Article
| Title: | Consecutive square-free values of the type $x^{2}+y^{2}+z^{2}+k$, $x^{2}+y^{2}+z^{2}+k+1$ (English) |
| Author: | Feng, Ya-Fang |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 73 |
| Issue: | 1 |
| Year: | 2023 |
| Pages: | 297-310 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We show that for any given integer $k$ there exist infinitely many consecutive square-free numbers of the type $x^{2}+y^{2}+z^{2}+k$, $x^{2}+y^{2}+z^{2}+k+1$. We also establish an asymptotic formula for $1\leq x, y, z \leq H$ such that $x^{2}+y^{2}+z^{2}+k$, $x^{2}+y^{2}+z^{2}+k+1$ are square-free. The method we used in this paper is due to Tolev. (English) |
| Keyword: | square-free number |
| Keyword: | Salié sum |
| Keyword: | Gauss sum |
| MSC: | 11L05 |
| MSC: | 11L40 |
| MSC: | 11N37 |
| idZBL: | Zbl 07655769 |
| idMR: | MR4541103 |
| DOI: | 10.21136/CMJ.2022.0154-22 |
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| Date available: | 2023-02-03T11:15:56Z |
| Last updated: | 2023-09-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151518 |
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| Reference: | [2] Dimitrov, S.: On the number of pairs of positive integers $x,y\leq H$ such that $x^2+y^2+1$, $x^2+y^2+2$ are square-free.Acta Arith. 194 (2020), 281-294. Zbl 1469.11263, MR 4096105, 10.4064/aa190118-25-7 |
| Reference: | [3] Dimitrov, S.: Pairs of square-free values of the type $n^2+1$, $n^2+2$.Czech. Math. J. 71 (2021), 991-1009. Zbl 07442468, MR 4339105, 10.21136/CMJ.2021.0165-20 |
| Reference: | [4] Estermann, T.: A new application of the Hardy-Littlewood-Kloosterman method.Proc. Lond. Math. Soc., III. Ser. 12 (1962), 425-444. Zbl 0105.03606, MR 0137677, 10.1112/plms/s3-12.1.425 |
| Reference: | [5] Heath-Brown, D. R.: The square sieve and consecutive square-free numbers.Math. Ann. 266 (1984), 251-259. Zbl 0514.10038, MR 0730168, 10.1007/BF01475576 |
| Reference: | [6] Louvel, B.: The first moment of Salié sums.Monatsh. Math. 168 (2012), 523-543. Zbl 1314.11050, MR 2993962, 10.1007/s00605-011-0366-5 |
| Reference: | [7] Reuss, T.: Pairs of $k$-free numbers, consecutive square-full numbers.Available at https://arxiv.org/abs/1212.3150v2 (2012), 28 pages. |
| Reference: | [8] Tolev, D. I.: On the number of pairs of positive integers $x,y\leq H$ such that $x^2+y^2+1$ is squarefree.Monatsh. Math. 165 (2012), 557-567. Zbl 1297.11118, MR 2891268, 10.1007/s00605-010-0246-4 |
| Reference: | [9] Zhou, G.-L., Ding, Y.: On the square-free values of the polynomial $x^2+y^2+z^2+k$.J. Number Theory 236 (2022), 308-322. Zbl 07493027, MR 4395352, 10.1016/j.jnt.2021.07.022 |
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