Nathaniel Dean [6] asks the following: is it possible to find four non-concyclic 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 (3 copies of) an elliptic surface. In doing so, we provide a detailed description of the correspondence, the group action and the

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... roup 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 non-concyclic (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. A Brief History of the Problem. We say that a collection of points in S ⊂ R n are 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 n , 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 2 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 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 1991 Mathematics Subject Classification. 14G05, 11G05, 11D25.