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Question : Are $$(x,y)=(-2,\pm1),(0,\pm3),(3,\pm6), (6,\pm15),(40,\pm253)$$ the only integer solutions of $\ y^2=x^3+9$ ?

Motivation : I've known that Euler proved that $(x,y)=(3,\pm5)$ are the only integer solutions of $y^2=x^3-2$. This got me interested in the equations $y^2=x^3+k$ where $k$ is an integer. In the $k=9$ case, I can neither find the other solutions nor prove that there is no other solution. Can anyone help?

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Just curious, how did you find those integer solutions? –  imranfat Oct 14 '13 at 15:54
@imranfat It's WolframAlpha I guess... –  user93957 Oct 14 '13 at 15:55
Well, then I would suspect if Wolfram didn't give any more integer solutions, then..... –  imranfat Oct 14 '13 at 15:56
@imranfat The OP asked for a proof, we know that $(x,y)=(-2,\pm1),(0,\pm3),(3,\pm6), (6,\pm15),(40,\pm253)$ are the only integer solutions to $y^2=x^3+9$ but the OP wants to prove that. –  user93957 Oct 14 '13 at 15:58
There can be up to 12 integer solutions more (Mazur's Theorem), including the point at infinite (=neutral point of the Poincare's group of the elliptic curve) –  DonAntonio Oct 14 '13 at 16:01

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