The number of primes not in a numerical semigroup
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912435262193664 |
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| author | Chen, Yong-Gao Zhu, Hui |
| author_facet | Chen, Yong-Gao Zhu, Hui |
| contents | For two coprime positive integers $a$ and $b$,let $π^* (a, b)$ be the number of primes that cannot be represented as $au+bv$, where $u$ and $v$ are nonnegative integers. It is clear that $π^* (a, b)\le π(ab-a-b)$, where $π(x)$ denotes the number of primes not exceeding $x$. In this paper, we prove that $π^* (a, b)\ge 0.04π(ab-a-b)$ and pose following conjecture: $π^* (a, b)\ge \frac 12 π(ab-a-b)$. This conjecture is confirmed for $1\le a\le 10$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_03625 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The number of primes not in a numerical semigroup Chen, Yong-Gao Zhu, Hui Number Theory 11D07, 11N13, 11Y35 For two coprime positive integers $a$ and $b$,let $π^* (a, b)$ be the number of primes that cannot be represented as $au+bv$, where $u$ and $v$ are nonnegative integers. It is clear that $π^* (a, b)\le π(ab-a-b)$, where $π(x)$ denotes the number of primes not exceeding $x$. In this paper, we prove that $π^* (a, b)\ge 0.04π(ab-a-b)$ and pose following conjecture: $π^* (a, b)\ge \frac 12 π(ab-a-b)$. This conjecture is confirmed for $1\le a\le 10$. |
| title | The number of primes not in a numerical semigroup |
| topic | Number Theory 11D07, 11N13, 11Y35 |
| url | https://arxiv.org/abs/2506.03625 |