diophantine equations $x^3-2y^3=1$

Question

I'm not familiar with diophantine equations. At most my approaches doesn't give results. I need to solve the following equation $$x^3-2y^3=1$$ Where $x,y,z\in\mathbb{Z}$ I know $x=-1,y=-1.x=1,y=0,$ Aware of any other integer solutions. Prove

Answer 1

You have found the only solutions. These things are not so easy to prove. This one is done in Mordell's book, Diophantine Equations, in Chapter 15.