I have been reading “Number Theory I: Fermat’s Dream” by Kazuya Kato, Nobushige Kurokawa, and Takeshi Saito. In Chapter 1: Rational Points on Elliptic Curves, the authors state two propositions:

Proposition 1.1. There is no integral solution (x,y,z) to x^4 + y^4 = z^4 satisfying xyz \neq 0.

and

Proposition 1.2. The only rational solutions to y^2=x^3-x are
(x,y)=(0,0) and (\pm1,0).

They then show how 1.2 implies 1.1:

Suppose (x,y,z) is an integral solution to x^4 + y^4 = z^4 satisfying xyz \neq 0. We rewrite the equation as

x^4 = z^4 - y^4

We then multiply each side by \dfrac{z^2}{y^6} to get

\dfrac{x^4z^2}{y^6} = \dfrac{z^6}{y^6} - \dfrac{z^2}{y^2}
\left(\dfrac{x^2z}{y^3}\right)^2=\left(\dfrac{z^2}{y^2}\right)^3-\left(\dfrac{z^2}{y^2}\right).

Setting Y=\dfrac{x^2z}{y^3},\, X=\dfrac{z^2}{y^2}, we have
Y^2=X^3-X.

Note that X,Y are rational numbers. The only rational points of this elliptic curve are (0,0), (\pm1,0). In either case, 0=Y=\dfrac{x^2z}{y^3} implies either x=0 or z=0, contradicting our assumption. So there are only trivial integral solutions (x,y,z) to x^4 + y^4 = z^4.

We can extend this argument to prove that there are no non-trivial integral solutions to x^n + y^n = z^n, for n\equiv 0\, ({\rm mod}\, 4).

Write n=4k.

x^n + y^n = z^n
x^{4k} + y^{4k} = z^{4k}
x^{4k} = z^{4k} - y^{4k}

We multiply each side by \dfrac{z^{2k}}{y^{6k}} to get
\dfrac{x^{4k}z^{2k}}{y^{6k}} = \dfrac{z^{6k}}{y^{6k}} - \dfrac{z^{2k}}{y^{2k}}
\left(\dfrac{x^{2k}z^k}{y^{3k}}\right)^2=\left(\dfrac{z^{2k}}{y^{2k}}\right)^3-\left(\dfrac{z^{2k}}{y^{2k}}\right).

Setting Y=\dfrac{x^{2k}z^k}{y^{3k}},\, X=\dfrac{z^{2k}}{y^{2k}}, we have
Y^2=X^3-X.

So we find that there are only trivial integral solutions (x,y,z) to x^n + y^n = z^n for n\equiv 0\, ({\rm mod}\, 4).

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