The Diophantine Frobenius Problem revisited
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915488538296320 |
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| author | Ding, Yuchen Wang, Weijia Zhang, Hao |
| author_facet | Ding, Yuchen Wang, Weijia Zhang, Hao |
| contents | Let $k\ge 2$ and $a_1, a_2, \cdots, a_k$ be positive integers with \[ \gcd(a_1, a_2, \cdots, a_k)=1. \] It is proved that there exists a positive integer $G_{a_1, a_2, \cdots, a_k}$ such that every integer $n$ strictly greater than it can be represented as the form \[ n=a_1x_1+a_2x_2+\cdots+a_kx_k, \quad (x_1, x_2, \cdots, x_k\in\mathbb{Z}_{\ge 0},~\gcd(x_1, x_2, \cdots, x_k)=1). \] We then investigate the size of $G_{a_1, a_2}$ explicitly. Our result strengthens the primality requirement of $x$'s in the classical Diophantine Frobenius Problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_08599 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Diophantine Frobenius Problem revisited Ding, Yuchen Wang, Weijia Zhang, Hao Number Theory Combinatorics Let $k\ge 2$ and $a_1, a_2, \cdots, a_k$ be positive integers with \[ \gcd(a_1, a_2, \cdots, a_k)=1. \] It is proved that there exists a positive integer $G_{a_1, a_2, \cdots, a_k}$ such that every integer $n$ strictly greater than it can be represented as the form \[ n=a_1x_1+a_2x_2+\cdots+a_kx_k, \quad (x_1, x_2, \cdots, x_k\in\mathbb{Z}_{\ge 0},~\gcd(x_1, x_2, \cdots, x_k)=1). \] We then investigate the size of $G_{a_1, a_2}$ explicitly. Our result strengthens the primality requirement of $x$'s in the classical Diophantine Frobenius Problem. |
| title | The Diophantine Frobenius Problem revisited |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2509.08599 |