I am asked to find constants $A$ and $B$ such that any solution of: $x^3+2y^3=m$ satisfies $\max\{|x|,|y|\} \leq A|m|^B$.
I am told to use the fact that: $ \left| \frac{p}{q} - \sqrt[3]{2} \right| \geq \frac{1}{4|q|^{2.45}} $ for all $(p,q) \in \mathbb{Z}\times \mathbb{Z} \setminus \{0\}$.
However, I don't know where to begin. Clearly, we have that $ \max\{|x|,|y|\} \leq A|m|^B = A|x^3+2y^3|^B \leq A(|x|^3+2|y|^3)^B$
But I have no idea how to continue.
I need to bound $\max \{|x|,|y|\}$ by some exponential function, and feel that I need to use that fact that the convergents $p_n/q_n$ of $\sqrt[3]{2}$ provide solutions to the equation above.
I have tried setting $q = y$ or $q=m$ but didn't see anything helpful from the algebra. Any advice for general strategy in approaching this problem, would be appreciated.