On Thue's equation

• DOI: 10.1007/bf01405092
• Publication Date: 2005-04-03T15:59:08Z
• Authors: E. Bombieri; W. M. Schmidt
• Keywords: 510.mathematics; 0101 mathematics; 01 natural sciences; Article

and more precisely for the number of primitive solutions to (1.1), that is, solutions in coprime integers x, y. The first important results on the number of solutions of Thue's equation were obtained by C.L. Siegel, in the case r = 3 and in the binomial case F ( x , y ) = a x r b y r. In view of his results and comments, we should attribute to Siegel the problem to decide whether the number of primitive solutions to (1.1), for irreducible F of degree r > 3, could be bounded by a function depending only on r and h, but otherwise independent of F. In 1983, Siegel's question was answered in the affirmative by J.-H. Evertse [2]. As a special case of his results (he also treats equations in number fields), Evertse obtains the bound