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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

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    B$y$ Thue-Siegel-Roth, there are only finitely many solutions. Page 154, LeVeque, Topics in Number Theory, volume 2.2012-04-19

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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.