Thue equation

A Thue equation is a Diophantine equation of the form f(x, y) = m, where f is an irreducible homogeneous polynomial of degree at least three and m is a fixed nonzero integer. In 1909, Axel Thue proved that such equations have only finitely many integer solutions — a result that launched the field of Diophantine approximation and transformed the study of higher-degree Diophantine problems.

Thue's method was radical. Before his work, the standard approach to proving finiteness was to find an explicit bound on the size of solutions and then check all possibilities. Thue showed that one could prove finiteness without producing any bound at all, using an approximation theorem for algebraic numbers that was itself non-constructive. The result was met with both admiration and frustration: mathematicians could now know that solutions were finite in number without having any procedure to find them.

The techniques Thue introduced — particularly the use of auxiliary polynomials and the approximation of algebraic numbers by rationals — became the template for a century of work by Siegel, Roth, and Baker. Thue equations are now understood as the simplest case where Diophantine methods transcend brute-force enumeration and enter the realm of approximation theory.