Points on $y=x^2$ at rational distance

Abstract:

Nathaniel Dean asks the following: Is it possible to find four nonconcyclic points on the parabola $y=x^2$ 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 surface. In doing so, we provide a detailed description of the correspondence, the group action and the group structure of the elliptic curves making up the (good) fibers of the surface. We find for example that each elliptic curve must contain a point of order 4. The main result is that there are infinitely many rational distance sets of four nonconcyclic (rational) points on $y=x^2$. We begin by giving a brief history of the problem and by placing the problem in the context of a more general, long-standing open problem. We conclude by giving several examples of solutions to the problem and by offering some suggestions for further work.