Fermat's Last Theorem: Contemplation.

Question

So my friend and I are studying elliptic curves and Fermat's Last Theorem appeared several times in the subject matter. So the proof of Fermat's Last Theorem was settled by Andrew Wiles with the work of many prominent mathematicians which I won't list because the list is rather long! At any rate, my friend suggested an alternative simple way to "proof" the last theorem. I am very skeptical and pessimistic about he's argument and; in fact, I believe there is a fundamental error somewhere in his logic that I haven't caught. However, his line of attack seems attractive at first. Here is his argument:

It has been established, by Sophie Germain in the 1800's, that $x^{3}+y^{3} \neq z^{3}$ for integral integers $x,y,$ and $z.$

Now this, equivalently, means that $(\frac{x}{z})^{3}+(\frac{y}{z})^{3}\neq 1$. Hence, it "must" be the case that either $(\frac{x}{z})^{3}+(\frac{y}{z})^{3} < 1$ or $(\frac{x}{z})^{3}+(\frac{y}{z})^{3} > 1$.

Suppose the former. Assume that the $(\frac{x}{z})$ and $\frac{y}{z}$ are positive. Then we have that $\frac{x}{z}<1$ and $\frac{y}{z}<1$. Thusly, it can be shown inductively that

$*$ $\quad$ $(\frac{x}{z})^{n} < (\frac{x}{z})^{n-1}< \cdots <(\frac{x}{z})^{3}<(\frac{x}{z})^{2}<\frac{x}{z}<1,$ for any integer $n>0.$ Similarly, we have

$\dagger$ $\quad$ $(\frac{y}{z})^{n} < (\frac{y}{z})^{n-1}< \cdots <(\frac{y}{z})^{3}<(\frac{y}{z})^{2}<\frac{y}{z}<1,$ for any integer $n>0.$

Now, we know that for some appropriate $x,y,z \in \mathbb{Z}$, $x^{2}+y^{2}=z^{2}$ so that, in particular $(\frac{x}{z})^{2}+(\frac{y}{z})^{2}=1$.

Then if we sum up $\dagger$ and $*$, we get

$**$ $\quad \quad \quad$$(\frac{x}{z})^{n}+(\frac{y}{z})^{n} < \cdots <(\frac{x}{z})^{3}+(\frac{y}{z})^{3}<(\frac{x}{z})^{2}+(\frac{y}{z})^{2}<1,$ for any integer $n>0.$

Then he claims, here, that as $x^{3}+y^{3}=z^{3}$ has no solution, then

$(\frac{x}{z})^{3}+(\frac{y}{z})^{3}< 1$, and so by $**$ $(\frac{x}{z})^{4}+(\frac{y}{z})^{4}<1$,...,$(\frac{x}{z})^{n}+(\frac{y}{z})^{n}<1$, and therefore, $x^{n}+y^{n}\neq z^{n}$ for $n>2$.

His argument seems flawed, but I am not sure where exactly. I am sure the error is trivial but I can't seem to find it. Any help or solution to finding the error is greatly appreciated. Thanks!

Answer 1

Your friend is perfectly correct up until he's decided he proved Fermat's Last Theorem.

For any positive integers $x,y,z$, we know that $x^3+y^3 \neq z^3$. So $\left(\frac{x}{z}\right)^3 + (\frac{y}{z})^3 \neq 1$. Now if $\left(\frac{x}{z}\right)^3 + (\frac{y}{z})^3 < 1$, of course we have $\frac{x}{z} < 1$, $\frac{y}{z} < 1$, so that we have by his inductive argument $$\left(\frac{x}{z}\right)^n + \left(\frac{y}{z}\right)^n < \left(\frac{x}{z}\right)^3 + \left(\frac{y}{z}\right)^3 < 1,$$ and hence that $x^n+y^n<z^n$ for all $n>3$. So he's proven that if $x^3+y^3<z^3$, then $x^n+y^n<z^n$ for all $n>3$. But he's said nothing at all about the case where $x^3+y^3>z^3$! This case is, of course, much harder, and his methods do nothing for us here. So he has not proven FLT after all.