Abstract
We establish upper bounds for the number of primitive integer solutions to inequalities of the shape \(0<|F(x, y)| \le h\), where \(F(x , y) =(\alpha x + \beta y)^r -(\gamma x + \delta y)^r \in \mathbb {Z}[x ,y]\), \(\alpha \), \(\beta \), \(\gamma \) and \(\delta \) are algebraic constants with \(\alpha \delta -\beta \gamma \ne 0\), and \(r \ge 5\) and h are integers. As an important application, we pay special attention to binomial Thue’s inequalities \(|ax^r - by^r| \le c\). The proofs are based on the hypergeometric method of Thue and Siegel and its refinement by Evertse.
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Acknowledgements
The authors are indebted to the anonymous referee for reading this manuscript carefully and providing several insightful comments which improved the content and the presentation. In particular, the referee’s remarks improved our Theorems 1.7 and 1.8, as well as Lemma 7.2 and some subsequent lemmas. The present work was started when N. Saradha visited the University of Oregon from June 15 to June 26, 2015. She would like to thank Shabnam Akhtari for the invitation. The computations in the paper were done by N. Saradha and Divyum Sharma at TIFR, Mumbai. They thank their home institution for providing computing facilities.
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Shabnam Akhtari’s research was partly supported by the National Science Foundation Grant DMS-1601837.
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Akhtari, S., Saradha, N. & Sharma, D. Thue’s inequalities and the hypergeometric method. Ramanujan J 45, 521–567 (2018). https://doi.org/10.1007/s11139-017-9887-4
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DOI: https://doi.org/10.1007/s11139-017-9887-4