On the number of solutions of binomial Thue inequalities

Abstract

Abstract Let a , b and n be positive integers with n ⩾ 3 and consider the binomial Thue inequality | a x n − b y n | ⩽ 3 . In this paper, we extend a result of the first author and prove that, apart from finitely many explicitly given exceptions, this inequality has at most a single solution in positive integers x and y . In the proof, we combine lower bounds for linear forms in logarithms of algebraic numbers with the hypergeometric method of Thue-Siegel and an assortment of techniques from computational Diophantine approximation.