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I've been reading about Weierstrass equations and shifted Weierstrass equations and Mordell curve and elliptic curves, but so far I haven't been able to transform my equation to any of this type. Could you give a direction to where I should be looking? Thanks.

user75619
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  • You want all solutions? Because taking $z$ to be anything, with $x=5+8z$ and $y=\pm(1+z)$ are apparent solutions. – 2'5 9'2 Jul 18 '14 at 20:50
  • Yeah, all solutions. As I understand, although I'm new to this, given an elliptic curve, you can iterate through an infinite set of solutions. I want something like that. Or as close as possible. – user75619 Jul 18 '14 at 20:52
  • It seems elliptic curve is not the best example... Good approximation of what I want would be the general solution to $X^2 + Y^2 = Z^2$. – user75619 Jul 18 '14 at 21:07
  • That equation is special, because it is quadratic. When you work this so that you only have one variable in play, you are in the special situation where knowing one root is rational automatically implies the other is rational. With an equation that is cubic in nature, I'm not optimistic the technique will still be helpful. – 2'5 9'2 Jul 18 '14 at 21:28
  • It sounds like you want a rational parameterization of your curve. – Semiclassical Jul 18 '14 at 21:28
  • Yeah, sounds like that. By setting $t = 1 + z$, it can be transformed into $xy^2 = t^2 (8t - 3)$, which is a little bit more good looking. I understand that my situation is not as simple as quadratic equation, but then Euler managed to parameterize $A^3 + B^3 = C^2$. I'm trying to find his parameterization now, may be it could give me a hint. – user75619 Jul 18 '14 at 21:33
  • why this particular diophantine equation? – Will Jagy Jul 18 '14 at 21:44
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    It's from one of those coding challenge web-sites - TopCoder, Project Euler, CodeChef, etc. I reduced the problem to this equation. Now I need to find solutions for different values of $z$. The problem is that factorizing the right side for each value of $z$ and then iterating through possible values of $x$ and $y$ on the left side takes too much time. I'm trying to find an alternative solution, so I thought may be it could parameterized. – user75619 Jul 18 '14 at 21:51
  • Finding all solutions is equivalent to finding the prime factorisation of $5+8z$, for all $z$. That's a ridiculous task. – Bart Michels Jul 27 '14 at 09:18

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