  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.
