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 that a Thue equation can have only finitely many solutions in integers x and y, a result known as Thue's theorem,

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: 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.