# Are rational points dense on every circle in the coordinate plane?

Are rational points dense on every circle in the coordinate plane?

First thing first I know that rational points are dense on the unit circle.
However, I am not so sure how to show that rational points are not dense on every circle.

We can find some more specific examples. Specifically, any rational point $$(a,b)(a, b)$$ on a circle of radius $$rr$$ centered at the origin satisfies $$a2+b2=r2a^2+b^2=r^2$$. In particular, $$r2r^2$$ must be rational. There are also radii whose squares are rational where there are no rational points. Clearing denominators, say multiplying by some $$c2c^2$$ to do so, we have that $$c2r2c^2r^2$$ is a sum of two squares. If $$r2r^2$$ is an integer, then $$r2r^2$$ must be a sum of two squares, since an integer is a sum of two squares if and only if its prime factorization doesn’t contain an odd power of a prime congruent to $$33$$ mod $$44$$. $$r2r^2$$ was arbitrary, so if we choose it not to be a sum of two squares we get circles with no rational points.