New small gaps between squarefree numbers
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866914800915709952 |
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| author | Chan, Tsz Ho |
| author_facet | Chan, Tsz Ho |
| contents | In this paper, we show that, for some constant $C > 0$, the interval $(x, x + C x^{5/26}]$ always contains a squarefree number when $x$ is sufficiently large (in terms of $C$). Our improvement comes from establishing asymptotic relations between the shifts $a$ and $b$ when $m n^2 \approx (m - a) (n + b)^2$ We apply them to study quadruples $(m + a_1) (n - b_1)^2 \approx m n^2 \approx (m - a_2)(n + b_2)^2 \approx (m - a_2 - a_3)(n + b_2 + b_3)^2$ and generalize Roth differencing and Filaseta-Trifonov differencing by allowing $b_1$ to be different from $b_3$. We also introduce a new differencing and exploit the interplay among these three differencings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_09990 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | New small gaps between squarefree numbers Chan, Tsz Ho Number Theory In this paper, we show that, for some constant $C > 0$, the interval $(x, x + C x^{5/26}]$ always contains a squarefree number when $x$ is sufficiently large (in terms of $C$). Our improvement comes from establishing asymptotic relations between the shifts $a$ and $b$ when $m n^2 \approx (m - a) (n + b)^2$ We apply them to study quadruples $(m + a_1) (n - b_1)^2 \approx m n^2 \approx (m - a_2)(n + b_2)^2 \approx (m - a_2 - a_3)(n + b_2 + b_3)^2$ and generalize Roth differencing and Filaseta-Trifonov differencing by allowing $b_1$ to be different from $b_3$. We also introduce a new differencing and exploit the interplay among these three differencings. |
| title | New small gaps between squarefree numbers |
| topic | Number Theory |
| url | https://arxiv.org/abs/2110.09990 |