(This question is on page 236 of Falko Lorenz's Algebra Volume 1: Fields and Galois Theory, exercise 4.5)
Prove that $Y^2 = X^3 - 2$ has exactly one solution in the natural numbers.
Hint: use the fact that $\mathbb{Z}[\sqrt{-2}]$ is a Euclidean domain, and that in any unique factorization domain $R$, if $\alpha_1,\dots ,\alpha_n$ are pairwise rel. prime in $R$ and their product is an $m$-th power in $R$, each $\alpha_i$ is associated to an $m$-th power in $R$.
Here's what I know. Since $\mathbb{Z}[\sqrt{-2}]$ is a Euclidean domain, then it is also a unique factorization domain. As well, $X^3=Y^2+2=Y^2-\sqrt{-2}^2=(Y-\sqrt{-2})(Y+\sqrt{-2})$. Clues for the clueless?