Solving the Mordell equation $y^2 = x^3 − 2$; what would be a general strategy? I am looking at the solution provided in my lecture notes for solving this particular Mordell equation: $$y^2 = x^3 − 2$$ which factors into: $$ (y- \sqrt {-2})(y+ \sqrt {-2}) = x^3.$$ In the lecture notes it proves that these two factors are coprime, meaning that the ideal $< y- \sqrt {-2},y+ \sqrt {-2} >$ is the unit ideal $O_K$ with $K= \mathbb{Q}(\sqrt{-2})$. The proof is shown with the part i do not understand in bold. 'First we check whether the factors on the left-hand side are coprime. **Any common factor would have to divide $2 \sqrt{−2}$ and thus be a power of the prime $ <\sqrt{−2}>$ above 2.** This would force $y$ to be even; but if $y$ is even, then $x$ is also even and hence $x^3 − 2$ is $2$ mod $4$, which is a contradiction' what would be the general approach for showing that the two factors are coprime for a Mordell equation?

If something divides $a$ and $b$, it divides $a-b$. So a common factor of $y+\sqrt{-2}$ and $y-\sqrt{-2}$ divides $(y+\sqrt{-2}) - (y-\sqrt{-2}) = 2\sqrt{-2}$.