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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.13867 |
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| _version_ | 1866929761973960704 |
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| author | Youmbai, Ahmed El Amine Zargar, Arman Shamsi Voznyy, Maksym |
| author_facet | Youmbai, Ahmed El Amine Zargar, Arman Shamsi Voznyy, Maksym |
| contents | Let $S_l(M,N)$ denote a set of $\ell$ triples of positive integers having the same sum $M$ and the same product $N$. For each $2\leq\ell\leq 4$ we establish a connection between a subset of $S_l(M,N)$ with (integral) parametric elements and a family of elliptic curves. When $\ell=2$ and $3$, we use certain known subsets of $S_l(M,N)$ with parametric elements and respectively find families of elliptic curves of generic rank $\geq 5$ and $\geq 6$, while for $\ell=4$ we first obtain a subset of $S_l(M,N)$ with parametric elements, then construct a family of elliptic curves of generic rank $\geq 8$. Finally, we perform a computer search within these families to find specific curves with rank $\geq 11$ and in particular we found two curves of rank $14$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_13867 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Partitions into Triples with Equal Products and Families of Elliptic Curves Youmbai, Ahmed El Amine Zargar, Arman Shamsi Voznyy, Maksym Number Theory 14H52, 11D25 (Primary), 11D09, 05A17 (Secondary) Let $S_l(M,N)$ denote a set of $\ell$ triples of positive integers having the same sum $M$ and the same product $N$. For each $2\leq\ell\leq 4$ we establish a connection between a subset of $S_l(M,N)$ with (integral) parametric elements and a family of elliptic curves. When $\ell=2$ and $3$, we use certain known subsets of $S_l(M,N)$ with parametric elements and respectively find families of elliptic curves of generic rank $\geq 5$ and $\geq 6$, while for $\ell=4$ we first obtain a subset of $S_l(M,N)$ with parametric elements, then construct a family of elliptic curves of generic rank $\geq 8$. Finally, we perform a computer search within these families to find specific curves with rank $\geq 11$ and in particular we found two curves of rank $14$. |
| title | Partitions into Triples with Equal Products and Families of Elliptic Curves |
| topic | Number Theory 14H52, 11D25 (Primary), 11D09, 05A17 (Secondary) |
| url | https://arxiv.org/abs/2408.13867 |