Failure of an elementary 'proof' of Fermat's Last Theorem? Can someone explain to me why this does not constitute a proof of Fermat's Last Theorem, please?
Basically, using something I've read online, it appears you can write an equation for $(a, b, c)$ to find solutions for equations $a^n + b^n = c^n$ in the form of
$$a = (u^2 - v^2)^{2/n}, \: b = (2uv)^{2/n},\: c = (u^2 + v^2)^{2/n}$$
which won't produce integer solutions for $n > 2$.
Could someone explain to me the mistake in this "proof" please?
The full work is here (there's no downloading required because it's just a PDF).
Thanks in advance
 A: The argument seems to implicitly assume that $u$ and $v$ are integers.  This assumption is not justified, as explained below.  But even accepting this assumption, there is a fatal flaw in the final step of the argument.  The argument given is that $b$ cannot be an integer if $n>2$, because of the irrational factor $2^{2/n}$ appearing in the expression $b=(2uv)^{2/n}$.  However, this factor might combine with irrational parts of $u^{2/n}$ and $v^{2/n}$ to produce an integer.  For instance, take $n=3$, $u=4$, and $v=1$.  
Moreover, the argument never actually defines exactly what $u$ and $v$ are.  Instead, it defines $$k=\frac{y^{n/2}}{1+x^{n/2}}$$ and later writes $k=\frac{u}{v}$.  So in order to be able to choose $u$ and $v$ to be integers, you would need to know $k$ is rational.  From the definition of $k$, there is no reason to expect it to be rational if $n$ is odd, since its definition involves taking square roots of $x$ and $y$.  There is no problem if $n$ is even, but the issue raised in the previous paragraph is still a problem then.
