Diophantine tuples and Integral Ideals of $\mathbb{Q}(\sqrt{d})$
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| Format: | Preprint |
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2025
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| _version_ | 1866915484198240256 |
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| author | Chakraborty, Kalyan Gupta, Shubham Krishnamoorthy, Krishnarjun |
| author_facet | Chakraborty, Kalyan Gupta, Shubham Krishnamoorthy, Krishnarjun |
| contents | Suppose $n$ is the fundamental discriminant associated with a quadratic extension of $\mathbb{Q}$. We show that for every Diophantine $m$-tuple $ \{t_1, t_2, \ldots, t_m\} $ with the property $ D(n) $, there exists integral ideals $ \mathfrak{t}_1, \mathfrak{t}_2, \ldots, \mathfrak{t}_m $ of $ \mathbb{Q}(\sqrt{n}) $ and $c\in \{1,2\}$ such that $ t_i= c\mathcal{N}(\mathfrak{t}_i) $ for $ i=1,2, \ldots, m $. Here, $ \mathcal{N}(\cdot) $ denotes the norm map from $\mathbb{Q}(\sqrt{n})$ to $\mathbb{Q}$. Moreover, we explicitly construct the above ideals for Diophantine pairs $\{a_1, a_2\}$ whenever $\gcd(a_1, a_2) = 1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_06522 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Diophantine tuples and Integral Ideals of $\mathbb{Q}(\sqrt{d})$ Chakraborty, Kalyan Gupta, Shubham Krishnamoorthy, Krishnarjun Number Theory Combinatorics 11D09, 11R11 Suppose $n$ is the fundamental discriminant associated with a quadratic extension of $\mathbb{Q}$. We show that for every Diophantine $m$-tuple $ \{t_1, t_2, \ldots, t_m\} $ with the property $ D(n) $, there exists integral ideals $ \mathfrak{t}_1, \mathfrak{t}_2, \ldots, \mathfrak{t}_m $ of $ \mathbb{Q}(\sqrt{n}) $ and $c\in \{1,2\}$ such that $ t_i= c\mathcal{N}(\mathfrak{t}_i) $ for $ i=1,2, \ldots, m $. Here, $ \mathcal{N}(\cdot) $ denotes the norm map from $\mathbb{Q}(\sqrt{n})$ to $\mathbb{Q}$. Moreover, we explicitly construct the above ideals for Diophantine pairs $\{a_1, a_2\}$ whenever $\gcd(a_1, a_2) = 1$. |
| title | Diophantine tuples and Integral Ideals of $\mathbb{Q}(\sqrt{d})$ |
| topic | Number Theory Combinatorics 11D09, 11R11 |
| url | https://arxiv.org/abs/2509.06522 |