Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The cardinality of some symmetric differences


Authors: Po-Yi Huang, Wen-Fong Ke and Günter F. Pilz
Journal: Proc. Amer. Math. Soc. 138 (2010), 787-797
MSC (2010): Primary 05A05; Secondary 11N05, 94B05
Published electronically: October 23, 2009
MathSciNet review: 2566544
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we prove that for positive integers $ k$ and $ n$, the cardinality of the symmetric differences of $ \{1,2,\dots,k\}$, $ \{2,4,\dots,2k\}$, $ \{3,6,\dots,3k\}$, ..., $ \{n,2n,\dots,kn\}$ is at least $ k$ or $ n,$ whichever is larger. This solved a problem raised by Pilz in which binary composition codes were studied.


References [Enhancements On Off] (What's this?)

  • 1. The GAP Group, GAP--Groups, Algorithms, and Programming, Version 4.4.12, 2008, http://www.gap-system.org.
  • 2. M. Nair, On Chebyshev-type inequalities for primes, Amer. Math. Monthly 89 (1982), no. 2, 126–129. MR 643279, 10.2307/2320934
  • 3. Günter Pilz, On polynomial near-ring codes, Contributions to general algebra, 8 (Linz, 1991) Hölder-Pichler-Tempsky, Vienna, 1992, pp. 233–238. MR 1281844
  • 4. Guy Robin, Estimation de la fonction de Tchebychef 𝜃 sur le 𝑘-ième nombre premier et grandes valeurs de la fonction 𝜔(𝑛) nombre de diviseurs premiers de 𝑛, Acta Arith. 42 (1983), no. 4, 367–389 (French). MR 736719
  • 5. J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 0137689

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 05A05, 11N05, 94B05

Retrieve articles in all journals with MSC (2010): 05A05, 11N05, 94B05


Additional Information

Po-Yi Huang
Affiliation: Department of Mathematics and National Center for Theoretical Sciences (South), National Cheng Kung University, 1 University Road, Tainan 701, Taiwan
Email: pyhuang@mail.ncku.edu.tw

Wen-Fong Ke
Affiliation: Department of Mathematics and National Center for Theoretical Sciences (South), National Cheng Kung University, 1 University Road, Tainan 701, Taiwan
Email: wfke@mail.ncku.edu.tw

Günter F. Pilz
Affiliation: Department of Algebra, Johannes Kepler Universität Linz, Altenberger Strasse 69, 4040 Linz, Austria
Email: guenter.pilz@jku.at

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10189-2
Received by editor(s): June 1, 2009
Published electronically: October 23, 2009
Additional Notes: The first author was supported by the National Science Council, Taiwan, grant #96-2115-M-006-003-MY3
The second author was partially supported by the National Science Council, Taiwan, grant #97-2923-M-006-001-MY2
The third author was supported by grant P19463 of the Austrian National Science Fund (FWF)
Communicated by: Jim Haglund
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.