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.

How would one come about answering this.
Any hits are appreciate it.

Answer

They’re not. No two different circles centered at the origin contain any of the same points. There are uncountably many circles (specifically, one for each real number, corresponding to the radius) , so most circles contain no rational points at all.

We can find some more specific examples. Specifically, any rational point (a,b) on a circle of radius r centered at the origin satisfies a2+b2=r2. In particular, r2 must be rational. There are also radii whose squares are rational where there are no rational points. Clearing denominators, say multiplying by some c2 to do so, we have that c2r2 is a sum of two squares. If r2 is an integer, then r2 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 3 mod 4. r2 was arbitrary, so if we choose it not to be a sum of two squares we get circles with no rational points.

Attribution
Source : Link , Question Author : Hidaw , Answer Author : Matt Samuel

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