<s>[INST] Find all the solutions of diophantine eq: $x^3-2xy^2+y^3-s^2=0$ Given $x,y,s$ are natural numbers:
$$x^3-2xy^2+y^3-s^2=0$$
I found the solutions using wolfram alpha
$$(x,y,s) = (1,2,1), (6,10,4), (4,8,8)$$
But how do I prove these are the only solutions? Any tips or reference to papers that study this diophantine equation would be much appreciated.
 [/INST] There is a beautiful parameterization by Lagrange and Legendre:
$$x^3 + a x^2y + b x y^2 + c y^3 = z^2\tag1$$
where,
$$x = u^4 - 2b u^2 v^2 - 8c u v^3 + (b^2 - 4a c)v^4$$
$$y = 4v(u^3 + a u^2v + b u v^2 + c v^3)$$
and $z$ is a $6$-deg polynomial. 
In your case, with $a,b,c = 0,-2,1$, we have,
$$x^3 - 2x y^2 + y^3 = z^2$$
where,
$$x = u^4 + 4 u^2 v^2 - 8 u v^3 + 4 v^4$$
$$y = 4 v (u^3 - 2 u v^2 + v^3)$$
$$z = u(u^5 - 10 u^3 v^2 + 20 u^2 v^3 - 20 u v^4 + 8 v^5)$$
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