Improved lower bounds for strong $n$-conjectures
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911058237587456 |
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| author | Hölzl, Rupert Kleine, Sören Stephan, Frank |
| author_facet | Hölzl, Rupert Kleine, Sören Stephan, Frank |
| contents | The well-known $abc$-conjecture concerns triples $(a,b,c)$ of non-zero integers that are coprime and satisfy ${a+b+c=0}$. The strong $n$-conjecture is a generalisation to $n$ summands where integer solutions of the equation ${a_1 + \ldots + a_n = 0}$ are considered such that the $a_i$ are pairwise coprime and satisfy a certain subsum condition. Ramaekers studied a variant of this conjecture with a slightly different set of conditions. He conjectured that in this setting the limit superior of the so-called qualities of the admissible solutions equals $1$ for any $n$.
In this article, we follow results of Konyagin and Browkin. We restrict to a smaller, and thus more demanding, set of solutions, and improve the known lower bounds on the limit superior: for ${n \geq 6}$ we achieve a lower bound of $\frac54$; for odd $n \geq 5$ we even achieve $\frac53$. In particular, Ramaekers's conjecture is false for every ${n \ge 5}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13439 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improved lower bounds for strong $n$-conjectures Hölzl, Rupert Kleine, Sören Stephan, Frank Number Theory 11D04, 11D72 The well-known $abc$-conjecture concerns triples $(a,b,c)$ of non-zero integers that are coprime and satisfy ${a+b+c=0}$. The strong $n$-conjecture is a generalisation to $n$ summands where integer solutions of the equation ${a_1 + \ldots + a_n = 0}$ are considered such that the $a_i$ are pairwise coprime and satisfy a certain subsum condition. Ramaekers studied a variant of this conjecture with a slightly different set of conditions. He conjectured that in this setting the limit superior of the so-called qualities of the admissible solutions equals $1$ for any $n$. In this article, we follow results of Konyagin and Browkin. We restrict to a smaller, and thus more demanding, set of solutions, and improve the known lower bounds on the limit superior: for ${n \geq 6}$ we achieve a lower bound of $\frac54$; for odd $n \geq 5$ we even achieve $\frac53$. In particular, Ramaekers's conjecture is false for every ${n \ge 5}$. |
| title | Improved lower bounds for strong $n$-conjectures |
| topic | Number Theory 11D04, 11D72 |
| url | https://arxiv.org/abs/2409.13439 |