Solve the integer equation $x^3+17=y^2$
In the book The Arithmetic of Elliptic curves of Silverman, in chapter 3, example 2.4, the author sad that this equation has eight solutions $(-2,3),(-1,4),(2,5),(4,9),(8,23),(43,282),(52,375)$ and $(5234,378661)$.
But in Elementary Number Theory of J Uspensky, page 400, the authors said that, these solutions is exactly all solutions of this Mordell equation.
I have try to solve it by follow some special case such as: First, I have an expression $$(y-\sqrt{17})(y+\sqrt{17})=x^3.$$ Moreover, $Z[\dfrac{1+\sqrt{17}}{2}]$ is an euclidean domain and thus UFD. Next, we must find the GCD of $y-\sqrt{17}$ and $y+\sqrt{17}$ in $Z[\dfrac{1+\sqrt{17}}{2}]$, we call it $d$. It is obvious $d$ is a divisor of $2\sqrt{17}$. I'm stuck here. Somebody can give me some hint. Thanks!