TITLE: Cubic Diophantine
QUESTION [5 upvotes]: I cannot seem to find a solution to the following Diophantine Equation: $x^2-y^3=2$, where $x,y \in \mathbb{Z}.$ I thought that I could maybe reduce it to a simpler equation , maybe check for the extension $\mathbb{Q}\left[\sqrt{2}\right],$ but nothing that I have tried seems to work. Perhaps it is a known equation/curve I am not aware of? I was first introduced to the problem when my friend pointed out that he could not solve it. I would appreciate your help. Thanks, in advance.

REPLY [2 votes]: This is an elliptic curve, and can be solved by a computer algebra system.
I use the following code on sage cell server, and you can also try it yourself.
E = EllipticCurve([0, 2])
print(E.integral_points())

The output:
[(-1 : 1 : 1)]

Therefore this is (up to sign) the only non-trivial solution.
An interesting fact is that this curve actually has Mordell-Weil rank $1$, meaning that it has infinitely many rational points.
Examples:
(17/4 : -71/8 : 1)
(127/441 : 13175/9261 : 1)
(66113/80656 : -36583777/22906304 : 1)

etc.
All these calculations are done with Sage.
How does Sage do that, and how can Sage be sure that the results are correct? This is a much deeper subject. A good starting point is the GTM book "The Arithmetic of Elliptic Curves" by Joe Silverman.