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Thue equation
In mathematics, a Thue equation is a Diophantine equation of the form
ｆ(x,y) = r,
where ｆ 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 ｆ. A stronger result holds, that if K is the field generated by the roots of ｆ then the equation has only finitely many solutions with x and y integers of K and again these may be effectively determined.[2]
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