# find all integer solutions of $y^2=x^3-2$ [duplicate]

• This equation is a special case of the Mordell equation, $y^2 =x^3 +A$. A tremendous amount of work has been done on the Mordell equation, and solutions have been tabulated for large ranges of values of $A$, for example, here. – Alex Ravsky Apr 3 '15 at 5:58
• To add a bit, the obvious solutions $x=\pm 5$, $y=3$ are the only ones, and the proof I am thinking of uses the fact that $\mathbb{Z}[\sqrt{-2}]$ has unique factorization. – André Nicolas Apr 3 '15 at 5:59
• @AndréNicolas: Sorry, but how do you get the solutions $(5, 3)$ and $(-5, 3)$? They do not solve the equation $y^{2} = x^{3} - 2$. Should not it be $(3, 5)$? And $(-3, 5)$ does not work because $(-3)^{3} - 2 = -29 \neq 5^{2}$. – Benicio Apr 3 '15 at 6:02
• Sorry, I should have written $y=\pm 5$, $x=3$. – André Nicolas Apr 3 '15 at 6:05