I do not know the search space for all the possible centers, and this becomes the bottleneck for both problems.
Any references and hints are appreciated.