Infinitely Many Counter Examples of a Conjecture of Franušić and Jadrijević
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917981534027776 |
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| author | Gupta, Shubham |
| author_facet | Gupta, Shubham |
| contents | Let $d$ be a square-free integer such that $d \equiv 15 \pmod{60}$ and the Pell's equation $x^2 - dy^2 = -6$ is solvable in rational integers $x$ and $y$. In this paper, we prove that there exist infinitely many Diophantine quadruples in $\mathbb{Z}[\sqrt{d}]$ with the property $D(n)$ for certain $n$'s. As an application of it, we `unconditionally' prove the existence of infinitely many rings $\mathbb{Z}[\sqrt{d}]$ for which the conjecture of Franušić and Jadrijević (Conjecture 1.1) does `not' hold. This conjecture states a relationship between the existence of a Diophantine quadruple in $\mathcal{R}$ with the property $D(n)$ and the representability of $n$ as a difference of two squares in $\mathcal{R}$, where $\mathcal{R}$ is a commutative ring with unity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_07026 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Infinitely Many Counter Examples of a Conjecture of Franušić and Jadrijević Gupta, Shubham Number Theory 11D09, 11R11 Let $d$ be a square-free integer such that $d \equiv 15 \pmod{60}$ and the Pell's equation $x^2 - dy^2 = -6$ is solvable in rational integers $x$ and $y$. In this paper, we prove that there exist infinitely many Diophantine quadruples in $\mathbb{Z}[\sqrt{d}]$ with the property $D(n)$ for certain $n$'s. As an application of it, we `unconditionally' prove the existence of infinitely many rings $\mathbb{Z}[\sqrt{d}]$ for which the conjecture of Franušić and Jadrijević (Conjecture 1.1) does `not' hold. This conjecture states a relationship between the existence of a Diophantine quadruple in $\mathcal{R}$ with the property $D(n)$ and the representability of $n$ as a difference of two squares in $\mathcal{R}$, where $\mathcal{R}$ is a commutative ring with unity. |
| title | Infinitely Many Counter Examples of a Conjecture of Franušić and Jadrijević |
| topic | Number Theory 11D09, 11R11 |
| url | https://arxiv.org/abs/2504.07026 |