Diophantine equation $x^2 + y^2 = z^3$ I have found all solutions to the Diophantine equation $x^2 + y^2 = z^3$ when 
$z$ is odd. I am having some difficulty finding the solutions when $z$ is even. I am asking for a proof that provides the solutions where $z$ is even. I want the proof to be elementary and use only Number theory and perhaps Calculus or basic ideas about groups and rings.
Unfortunately, there isn't (apparently) one complete polynomial parameterization to

$$x^2+y^2 = z^k\tag1$$

when $k&gt;2$. For $k=2$, the complete solution is,

$$x,\,y,\,z = (a^2-b^2)s,\; (2ab)s,\; (a^2+b^2)s$$

where $s$ is a scaling factor. Using complex numbers $a+b i$, one can generalize the method. For $k=3$, it is

$$x,\,y,\,z = (a^3 - 3a b^2)s^3,\; (3a^2 b - b^3)s^3,\; (a^2+b^2)s^2\tag2$$

but you can no longer find rational $a,b,s$ for certain solutions. For example,

$\hskip2.7in$ $9^2+46^2 = 13^3\quad$ Yes 

$\hskip2.7in$ $58^2+145^2=29^3\quad$ No 

(You can click on the Yes/No links for Walpha output.) A related discussion can be found in this post while an alternative method is described here. For the case $k=3$, if $a^2+b^2=c^3$, then an infinite more can be found as,

$$(a u^3 + 3 b u^2 v - 3 a u v^2 - b v^3)^2 + (b u^3 - 3 a u^2 v - 3 b u v^2 + 
        a v^3)^2 = c^3(u^2+v^2)^3\tag3$$

which should provide some solutions not covered by $(2)$.
Usually a view.  http://www.artofproblemsolving.com/community/c3046h1124891_almost_pythagoras
http://www.artofproblemsolving.com/community/c3046h1049554__

You can write the solution in this form.  http://www.artofproblemsolving.com/community/c3046h1054060_cubes_with_squares

http://www.artofproblemsolving.com/community/c3046h1048815___
http://www.artofproblemsolving.com/community/c3046h1047876____

But usually use the standard simple approach. In the equation.

$$X^2+Y^2=Z^3$$

$$X=ab+cd$$

$$Y=cb-ad$$

And receive such record.

$$(a^2+c^2)(b^2+d^2)=Z*Z^2$$

$$b^2+d^2=Z^2$$

$$Z=a^2+c^2$$

So.

$$d=a^2-c^2$$

$$b=2ac$$

Then the decision on the record.

$$X=3ca^2-c^3$$

$$Y=3ac^2-a^3$$

$$Z=a^2+c^2$$