Josh Nichols-Barrer wrote a delightful proof of Fermat's Last Theorem (and much more) here:

https://groups.google.com/d/msg/rec.humor/wUZ9gBmMchM/V9OS_or6gIQJ

In a nutshell: if $x^n+y^n=z^n$ then by differentiating and dividing by $n$, we get $x^{n-1}+y^{n-1}=z^{n-1}$. There are no integer solutions to $x^0+y^0=z^0$, so by induction Fermat's Last Theorem holds. As corollaries, there are no Pythagorean triples, and also addition is a lie. (But this is just a summary of Josh's amusing post.)