| 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 |
| . |
| Category:
|
math |
| . |
| 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 |
| . |
| Date available:
|
2023-02-03T11:15:56Z |
| Last updated:
|
2025-04-07 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151518 |
| . |
| Reference:
|
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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:
|
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| 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 |
| . |
