Mordell's Equation Solutions

Question

I need to solve the equation $y^3 = x^2+5$, factoring we get $y^3 = (x+\sqrt{-5})(x-\sqrt{-5})$. Now considering the two ideals $(x\pm \sqrt{-5})$, I should show that they are coprime. Now I've been told that if I is prime and divides both of them, then it also divides $(2\sqrt{-5})$, ie $((x+\sqrt{-5})-(x-\sqrt{-5}))$, but why is it the case? Once i can prove this, I can easily solve the equation.

Answer 1

Keith Conrad has answered this with many examples in his article Examples of Mordell's equation; e.g., compare with Theorem $2.2$ and $2.3$, which is, for example $y^2=x^3-5$ and $y^2=x^3-6$. Rewriting this by exchanging $x$ and $y$ we also obtain your case $y^3-5=x^2$. It has no integer solutions. The solution is elementary arguing modulo $4$.