Pairs of square-free values of the type n 2 + 1 , n 2 + 2

Stoyan Dimitrov

Czechoslovak Mathematical Journal (2021)

  • Volume: 71, Issue: 4, page 991-1009
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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Dimitrov, Stoyan. "Pairs of square-free values of the type $n^2+1$, $n^2+2$." Czechoslovak Mathematical Journal 71.4 (2021): 991-1009. <http://eudml.org/doc/298177>.

@article{Dimitrov2021,
abstract = {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.},
author = {Dimitrov, Stoyan},
journal = {Czechoslovak Mathematical Journal},
keywords = {square-free number; asymptotic formula; Kloosterman sum},
language = {eng},
number = {4},
pages = {991-1009},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Pairs of square-free values of the type $n^2+1$, $n^2+2$},
url = {http://eudml.org/doc/298177},
volume = {71},
year = {2021},
}

TY - JOUR
AU - Dimitrov, Stoyan
TI - Pairs of square-free values of the type $n^2+1$, $n^2+2$
JO - Czechoslovak Mathematical Journal
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 71
IS - 4
SP - 991
EP - 1009
AB - 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.
LA - eng
KW - square-free number; asymptotic formula; Kloosterman sum
UR - http://eudml.org/doc/298177
ER -

References

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  3. Dimitrov, S. I., 10.17777/pjms2019.22.3.463, Proc. Jangjeon Math. Soc. 22 (2019), 463-470. (2019) Zbl1428.11163MR3994243DOI10.17777/pjms2019.22.3.463
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  9. Iwaniec, H., Kowalski, E., 10.1090/coll/053, Colloquium Publications 53. American Mathematical Society (2004). (2004) Zbl1059.11001MR2061214DOI10.1090/coll/053
  10. Reuss, T., Pairs of k -free numbers, consecutive square-full numbers, Available at https://arxiv.org/abs/1212.3150v2 (2014), 28 pages. (2014) 
  11. Tolev, D. I., 10.1112/S0024609305004753, Bull. Lond. Math. Soc. 37 (2005), 827-834. (2005) Zbl1099.11042MR2186715DOI10.1112/S0024609305004753
  12. Tolev, D. I., Lectures on Elementary and Analytic Number Theory. II, St. Kliment Ohridski University Press, Sofia (2016), Bulgarian. (2016) 

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