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Fermat's last theorem states that $X^n+Y^n=Z^n$ when $n$ is greater than or equal to $3$ has no solutions in integers other than the obvious ones. How do you go about proving this theorem? I realize it took the entirety of a guy's life to fully prove this, but why was it so complex?

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closed as too broad by Will Jagy, Potato, N. S., Dominic Michaelis, iostream007 Nov 16 '13 at 7:50

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs.If this question can be reworded to fit the rules in the help center, please edit the question.

Wikipedia has a decent summary. –  user61527 Nov 16 '13 at 4:09
I might add: It took the entirety of the lives of many mathematicians to fully prove the theorem, not just Andrew Wiles! –  Mike Miller Nov 16 '13 at 4:10
You may also enjoy a nice fluffy hour long BBC documentary. –  J. W. Perry Nov 16 '13 at 4:26
A better question might be why did people consider it such an interesting question. It is completely unclear to me what fascinates people about it - to me it looks like just any other Diophantine equation. So perspectives of this sort might be interesting. –  Timotej Nov 16 '13 at 4:56
Does anyone seriously think this should be tagged as "elementary-number-theory"? –  Paramanand Singh Nov 16 '13 at 17:52

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up vote 8 down vote accepted

I will not answer the question of "how one can go about to prove this theorem", for obvious reasons. However, I can say something about the question: "why is it so difficult"? The answer is simple: for a single exponent, the problem is already fantastically difficult.

Have you ever tried showing that $x^3+y^3 =z^3$ has no nontrivial rational solutions? If you are very clever, you might discover a proof on your own.

If you are as clever as Kummer, you might be able to prove it for $x^{17} + x^{17} = z^{17}$. However, you'll need to reinvent the deepest and most beautiful number theory since quadratic reciprocity (class groups of cyclotomic fields, special values of $L$-functions...)

However, even Kummer could not do it for $x^{37} + y^{37} = z^{37}$. Unless I am mistaken, this case resisted until the general proof by Wiles. Even by itself, the non-existence of nontrivial rational solutions to $x^{37} + y^{37} = z^{37}$ is an extremely difficult problem. It is a problem that elementary number theory does not even come close to grazing. With this in mind, Wile's result is truly spectacular.

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Wiles proof uses - group theory, algebraic geometry, commutative algebra, and Galois theory.
There have been numerous attempts to prove this theorem before Wiles by simpler means as many partial proofs of exponents - $3 , 5 , 7 , 10$ etc where given .Look here.
Inspite of many attempts there was no simple way of proving it for a general case .So , Andrew Wiles opted to attempt to "count" and match elliptic curves to counted modular forms.
Thus this being very difficult to realise took the full educational career of Andrew Wiles

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