The distribution of powers of primes related to the Frobenius problem
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915080797421568 |
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| author | Huang, Enxun Zhu, Tengyou |
| author_facet | Huang, Enxun Zhu, Tengyou |
| contents | Let $1<c<d$ be two relatively prime integers, $g_{c,d}=cd-c-d$ and $\mathbb{P}$ is the set of primes. For any given integer $k \geq 1$, we prove that $$\#\left\{p^k\le g_{c,d}:p\in \mathbb{P}, ~p^k=cx+dy,~x,y\in \mathbb{Z}_{\geqslant0} \right\}\sim \frac{k}{k+1}\frac{g^{1/k}}{\log g} \quad (\text{as}~c\rightarrow\infty),$$ which gives an extension of a recent result of Ding, Zhai and Zhao. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18898 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The distribution of powers of primes related to the Frobenius problem Huang, Enxun Zhu, Tengyou Number Theory Let $1<c<d$ be two relatively prime integers, $g_{c,d}=cd-c-d$ and $\mathbb{P}$ is the set of primes. For any given integer $k \geq 1$, we prove that $$\#\left\{p^k\le g_{c,d}:p\in \mathbb{P}, ~p^k=cx+dy,~x,y\in \mathbb{Z}_{\geqslant0} \right\}\sim \frac{k}{k+1}\frac{g^{1/k}}{\log g} \quad (\text{as}~c\rightarrow\infty),$$ which gives an extension of a recent result of Ding, Zhai and Zhao. |
| title | The distribution of powers of primes related to the Frobenius problem |
| topic | Number Theory |
| url | https://arxiv.org/abs/2412.18898 |