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 to satisfying

and

**Proposition 1.2.** The only rational solutions to are

and

They then show how 1.2 implies 1.1:

Suppose is an integral solution to satisfying We rewrite the equation as

We then multiply each side by to get

Setting , we have

Note that *X,Y* are rational numbers. The only rational points of this elliptic curve are . In either case, implies either or , contradicting our assumption. So there are only trivial integral solutions to .

We can extend this argument to prove that there are no non-trivial integral solutions to , for .

Write .

We multiply each side by to get

Setting , we have

So we find that there are only trivial integral solutions to for .

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