# Will this rational parametric solution generate all possible solutions?

I was looking for integer solutions to this equation: $$a^3+b^3+c^3-3abc=d^3$$ And found a parametric solution. Given u, v, w : $$\begin{cases} a=3\left(u^2v+v^2w+w^2u\right)\\ b=3\left(uv^2+vw^2+wu^2\right)\\ c=u^3+v^3+w^3+6uvw\\ d=u^3+v^3+w^3-3uvw \end{cases}$$

A Natural Extension of the Pythagorean Equation to Higher Dimensions http://www.math.grinnell.edu/~chamberl/papers/pythagorean.pdf

• – individ Dec 5 '19 at 14:22

$$\begin{cases} a=3\left(u^2v+v^2w+w^2u\right)\\ b=3\left(uv^2+vw^2+wu^2\right)\\ c=u^3+v^3+w^3+6uvw\\ d=u^3+v^3+w^3-3uvw \end{cases}$$
$$(a,b,c,d)=(2,4,3,3)$$