TITLE: When is $(-1+\sqrt[3]{2})^n$ of the form $a+b\sqrt[3]{2}$?
QUESTION [6 upvotes]: When is $(-1+\sqrt[3]{2})^n$ of the form $a+b\sqrt[3]{2}$ ($n$ being an integer) , i .e.,  when does $(-1+\sqrt[3]{2})^n$ not have a non-zero term in $\sqrt[3]{4}$. As you might have noticed, I'm interested in solving the diophantine equation $x^3-2y^3=1$ using this specific method. Is there any way I can use Skolem's p-adic method here?

REPLY [13 votes]: In section 4 of my book on the Catalan conjecture, I present such a proof (for $p=3$).  I took it from Bill McCallum's 1977 honours project at the University of Sydney