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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. – Gerry Myerson Nov 15 '12 at 5:24
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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 $n = 3$. – KCd Nov 18 '12 at 9:04

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Proof based on method of infinite descent: Method of infinite descent and proof of Fermat's last theorem for $n=3$. Canadian Journal on Computing in Mathematics, Natural Sciences, Engineering and Medicine (CMNSEM), Vol.1, No.6, Septermber 2010, pp. 181-186.

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