TITLE: Is there an elementary proof for Fermat's last theorem?
QUESTION [22 upvotes]: I wonder is there an elementary proof for Fermat's last theorem. Why it's so difficult to prove this theorem by elementary method? 
Thanks,

REPLY [23 votes]: Fermat's Last Theorem, although elementary to state, is a very subtle problem.
The general $ABC$ conjecture (still unproved) states (roughly) that if $C = A + B$ (with $A, B, C$ coprime integers), then it is not possible for $A$, $B$, and $C$ to be simultaneously divisible by high powers of integers. 
Fermat's equation considers the special case when $A$, $B$, and $C$ are all taken to be perfect $n$th powers.
The $ABC$ conjecture in general, and Fermat in particular, are then subtle problems relating the additive and multiplicative nature of the integers, and so
there it is perhaps not too surprising that they are difficult to prove (or, in 
the case of $ABC$, that it remains unproved!).
Another famous conjecture relating the additive and multiplicative nature of the
integers is the Goldbach conjecture.  This is, if you like, the "opposite" of $ABC$; it states that for an even integer $N$, we may write $N = p_1 + p_2,$ where $p_1$ and $p_2$ are prime (which is a kind of "opposite" to being divisible by a high perfect power).  It also resists proof.
At a technical level, the tools that have been brought to bear on Fermat, and on $ABC$, are quite different from the tools that have been brought to bear on Goldbach, so perhaps one shouldn't take a comparison between them too seriously.
But they do share the common element of getting at something quite deep about the interrelationship between the additive and mutliplicative structure of the integers, and this is what makes them difficult (or so it seems to me). 
[Added September 2012:] Shinichi Mochizuki has very recently claimed a proof of the ABC conjecture.