Thue equation

Builds on Diophantine equation Irreducible polynomial

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 over $\Q$ . Let $m$ be a nonzero integer. The Thue equation for $f$ and $m$ is the Diophantine equation $f(x, y) = m$ 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 $(17, 12)$ .

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`3601Z7V838`
URI	`m2://3601Z7V838`
Kind	definition
Writing	W0 stub
MSC	11D59, 11D41
Defines	Thue equation
Uses	Diophantine equation, Irreducible polynomial
Changed	2 November 2026

Thue equation

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here