Is it possible to find a circle with exactly 5 rational points

arithmetic-geometrygeometryirrational-numbersrational numbers

Can we find a circle in $\mathbb{R}^2$ with exactly 5 points with rational coordinates?

What is obvious is that a circle with a rational center and a rational radius has infinitely many rational points. And that a circle with a radius whose square is irrational and a rational center has no rational points.

I tried finding a circle with 1 rational point but to no avail.

Best Answer

Claim: If the circle has (at least) 3 rational points, then

  1. the center is rational (so we can translate to the origin), and
  2. the set of rational points is infinite, actually dense.

So the answer is no.

If you're stuck proving these statements, show what you've tried.

1) (If you don't have a slick argument,) You can find the coordinates of the center by taking the intersection of the perpendicular bisectors.

2) From a rational point on the circle, take a line with rational slope. Show that it intersects the circle again at a rational point.

Notes:

  • I know of a circle with exactly 0 rational points.
  • I know of a circle with exactly 2 rational points -> This is similar to (arguably identical to) 2009 Putnam B1, which is why the above approach was so familiar to me.
  • Lulu provided an example of a circle with exactly 1 rational points in the comments.
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