Here's a question from an old examination paper:
Find all $(x,y)$ in $\mathbf{Z}^{2}$ where $y$ is odd and $y^2=x^3-4$.
Find all $(x,y)$ in $\mathbf{Z}^{2}$ with $y$ even and $y^2=x^3 -4$.
When $(x,y)$ in $\mathbf{Z}^{2}$ where $y=2Y$ is even and $y^2=x^3-4$, show that $x=2X$ with $X, Y$ odd and that $\gcd(Y+i,Y-i) = 1+i$.
An older student who has taken the exam already told us that we should look at $\mathbf{Z}[i]$ but I don't see where to go with this information. Help is greatly appreciated.