I have the equation: $$y^2=x^3-2$$ It seems to be deceivingly simple, yet I simply cannot crack it. It is obviously equivalent to finding a perfect cube that is two more than a perfect square, and a brute force check shows no solutions other than $y=5$ and $x=3$ under 10,000. However, I can't prove it.
Are there other integer solutions to this equation? If so, how many? If not, can you prove that there aren't?
Bonus: What about the more general equation" $$y^2=x^3-c$$ Where $c$ is a positive integer?