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| Format: | Preprint |
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2026
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| Accès en ligne: | https://arxiv.org/abs/2604.23482 |
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| _version_ | 1866915958900129792 |
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| author | Das, Shamik De, Debajyoti Mondal, Sudipa |
| author_facet | Das, Shamik De, Debajyoti Mondal, Sudipa |
| contents | In this article, we study necessary conditions for certain square-free integers to be congruent numbers. Our method uses divisibility properties of class numbers of related imaginary quadratic fields. We first consider positive square-free integers of the form $n = p_1 p_2 \cdots p_t q,$ where each prime $p_i \equiv 5 \pmod{8}$ and $q \equiv 7 \pmod{8}$. We show that if such an integer $n$ is a congruent number, then the class number $h(-n)$ of the quadratic field $\mathbb{Q}(\sqrt{-n})$ satisfies a specific divisibility condition. Furthermore, we provide quantitative lower bounds on the number of non-congruent numbers of this form. Next, we study integers of the form $n = p_1 p_2 \cdots p_t q,$ with $p_i \equiv 5 \pmod{8}$ and $q \equiv 3 \pmod{8}$. Assuming that $n$ is a congruent number, we obtain a congruence modulo powers of $2$ between the class numbers of the fields $\mathbb{Q}(\sqrt{-n})$ and $\mathbb{Q}\!\left(\sqrt{-p_1 p_2 \cdots p_t}\right)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23482 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $2$-Selmer groups, $2$-class groups, and congruent numbers Das, Shamik De, Debajyoti Mondal, Sudipa Number Theory In this article, we study necessary conditions for certain square-free integers to be congruent numbers. Our method uses divisibility properties of class numbers of related imaginary quadratic fields. We first consider positive square-free integers of the form $n = p_1 p_2 \cdots p_t q,$ where each prime $p_i \equiv 5 \pmod{8}$ and $q \equiv 7 \pmod{8}$. We show that if such an integer $n$ is a congruent number, then the class number $h(-n)$ of the quadratic field $\mathbb{Q}(\sqrt{-n})$ satisfies a specific divisibility condition. Furthermore, we provide quantitative lower bounds on the number of non-congruent numbers of this form. Next, we study integers of the form $n = p_1 p_2 \cdots p_t q,$ with $p_i \equiv 5 \pmod{8}$ and $q \equiv 3 \pmod{8}$. Assuming that $n$ is a congruent number, we obtain a congruence modulo powers of $2$ between the class numbers of the fields $\mathbb{Q}(\sqrt{-n})$ and $\mathbb{Q}\!\left(\sqrt{-p_1 p_2 \cdots p_t}\right)$. |
| title | $2$-Selmer groups, $2$-class groups, and congruent numbers |
| topic | Number Theory |
| url | https://arxiv.org/abs/2604.23482 |