Thue's theorem

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Thue's theorem is a mathematical theorem first proved by Axel Thue in 1909.A. Thue, Über Annäherungswerte algebraischer Zahlen, Journal für die reine und angewandte Mathematik, 135, pages 284-305 (1909)

Statement of Thue's theorem

If f is a bivariate form with rational coefficients which is irreducible over the rational numbers and has degree greater than or equal to 3, and r is a rational number other than 0, then the equation

f(x,y)=r

has only finitely many solutions in integers x and y.

An equation of the form f(x,y)=r, where f and r are as in the theorem, is called a Thue equation. Hence the theorem can be stated concisely as: Any Thue equation has only finitely many integer solutions.

References

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