| Title:
|
Pairs of square-free values of the type $n^2+1$, $n^2+2$ (English) |
| Author:
|
Dimitrov, Stoyan |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
71 |
| Issue:
|
4 |
| Year:
|
2021 |
| Pages:
|
991-1009 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We show that there exist infinitely many consecutive square-free numbers of the form $n^2+1$, $n^2+2$. We also establish an asymptotic formula for the number of such square-free pairs when $n$ does not exceed given sufficiently large positive number. (English) |
| Keyword:
|
square-free number |
| Keyword:
|
asymptotic formula |
| Keyword:
|
Kloosterman sum |
| MSC:
|
11L05 |
| MSC:
|
11N25 |
| MSC:
|
11N37 |
| idZBL:
|
Zbl 07442468 |
| idMR:
|
MR4339105 |
| DOI:
|
10.21136/CMJ.2021.0165-20 |
| . |
| Date available:
|
2021-11-08T15:58:12Z |
| Last updated:
|
2024-01-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149232 |
| . |
| Reference:
|
[1] Carlitz, L.: On a problem in additive arithmetic. II.Q. J. Math., Oxf. Ser. 3 (1932), 273-290. Zbl 0006.10401, 10.1093/qmath/os-3.1.273 |
| Reference:
|
[2] Dimitrov, S. I.: Consecutive square-free numbers of the form $[n^c],[n^c]+1$.JP J. Algebra Number Theory Appl. 40 (2018), 945-956. Zbl 1417.11145, 10.17654/NT040060945 |
| Reference:
|
[3] Dimitrov, S. I.: On the distribution of consecutive square-free numbers of the form $[\alpha n],[\alpha n]+1$.Proc. Jangjeon Math. Soc. 22 (2019), 463-470. Zbl 1428.11163, MR 3994243, 10.17777/pjms2019.22.3.463 |
| Reference:
|
[4] Dimitrov, S. I.: Consecutive square-free values of the form $[\alpha p],[\alpha p]+1$.Proc. Jangjeon Math. Soc. 23 (2020), 519-524. MR 4169549 |
| Reference:
|
[5] Dimitrov, S. I.: 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 07221818, MR 4096105, 10.4064/aa190118-25-7 |
| Reference:
|
[6] Estermann, T.: Einige Sätze über quadratfreie Zahlen.Math. Ann. 105 (1931), 653-662 German. Zbl 0003.15001, MR 1512732, 10.1007/BF01455836 |
| Reference:
|
[7] 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:
|
[8] Heath-Brown, D. R.: Square-free values of $n^2+1$.Acta Arith. 155 (2012), 1-13. Zbl 1312.11077, MR 2982423, 10.4064/aa155-1-1 |
| Reference:
|
[9] Iwaniec, H., Kowalski, E.: Analytic Number Theory.Colloquium Publications 53. American Mathematical Society (2004). Zbl 1059.11001, MR 2061214, 10.1090/coll/053 |
| Reference:
|
[10] Reuss, T.: Pairs of $k$-free numbers, consecutive square-full numbers.Available at https://arxiv.org/abs/1212.3150v2 (2014), 28 pages. |
| Reference:
|
[11] Tolev, D. I.: On the exponential sum with square-free numbers.Bull. Lond. Math. Soc. 37 (2005), 827-834. Zbl 1099.11042, MR 2186715, 10.1112/S0024609305004753 |
| Reference:
|
[12] Tolev, D. I.: Lectures on Elementary and Analytic Number Theory. II.St. Kliment Ohridski University Press, Sofia (2016), Bulgarian. |
| . |
