How to generate primitive solutions to the equation $a^3 + b^3 = c^2$

Question

The solution for this is that we are supposed to pick numbers x and y, then we can substitute them in the equation and obtain some z, which we then multiply the left side of the equation with to generate our a and b. After that we check if the solution is primitive by checking if they share the same square.

Are there any characteristics of x and y which would make (a, b, c) a primitive solution? or is it purely a matter of trial and error when it comes to solving it that way?

EDIT: By 'primitive' it means we cannot express a,b,c in this form:

$a = n^2 x$

$b = n^2 y$

$c = n^3 z$

Answer 1

Here's a way to generate lots of primitive solutions, though not all of them.

$A = 4r^4 - 4p^3r$

$B = 8pr^3 + p^4$

$C = 20p^3r^3 - p^6 + 8r^6$

Then $A^3 + B^3 = C^2$