Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
square-free number; asymptotic formula; Kloosterman sum
square-free number; asymptotic formula; Kloosterman sum
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.
References:
[1] Carlitz, L.: On a problem in additive arithmetic. II. Q. J. Math., Oxf. Ser. 3 (1932), 273-290. DOI 10.1093/qmath/os-3.1.273 | Zbl 0006.10401
[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. DOI 10.17654/NT040060945 | Zbl 1417.11145
[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. DOI 10.17777/pjms2019.22.3.463 | MR 3994243 | Zbl 1428.11163
[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
[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. DOI 10.4064/aa190118-25-7 | MR 4096105 | Zbl 07221818
[6] Estermann, T.: Einige Sätze über quadratfreie Zahlen. Math. Ann. 105 (1931), 653-662 German. DOI 10.1007/BF01455836 | MR 1512732 | Zbl 0003.15001
[7] Heath-Brown, D. R.: The square sieve and consecutive square-free numbers. Math. Ann. 266 (1984), 251-259. DOI 10.1007/BF01475576 | MR 0730168 | Zbl 0514.10038
[8] Heath-Brown, D. R.: Square-free values of $n^2+1$. Acta Arith. 155 (2012), 1-13. DOI 10.4064/aa155-1-1 | MR 2982423 | Zbl 1312.11077
[9] Iwaniec, H., Kowalski, E.: Analytic Number Theory. Colloquium Publications 53. American Mathematical Society (2004). DOI 10.1090/coll/053 | MR 2061214 | Zbl 1059.11001
[10] Reuss, T.: Pairs of $k$-free numbers, consecutive square-full numbers. Available at https://arxiv.org/abs/1212.3150v2 (2014), 28 pages.
[11] Tolev, D. I.: On the exponential sum with square-free numbers. Bull. Lond. Math. Soc. 37 (2005), 827-834. DOI 10.1112/S0024609305004753 | MR 2186715 | Zbl 1099.11042
[12] Tolev, D. I.: Lectures on Elementary and Analytic Number Theory. II. St. Kliment Ohridski University Press, Sofia (2016), Bulgarian.
Search
Partner of
