How can I solve the parametric representation problem for x^3+y^3=u^3+v^3?

[gonzo]

Does anyone have any ideas on how to even start this problem? I am supposed to find a general solution in rational numbers for (aside from the trivial ones):

[itex]x^3+y^3=u^3+v^3[/itex]

Actually, I'm given the answer (which is really messy) and am supposed to show how to derive it. The book gives the hint to use a substitution x=X-Y, y=X+Y, u=U-V, and v=U+V and then factor the result in [itex]Q(\sqrt{-3})[/itex].

The hint was easy to enact, but led to another dead end. I figure you can start by trying to find integer solutions, but that doesn't help either. In fact, I can't even figure out how to make any progress at all. Not even a little bit.

Any clues out there at all?

 

[Gib Z]

Are the capitals related in anyway to the lowercase counter parts or is that just a coincidence?

 

[gonzo]

Are the capitals related in anyway to the lowercase counter parts or is that just a coincidence?

I don't understand this question? The capitals are a change of variable, and I gave the formula for the new variables and how they relate to the old ones in the post?

By the way, the new equation you get is:

[itex]X^3+3XY^2=U^3+3UV^2[/itex]

And factoring as the hint further suggests gives you:

[itex]X(X+Y\sqrt{-3})(X-Y\sqrt{-3})=U(U+V\sqrt{-3})(U-V\sqrt{-3})[/itex]

Which was the other dead end I hit. No idea what to do from there. In case it helps anyone get inspired, I'll write out the answer they give for x (small x), the others are similar. a,b and k are rational numbers:

[itex]x=k(1-(a-3b)(a^2+3b^2))[/itex]