Thue equation

Summary

A Thue equation is a Diophantine equation $f(x, y) = m$ with $m$ a nonzero integer. Here $f$ is an irreducible homogeneous polynomial in two variables, with integer coefficients and degree at least $3$. Axel Thue proved in 1909 that such an equation has only finitely many integer solutions. His proof came from a sharpening of Liouville's bound on rational approximation of algebraic numbers, the line that led to the Thue–Siegel–Roth theorem. Later, Baker's theorem gave explicit bounds on the solutions.