Hi everybody. I need to know if the cubic Thue equation $x^3 + x^2y + 3xy^2 - y^3 = \pm 1$ is completely solved. I know that there are effective algorithms to solve any cubic Thue equation and that some of them are implemented in computer programs. However, I think that since the coefficients of that equation are small, it may have already been discussed in the literature. Thank you.
For a cheaper solution, use Magma. For a free solution, use pari/gp:
(17:47) gp > thue(thueinit(x^3+x^2+3*x-1,1),1)
%2 = [[1, 0], [0, -1]]
Dec. 1, 2012: your equation has only two integer solutions:
$x=0,y=-1$ and $x=1,y=0$ if the right-hand-side equals $+1$,
or $x=-1,y=0$ and $x=0,y=1$ if the right-hand-side equals $-1$.
if you have access to
Dec. 22, 2012: the question has evolved a bit into the direction of the (monetary -- not computational) cost of the software used to solve the Thue equation; in that connection it might be of some interest to note that the algorithm implemented by