How to prove that the equation x^2+2=y^3 admits a unique solution in positive integer?



The equation x^2+2=y^3 admits a unique solution in positive integer
You don't need the general theory of Mordell's equation to handle this particular case. Since $\mathbb{Z}[\sqrt{-2}]$ is a Euclidean domain whose only units are $ \pm 1,$ it follows that there are integers $a$ and $b$ such that $x + \sqrt{-2} = (a+b\sqrt{-2})^{3}.$ Then $1 = 3a^{2}b -2b^{3}.$ Hence $b = \pm 1.$ The case $b = 1$ leads to $a = \pm 1$ and $y=3$. The case $b = -1$ leads to a contradiction.


