Question: What are the integer solutions to$$x^2+7=y^3\tag1$$
Through Wolfram Alpha, there seems to be only two solutions. Namely, $$\begin{align*} & (x,y)=(1,2)\\ & (x,y)=(181,32)\end{align*}\tag2$$ So I'm wondering about how would you find those solutions. And is there a way to use some sort of transformation to make $(1)$ into a more recognizable form.
And furthermore, is there a formula to determine other type solutions similar to what you already have?
According to this link in the Wikipedia article on the Mordell curve, your solutions are the only ones:
http://tnt.math.se.tmu.ac.jp/simath/MORDELL/MORDELL-
That file claims to have all solutions for $|x^2-y^3| \le 10000 $.
I also recommend this presentation, which was found by a Google search for "mordell equation":
http://www.math.uconn.edu/~kconrad/ross2008/mordell.pdf