Strong $n$-conjectures over rings of integers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915391481053184 |
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| author | Hölzl, Rupert Kleine, Sören Stephan, Frank |
| author_facet | Hölzl, Rupert Kleine, Sören Stephan, Frank |
| contents | We study diophantine equations of the form ${a_1 + \ldots + a_n = 0}$ where the $a_i$'s are assumed to be coprime and to satisfy certain subsum conditions. We are interested in the limit superior of the qualities of the admissible solutions of these equations, a question that in the case ${n = 3}$ is closely related to the famous $abc$-conjecture. In a previous article, we studied multiple versions of this problem over the ring of rational integers, summarising known results and proving stronger lower bounds. In this article we extend our work to the rings of the Gaussian integers and the Hurwitz quaternions, where a somewhat different picture emerges. In particular, we establish much stronger lower bounds on qualities than for the rational integers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_05296 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strong $n$-conjectures over rings of integers Hölzl, Rupert Kleine, Sören Stephan, Frank Number Theory 11D04, 11D72, 11R11, 11R52 We study diophantine equations of the form ${a_1 + \ldots + a_n = 0}$ where the $a_i$'s are assumed to be coprime and to satisfy certain subsum conditions. We are interested in the limit superior of the qualities of the admissible solutions of these equations, a question that in the case ${n = 3}$ is closely related to the famous $abc$-conjecture. In a previous article, we studied multiple versions of this problem over the ring of rational integers, summarising known results and proving stronger lower bounds. In this article we extend our work to the rings of the Gaussian integers and the Hurwitz quaternions, where a somewhat different picture emerges. In particular, we establish much stronger lower bounds on qualities than for the rational integers. |
| title | Strong $n$-conjectures over rings of integers |
| topic | Number Theory 11D04, 11D72, 11R11, 11R52 |
| url | https://arxiv.org/abs/2503.05296 |