This problem is working on my nerves. I`m trying to find all integer solutions to the equation [itex]x^2+4=y^3[/itex] using the PID of Gaussian integers Z. My thoughts. By inspection (2,2) is a solution. Suppose (x,y) is a solution. I write the equation as [itex](x+2i)(x-2i)=y^3[/itex]. I now look at the ideal (x+2i,x-2i)=(d) with d a generator. d divides x+2i and x-2i, so it also divides the difference 4i. What I want is to find conditions under which x+2i and x-2i are coprime in Z. Then I can show that (under the conditions) x+2i has to be a third power in Z and that no solutions exist (under this condition). Any help is appreciated.