On Rational Points on the Elliptic Curve E(q) : p2 + q2 = r2(1 + p2q2)

TL;DR

This paper investigates rational points on a specific elliptic curve E(q), proving that conjugate points cannot both be rational, thus contributing to the understanding of rational solutions on elliptic curves.

Contribution

It introduces the concept of conjugate points on E(q) and establishes that both cannot be rational simultaneously, providing new insights into rational solutions.

Findings

Conjugate points on E(q) cannot both be rational.

The paper characterizes rational points on the elliptic curve E(q).

It advances understanding of rational solutions on elliptic curves.

Abstract

We look at the elliptic curve E(q), where q is a fixed rational number. A point (p,r) on E(q) is called a rational point if both p and r are rational numbers. We introduce the concept of conjugate points and show that not both can be rational points.