TITLE: Solving the equation by going into a non-UFD
QUESTION [6 upvotes]: To solve $y^2 + 2 = x^3$ you can factor $(y - \sqrt{-2})(y + \sqrt{-2}) = x^3$ and then check that they are relatively prime and by unique factorization both must be cubes then you can solve it.
What about $y^2 + 5 = x^3$ which does not have unique factoring?

REPLY [2 votes]: No factorization in rings without unique factorization is necessary.  Look at the proof of Theorem 2.2 in http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/mordelleqn1.pdf.