In the book by Keith Devlin on the Millenium Problems - in Chapter 6 on the Birch and Swinnerton-Dyer Conjecture we find the following text:

"It is a fairly straightforward piece of algebraic reasoning to show that there is a right triangle with rational sides having an area d if and only the equation:

$\displaystyle y^2 = x^3 - d^2 x $

has rational solutions for x and y with $\displaystyle y \ne 0 $"

(Note: Devlin has defined d as a positive whole number earlier on)

Can someone please supply the straightforward algebra to show that:

There is a right triangle with rational sides having an area d if and only the equation:

$\displaystyle y^2 = x^3 - d^2 x $

has rational solutions for x and y with $\displaystyle y \ne 0 $"

Peter