In this post a proof of the following theorem is going to be sketched, following the treatment in Borevich and Shafarevich’s *Number Theory*. This sketch is by no means meant to be highly detailed and I am writing it mostly for my own purposes, so I avoid proving some things, even if they aren’t that straightforward.

**Thue’s theorem:** Suppose \(f(x,y)=a_0x^n+a_1x^{n-1}y+\dots+a_ny^n\) is a binary form which has degree \(n\geq 3\), is irreducible (i.e. \(f(x,1)\) is an irreducible polynomial in \(x\)) and \(f(x,1)\) has at least one nonreal root in \(\mathbb C\). Then for any nonzero integer \(c\) the equation \(f(x,y)=c\) has only finitely many integral solutions.

**Proof:** Suppose otherwise…