The theorem was first proved in 1929 by Carl Ludwig Siegel and was the first major result on Diophantine equations that depended only on the genus and not any special algebraic form of the equations.
Siegel's theorem on integral points: For a smooth algebraic curve C of genus g defined over a number field K, presented in affine space in a given coordinate system, there are only finitely many points on C with coordinates in the ring of integers O of K, provided g > 0.In 1926, Siegel proved the theorem effectively in the special case
, so that he proved this theorem conditionally, provided the Mordell's conjecture is true.
In 2002, Umberto Zannier and Pietro Corvaja gave a new proof by using a new method based on the subspace theorem.
(see effective results in number theory), since Thue's method in diophantine approximation also is ineffective in describing possible very good rational approximations to almost all algebraic numbers of degree