Almost cubes and fourth powers in short intervals
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915216378298368 |
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| author | Chan, Tsz Ho |
| author_facet | Chan, Tsz Ho |
| contents | In this paper, we study how short an interval $[x, x + x^θ]$ contains an integer of the form $n_1 n_2 n_3$ and $m_1 m_2 m_3 m_4$ with $n_1 \approx n_2 \approx n_3$ and $m_1 \approx m_2 \approx m_3 \approx m_4$. The new idea is to adopt a second moment method (usually used for almost all results) to deduce a result for all short intervals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_21490 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Almost cubes and fourth powers in short intervals Chan, Tsz Ho Number Theory In this paper, we study how short an interval $[x, x + x^θ]$ contains an integer of the form $n_1 n_2 n_3$ and $m_1 m_2 m_3 m_4$ with $n_1 \approx n_2 \approx n_3$ and $m_1 \approx m_2 \approx m_3 \approx m_4$. The new idea is to adopt a second moment method (usually used for almost all results) to deduce a result for all short intervals. |
| title | Almost cubes and fourth powers in short intervals |
| topic | Number Theory |
| url | https://arxiv.org/abs/2503.21490 |