# Thue equation

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

ƒ(x,y) = r,

where ƒ is an irreducible bivariate form of degree at least 3 over the rational numbers, and r is a nonzero rational number. It is named after Axel Thue who in 1909 proved a theorem, now called Thue's theorem, that a Thue equation has finitely many solutions in integers x and y.[1]

The Thue equation is solvable effectively: there is an explicit bound on the solutions x, y of the form ${\displaystyle (C_{1}r)^{C_{2}}}$ where constants C1 and C2 depend only on the form ƒ. A stronger result holds, that if K is the field generated by the roots of ƒ then the equation has only finitely many solutions with x and y integers of K and again these may be effectively determined.[2]

## Solving Thue equations

Solving a Thue equation can be described as an algorithm[3] ready for implementation in software. In particular, it is implemented in the following computer algebra systems: