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Acta Mathematica Sinica, English Series

, Volume 34, Issue 10, pp 1517–1530 | Cite as

On the Equation n1n2 = n3n4 Restricted to Factor Closed Sets

  • San Ying Shi
  • Michel Weber
Article

Abstract

We study the number of solutions N(B,F) of the diophantine equation n1n2 = n3n4, where 1 ≤ n1B, 1 ≤ n3B, n2, n4F and F ⊂ [1,B] is a factor closed set. We study more particularly the case when F = {m = \(p_1^{{\varepsilon _1}}\) · · · \(p_k^{{\varepsilon _k}}\), εj∈ {0, 1}, 1 ≤ j ≤ k}, p1,..., pk being distinct prime numbers.

Keywords

Diophantine equation arithmetical functions factor closed set 

MR(2010) Subject Classification

11D57 11D57 11A25 

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Notes

Acknowledgements

This work was carried out during the first author’s visit at Department of Mathematics, Kansas State University and supported by China Scholarship Council. We wish to thank the anonymous referee for a thorough reading of the paper and helpful remarks.

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Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsHefei University of TechnologyHefeiP. R. China
  2. 2.IRMA, UMR 7501Strasbourg CedexFrance

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