Thue equation

In mathematics, a Thue equation is a Diophantine equation of the form

$${\displaystyle f(x,y)=r,}$$

where ${\displaystyle f}$ is an irreducible bivariate form of degree at least 3 over the rational numbers, and ${\displaystyle r}$ is a nonzero rational number. It is named after Axel Thue, who in 1909 proved that a Thue equation can have only finitely many solutions in integers ${\displaystyle x}$ and ${\displaystyle y}$, a result known as Thue's theorem.^[1]

The Thue equation is solvable effectively: there is an explicit bound on the solutions ${\displaystyle x}$, ${\displaystyle y}$ of the form ${\displaystyle (C_{1}r)^{C_{2}}}$ where constants ${\displaystyle C_{1}}$ and ${\displaystyle C_{2}}$ depend only on the form ${\displaystyle f}$. A stronger result holds: if ${\displaystyle K}$ is the field generated by the roots of ${\displaystyle f}$, then the equation has only finitely many solutions with ${\displaystyle x}$ and ${\displaystyle y}$ integers of ${\displaystyle K}$, and again these may be effectively determined.^[2]