Does the above Diophantine equation have other integer solutions besides $(x,y)=(1,2)$ and $(x, y) = (0, -1)$?



Are there other integer solutions to the equation $9x^3 -1 = y^3$ besides $(x,y) =(1,2)$ and $(0, -1)$?
Theorem 6.4.30 in Cohen's *Number Theory: Volume I* asserts: For each nonzero integer $d$, there is at most one pair of integers $(X,Y)$ with $Y\ne 0$ and $X^3+dY^3=1$. Apply this with $X=-y$, $Y=x$, and $d=9$, to see that that there are no more solutions. Theorem 6.4.30 is attributed to Skolem.


