On the number of solutions of binomial Thue inequalities

Rábai, Zs. and Bennett, M. A. and Pink, I. (2013) On the number of solutions of binomial Thue inequalities. ELECTRONIC NOTES IN DISCRETE MATHEMATICS, 43. pp. 299-304. ISSN 1571-0653

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Abstract

Let a, b and n be positive integers with n ≥ 3 and consider the binomial Thue inequality |axn − byn| ≤ 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.

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