Area of a Triangle and Elliptic Curves - Birch and Swinnerton Dyer Conjecture

[Math Amateur]

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 if the equation:

y^2 = x^3 - d^2 x

has rational solutions for x and y with

(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 if the equation:

y^2 = x^3 - d^2 x

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


Peter

 

[morphism]

I think it's a bit disingenuous to say that this is straightforward. Sure, if someone gives you the substitutions then you can execute them with no difficulty. But coming up with these substitutions on your own is not so trivial (in my opinion).

Anyway, the key phrase here is "congruence number". See section 3 here for the details you're after.