Inspired by Marte Reece 2.0

Consider a circle with center $\left (\dfrac{1}{3};0\right )$ and radii $r=\dfrac{25}{3}$ . The equation of it: $\begin{aligned} \left (x-\dfrac{1}{3}\right )^2+y^2 & = \dfrac{25^2}{9} \\ \ \\ (3x-1)^2+(3y)^2 & = 5^4=625 \end{aligned}$

It is verifiable, that the equation has exactly five solutions: $(7;5),(7;-5),(-2;8),(-2;-8),(-8;0)$

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Note: I don't know the proof, but it is verifiable, that for $n \in \mathbb N$ there is a circle which goes through exactly $n$ of the grid points.

For $n=2k+1$ , consider a circle with center $\left (\dfrac{1}{3};0\right )$ and radii $r=\dfrac{5^k}{3}$ . This circle goes through exactly $n$ grid points.

For $n=2k$ , consider a cirlce with center $\left (\dfrac{1}{2};0\right )$ and radii $r=\dfrac{5^{\frac{k-1}{2}}}{2}$ . This circle goes through exactly $n$ of the grid points.