Thue's inequalities and the hypergeometric method

Abstract

Following a method originally due to Siegel, we establish upper bounds for the number of primitive integer solutions to inequalities of the shape 0<|F(x,y)|≤h, where F(x,y)=(αx+βy)r−(γx+δy)r∈Z[x,y], α, β, γ and δ are algebraic constants with αδ−βγ≠0, and r≥3 and h are integers. As an important application, we pay special attention to the binomial Thue's inequaities |axr−byr|≤c. The proofs are based on the hypergeometric method of Thue and Siegel and its refinement by Evertse.