论文标题
使用毕达哥拉斯三胞胎在抛物线上的有理距离设置
Rational distance sets on a parabola using Pythagorean triplets
论文作者
论文摘要
我们在Parabola $ y = x^2 $上研究$ n $ - 点有理距离集（$ \ textrm {rds}（n）$）。以前解决该问题的方法包括使用椭圆曲线和七翼链做出的努力，并以$ n \ leq 4 $的成功分析。我们通过在$ \ textrm {rds}（n）$ s和毕达哥拉斯三胞胎之间建立对应关系来扩展对任意$ n $的分析。我们的主要结果为任意$ n $的$ \ textrm {rds}（n）$ s的存在和性质提供了足够和必要的条件。我们的方法还导致了一种有效的计算算法来构建新的$ \ textrm {rds}（n）$ s，我们提供了$ \ textrm {rds}（n）$ s的多个新示例，为四个和五点。与毕达哥拉斯三胞胎的通信还有助于研究解决方案的密度，我们以$ n = 2 $和$ 3 $重现密度结果。
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 $3$.
