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arXiv · 2003.03298

Certain Diophantine Tuples in Imaginary Quadratic Fields

Shubham Gupta
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Abstract

Let $K$ be an imaginary quadratic field and $ \mathcal{O}_K$ be its ring of integers. A set $\{a_1, a_2, \cdots,a_m\} \subset \mathcal{O}_K\setminus\{0\}$ is called a Diophantine $m$-tuple in $\mathcal{O}_K$ with $D(-1)$ if $a_ia_j -1 = x_{ij}^2$, where $x_{ij} \in \mathcal{O}_K$ for all $i,j$ such that $1 \leq i < j \leq m$. Here we prove the non-existence of Diophantine $m$-tuples in $\mathcal{O}_K$ with $D(-1)$ for $m > 36$.

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Shubham Gupta. 2020-03-06. Certain Diophantine Tuples in Imaginary Quadratic Fields. https://arxiv.org/abs/2003.03298

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