A Theorem of Fermat on Congruent Number Curves

TL;DR

This paper provides an elementary proof, based on Fermat's theorem, that congruent number curves lack rational points of finite order, clarifying a key property related to the congruent number problem.

Contribution

It offers a new elementary proof, utilizing Fermat's theorem, demonstrating that congruent number curves do not have rational points of finite order.

Findings

Congruent number curves lack rational points of finite order.

Elementary proof based on Fermat's theorem.

Clarifies properties of rational points on these curves.

Abstract

A positive integer $A$ is called a congruent number if $A$ is the area of a right-angled triangle with three rational sides. Equivalently, $A$ is a congruent number if and only if the congruent number curve $y^2=x^3-A^2x$ has a rational point $(x,y)\in\mathbb Q^2$ with $y\neq 0$. Using a theorem of Fermat, we give an elementary proof for the fact that congruent number curves do not contain rational points of finite order.