Numbers with Four Close Factorizations
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Soggetti: | |
| Accesso online: | |
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| _version_ | 1866918114235514880 |
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| author | Chan, Tsz Ho Holmes, Laura Liu, Michael Villarreal, Jose |
| author_facet | Chan, Tsz Ho Holmes, Laura Liu, Michael Villarreal, Jose |
| contents | In this paper, we study numbers $n$ that can be factored in four different ways as $n = A B = (A + a_1) (B - b_1) = (A + a_2) (B - b_2) = (A + a_3) (B - b_3)$ with $B \le A$, $1 \le a_1 < a_2 < a_3 \le C$ and $1 \le b_1 < b_2 < b_3 \le C$. We obtain the optimal upper bound $A \le 0.04742 \ldots \cdot C^3 + O(C)$. The key idea is to transform the original question into generalized Pell equations $a x^2 - b y^2 = c$ and study their solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_02818 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Numbers with Four Close Factorizations Chan, Tsz Ho Holmes, Laura Liu, Michael Villarreal, Jose Number Theory In this paper, we study numbers $n$ that can be factored in four different ways as $n = A B = (A + a_1) (B - b_1) = (A + a_2) (B - b_2) = (A + a_3) (B - b_3)$ with $B \le A$, $1 \le a_1 < a_2 < a_3 \le C$ and $1 \le b_1 < b_2 < b_3 \le C$. We obtain the optimal upper bound $A \le 0.04742 \ldots \cdot C^3 + O(C)$. The key idea is to transform the original question into generalized Pell equations $a x^2 - b y^2 = c$ and study their solutions. |
| title | Numbers with Four Close Factorizations |
| topic | Number Theory |
| url | https://arxiv.org/abs/2508.02818 |