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Suppose $(x,y)\in\mathbb{Z}^2$ were a solution to $y^2=x^3-6$

Prove that there exists a principal ideal $\mathfrak{a}$ of $\mathbb{Z}[\sqrt{-6}]$ such that $(y+\sqrt{-6})=\mathfrak{a}^3$.

I have no idea what to do for this part of the proof.

pilgrim
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  • Can someone please fix the misspelling in the heading? There's got to be a way to do that without triggering the reopen question process. – Lisa Mar 16 '17 at 21:17

1 Answers1

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This is Theorem $2.3$ of this very nice article by Keith Conrad. The proof is quite elementary and starts by rewriting $y^2=x^3-6$ by $$ y^2 - 2 = x^3- 8 = (x- 2)(x^2 + 2x + 4). $$

Dietrich Burde
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