Almost primes between all squares
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908420607574016 |
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| author | Dudek, Adrian W. Johnston, Daniel R. |
| author_facet | Dudek, Adrian W. Johnston, Daniel R. |
| contents | We prove that for all $n\geq 1$ there exists a number between $n^2$ and $(n+1)^2$ with at most 4 prime factors. This is the first result of this kind that holds for every $n\geq 1$ rather than just sufficiently large $n$. Our approach relies on a recent computation by Sorenson and Webster, along with an explicit version of the linear sieve. As part of our proof, we also prove an explicit version of Kuhn's weighted sieve. This is done for generic sifting sets to enhance the future applicability of our methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_18048 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Almost primes between all squares Dudek, Adrian W. Johnston, Daniel R. Number Theory 11N36, 11N05 We prove that for all $n\geq 1$ there exists a number between $n^2$ and $(n+1)^2$ with at most 4 prime factors. This is the first result of this kind that holds for every $n\geq 1$ rather than just sufficiently large $n$. Our approach relies on a recent computation by Sorenson and Webster, along with an explicit version of the linear sieve. As part of our proof, we also prove an explicit version of Kuhn's weighted sieve. This is done for generic sifting sets to enhance the future applicability of our methods. |
| title | Almost primes between all squares |
| topic | Number Theory 11N36, 11N05 |
| url | https://arxiv.org/abs/2501.18048 |