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'Rational Round' printed from https://nrich.maths.org/
Show that for every integer $k$ the point $(x, y)$, where
$$x = {2k\over k^2 + 1}, \ y = {k^2 - 1\over k^2 + 1},$$
lies on the unit circle, $x^2 + y^2 =1$. That is, there are
infinitely many rational points on this circle.
Show that there are no rational points on the circle $x^2 + y^2
=3$.