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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.23450 |
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| _version_ | 1866918468292444160 |
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| author | Das, Shamik Mondal, Sudipa |
| author_facet | Das, Shamik Mondal, Sudipa |
| contents | A positive square-free integer is called a \textit{congruent number} if it arises as the area of a right triangle with rational side lengths. Let $ n = p_1p_2 \cdots p_t q $ be a square-free integer, where each $ p_i \equiv 1 \pmod{8} $ and $ q \equiv 3 \pmod{8} $, with the $ p_i $ and $ q $ being distinct primes. In this article, we present a congruence relation modulo powers of 2 between the 2-part of the class numbers of $ \mathbb{Q}(\sqrt{-n}) $ and $ \mathbb{Q}(\sqrt{-p_1p_2 \cdots p_t}) $, under the assumption that $ n $ is a congruent number, using a modified Rédei matrix. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23450 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A necessary condition for a congruent number of the form $8k+3$ Das, Shamik Mondal, Sudipa Number Theory A positive square-free integer is called a \textit{congruent number} if it arises as the area of a right triangle with rational side lengths. Let $ n = p_1p_2 \cdots p_t q $ be a square-free integer, where each $ p_i \equiv 1 \pmod{8} $ and $ q \equiv 3 \pmod{8} $, with the $ p_i $ and $ q $ being distinct primes. In this article, we present a congruence relation modulo powers of 2 between the 2-part of the class numbers of $ \mathbb{Q}(\sqrt{-n}) $ and $ \mathbb{Q}(\sqrt{-p_1p_2 \cdots p_t}) $, under the assumption that $ n $ is a congruent number, using a modified Rédei matrix. |
| title | A necessary condition for a congruent number of the form $8k+3$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2604.23450 |