Failure of an elementary 'proof' of Fermat's Last Theorem?

Solution 1:

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.