Rational distance sets on a parabola using Pythagorean triplets

Abstract

We study $N$-point rational distance sets ($\textrm{RDS}(N)$) on the parabola $y=x^2$. Previous approaches to the problem include efforts made using elliptic curves and diophantine chains, with successful analysis for $N\leq 4$. We extend the analysis for arbitrary $N$ by establishing a correspondence between $\textrm{RDS}(N)$s and Pythagorean triplets. Our main result gives sufficient and necessary conditions for the existence and nature of the $\textrm{RDS}(N)$s for arbitrary $N$. Our approach also leads to an efficient computational algorithm to construct new $\textrm{RDS}(N)$s, and we provide multiple new examples of $\textrm{RDS}(N)$s for four and five points. The correspondence with Pythagorean triplets also helps to study the density of the solutions and we reproduce density results for $N=2$ and $3$.