This is a VERY VERY hard problem and has not been solved until recently by Andrew Wiles.
1) n=4 can be solved by infinite descent.
2) Kummer provided a proof for certain primes (this uses extensive algebraic number theory)
3) all natural numbers: the proof relies heavily on the theory of modular elliptic curves, and it is very hard to comprehend.
2007-02-01 22:08
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