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arXiv · 1603.03340

Thue's inequalities and the hypergeometric method

Shabnam Akhtari·N. Saradha·Divyum Sharma
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Abstract

Following a method originally due to Siegel, we establish upper bounds for the number of primitive integer solutions to inequalities of the shape $0<|F(x, y)| \leq h$, 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 $\alpha\delta-\beta\gamma \neq 0$, and $r \geq 3$ and $h$ are integers. As an important application, we pay special attention to the binomial Thue's inequaities $|ax^r - by^r| \leq c$. The proofs are based on the hypergeometric method of Thue and Siegel and its refinement by Evertse.

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Shabnam Akhtari, N. Saradha, Divyum Sharma. 2016-03-10. Thue's inequalities and the hypergeometric method. https://arxiv.org/abs/1603.03340

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