POINTS ON HYPERBOLAS AT RATIONAL DISTANCE

Points on y = x2 at rational distance

Nathaniel Dean asks the following: Is it possible to find four nonconcyclic points on the parabola y = x2 such that each of the six distances between pairs of points is rational? We demonstrate that there is a correspondence between all rational points satisfying this condition and orbits under a particular group action of rational points on a fiber product of (three copies of) an elliptic surf...

POINTS ON y = x AT RATIONAL DISTANCE

Nathaniel Dean asks the following: Is it possible to find four nonconcyclic points on the parabola y = x2 such that each of the six distances between pairs of points is rational? We demonstrate that there is a correspondence between all rational points satisfying this condition and orbits under a particular group action of rational points on a fiber product of (three copies of) an elliptic surf...

Points at Rational Distance from the Vertices of a Unit Polygon

Geometric Properties of Points on Modular Hyperbolas

Given an integer n 2, let H n be the set H n = {(a, b) : ab ≡ 1 (mod n), 1 a, b n − 1} and let M (n) be the maximal difference of b − a for (a, b) ∈ H n. We prove that for almost all n, n − M (n) = O n 1/2+o(1). We also improve some previously known upper and lower bounds on the number of vertices of the convex closure of H n .

points at rational distance from the vertices of a unit polygon