Topology of Numbers · Problem 9

The key is to use a stereographic projection from a known rational point on the hyperbola. If you pick a rational point (like $(0, 1)$) and draw lines with rational slope through it, each line intersects the hyperbola at exactly one other point—and that intersection will have rational coordinates. This parametrization trick converts a geometric problem into an algebraic one. For part (b), notice that the $x$ and $y$ coordinates of these rational points are built from the same expressions that appear in Pythagorean triples; the hyperbola is secretly encoding the structure of integer right triangles.

The key is to use a stereographic projection from a known rational point on the hyperbola. If you pick a rational point (like $(0, 1)$) and draw lines with rational slope through it, each line intersects the hyperbola at exactly one other point—and that intersection will have rational coordinates. This parametrization trick converts a geometric problem into an algebraic one. For part (b), notice that the $x$ and $y$ coordinates of these rational points are built from the same expressions that appear in Pythagorean triples; the hyperbola is secretly encoding the structure of integer right triangles.