---
tag: 3601Z7V838
kind: definition
name: Thue equation
msc: [11D59, 11D41]
defines: 3Q6CFPQZDF
uses: [08B1RTW7VZ, 03FKX7Z5DC]
sources:
  - url: https://en.wikipedia.org/wiki/Thue_equation
---

## Definition

Let $n \ge 3$ be an integer and let
$$f(x, y) = a_n x^n + a_{n-1} x^{n-1} y + \dots + a_0 y^n$$
be a homogeneous polynomial of degree $n$ with integer coefficients $a_0, \dots, a_n$.
Assume that $f$ is [[irreducible|03FKX7Z5DC]] over $\Q$.
Let $m$ be a nonzero integer.
The *Thue equation* for $f$ and $m$ is the [[Diophantine equation|08B1RTW7VZ]] $f(x, y) = m$ in the unknowns $x, y \in \Z$.

## Remarks

Thue's theorem says that a Thue equation has only finitely many solutions $(x, y) \in \Z^2$.

Example: $x^3 - 2y^3 = 1$ is a Thue equation, since $x^3 - 2y^3$ is homogeneous of degree $3$ and irreducible over $\Q$, because $2$ is not the cube of a rational number.
It has the solutions $(1, 0)$ and $(-1, -1)$: indeed $(-1)^3 - 2(-1)^3 = 1$.

Non-example: $x^2 - 2y^2 = 1$ is not a Thue equation, because the degree is $2$.
It is a Pell equation, and it has infinitely many solutions, for example $(3, 2)$ and $(17, 12)$.