From: jr@redmink.demon.co.uk (John R Ramsden)
Subject: Re: x^3 + y^2 = z^2
Date: Tue, 31 Aug 1999 00:49:07 GMT
Newsgroups: sci.math

On Sat, 28 Aug 1999 02:21:20 GMT, kshaban@my-deja.com wrote:

>It is well known that one can generate all primitive Pythagorean
>triples (x^2 + y^2 = z^2) by using the following equation:
>
>	x=2ab, y=a^2-b^2, z=a^2+b^2, where (a,b)=1.
>
> My question is whether there are any known identities for generating
> the equation x^3 + y^2 = z^2.

A general rational solution, in terms of a parameter u, is:

             (u^2 - x).x   (u^2 + x).x
   y,  z  =  -----------,  -----------
                 2.u           2.u

From  this you can get every integer solution by replacing x, y, z
by v^2.x, v^3.y, v^3.z resp for a suitably chosen integer v.


Cheers

---

John R Ramsden             #   "No one who has not shared a submarine
                           #    with a camel can claim to have plumbed
 (jr@redmink.demon.co.uk)  #    the depths of human misery."
                           #
                           #      Ritter von Haske
                           #        "Adventures of a U-boat Commander".