Rational distance sets on a parabola using Pythagorean triplets

TL;DR

This paper establishes a new correspondence between rational distance sets on a parabola and Pythagorean triplets, providing conditions for their existence, an algorithm for construction, and new examples for N=4 and 5.

Contribution

It introduces a novel link between rational distance sets on a parabola and Pythagorean triplets, extending analysis to arbitrary N and enabling efficient construction.

Findings

Established necessary and sufficient conditions for RDS(N) existence.

Developed an algorithm to construct new RDS(N) examples.

Reproduced density results for solutions with N=2 and 3.

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 $3$.