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I have seen a proof for FLT, $n=3$ using factorisation in the ring of Eisenstein integers, but it's quite long and convoluted; I am wondering if there is a more 'advanced' proof which avoids infinite descent/messy repeated calculations.

What's your favorite proof of Fermat's last theorem for $n=3$, and where can it be found written down?

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    I don't think I've ever seen a proof that didn't use factorization in the Eisenstein integers. Mess is in the eyes of the beholder.2012-11-15
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    Suppose $a^3 + b^3 = c^3$ where $a, b, c$ are nonzero integers. Let $x = 4bc/a^3$ and $y = 4(a^3+2b^3)/a^3$. Then $y^2 = x^3 + 16$, which is the equation for an elliptic curve. If you could show the only rational points on this curve (besides the point at infinity) are $(0,\pm 4)$, then we get a contradiction since from our example $x \not= 0$. That would settle FLT for $n = 3$.2012-11-18

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