Diophantine Equations and Congruent Number Equation Solutions

TL;DR

This paper constructs pairs of rational right triangles sharing a side using solutions of the congruent number equation, providing parametrizations for various related Diophantine systems.

Contribution

It introduces new parametrizations for Diophantine systems based on rational solutions of the congruent number equation, linking them to Pythagorean triangles.

Findings

Pairs of rational right triangles with a common side are constructed.

Parametrizations for specific Diophantine systems are derived.

Solutions connect congruent number equations to Pythagorean triples.

Abstract

By using pairs of nontrivial rational solutions of congruent number equation $$ C_N:\;\;y^2=x^3-N^2x, $$ constructed are pairs of rational right (Pythagorean) triangles with one common side and the other sides equal to the sum and difference of the squares of the same rational numbers. The parametrizations are found for following Diophantine systems: \begin{align*} (p^2\pm q^2)^2-a^2 & =\square_{1,2}\,, \\[0.2cm] c^2-(p^2\pm q^2)^2 & =\square_{1,2}\,, \\[0.2cm] a^2+(p^2\pm q^2)^2 & =\square_{1,2}\,, \\[0.2cm] (p^2\pm q^2)^2-a^2 & =(r^2\pm s^2)^2. \end{align*}