Pairwise Rational Points on a Parabola

Abstract

This paper presents a solution to the following open problem in Number Theory and Geometry: How many points can you find on the (half) parabola $y=x^2$, $x>0$, so that the distance between any pair of them is rational? This problem sounds like a geometry problem, but it is likely to require techniques in number theory. That's because, to determine if the distance between (a,b) and (s,t) is rational, you need to test whether $(a-s)^2$ + $(b-t)^2$ is the square of a rational number...and such questions typically fall into the realm of number theory. Of course, since this is an open problem, no-one can claim to know just what field the problem lies in, since no-one knows the solution. I believe it is not even known if there are more than 5 points on the parabola which satisfy the condition. It is certainly not known if there is an infinite family with pairwise rational distances. We can quickly see that there are three such points in the following way: Pick two rational numbers, say 1/5 and 3/5, and let those be the distances between the pairs of points AB and BC. If you fix those distances and slide the points up the parabola, the distance AC will gradually increase, bounded above by 4/5. Since the rational numbers are dense in the reals, there will be many placements of the points so that AC is a rational distance. It is not too hard to show that if you have N points on the parabola with rational distances between them, then you can find N points on the parabola with integer distances between them. This problem is publicly posted on the DIMACS mathematical research website in order to search for solutions. This paper demonstrates that an infinite number of families each containing an arbitrarily large number of such pairwise rational points on the parabola can be found. A constructive algorithm for doing so is outlined in the paper.