A Theorem of Fermat on Congruent Number 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 $y^2=x^3-A^{2}x$ has a rational point $(x,y)\in {\mathbb{Q}}^2$ with $y\ne 0$. 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.

8 pages, one figure