Is there an intuitive explanation for Fermat's Last Theorem? Does a heuristic explanation exist why $x^n+y^n=z^n \;\forall x,y,n,z\in \mathbb{Z},n>2$ doesn't have any solutions? 
I'm not asking for an elementary proof or for an explanation of Wile's proof but maybe there is some kind of intuitive reasoning why the proposition should be true. Why did Fermat and many other mathematicians think that it was true? Did they just try a lot of different values?
 A: One explanation for why $x^n+y^n=z^n$ should not hold for $xyz\neq 0$ and $n>2$ comes from the abc-conjecture, in terms of the radical of an integer. Basically it says that if $a+b=c$ and $a$ and $b$ are high powers of an integer, then $c$ cannot be a high power of an integer. For example, if $a=2^{13}$ and $b=5^{13}$, then
$$
a+b=2^{13}+5^{13}=1220711317=7\cdot 53\cdot 131\cdot 25117=c,
$$
and $c$ is far from being a high power.
A: I don't know whether the early pioneers have thought this way, but they might have.      
If you write $ x^n - z^n = w $ and look at the prime-factorization of $w$ then you can use Fermat's little theorem to deduce that the exponents of the primefactors of $w$ grow only logarithmically with $n$. So it is very unlikely, that for larger $n$ we could have $w = p^n \cdot q^n \cdot \ldots \cdot u^n $ where $p \ldots u$ are the primefactors of $w$ - and this might have been very obvious to Fermat himself and also to Euler, who had analyzed this so-called "cyclotomic expression"s on the lhs and had generalized Fermat's little theorem to his "totient"-formula.       
So ... this might have been a heuristic evidence for them pioneers ...
A: For $n=3$ and $n=4$ it can be proved by infinite descent, as done by Euler and Fermat.
Lamé proposed a similar infinite descent approach when the cyclotomic integers have unique factorization. Alas, this does not always happen.
