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 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 = x2. 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. 1. A brief history of the problem We say that a collection of points in S ⊂ R is at rational distance if the distance between each pair of points is rational. We will call such a collection of points a rational distance set. For example, the rationals themselves form a rational distance subset of the reals. Therefore, if S is any line in R, S contains a dense set of points at rational distance. Furthermore, it was known to Euler that Proposition 1.1. Every circle contains a dense set of points at rational distance. Remark 1.2. Several proofs of this exist (see [1] for example). We follow the ideas articulated in [7]. Proof. To make the writing of the argument a bit cleaner, we identify R with the complex plane in the usual manner. Now observe that if two points in the complex plane, z and w, are at rational distance and have rational length, then since || z − 1 w || = ||z − w|| ||z|| · ||w|| , 1/z and 1/w are at rational distance as well. Now, consider a vertical line L. One can easily parameterize all points on L whose lengths and imaginary parts Received by the editor January 7, 2003 and, in revised form, February 4, 2003. 2000 Mathematics Subject Classification. Primary 14G05, 11G05, 11D25.