The cardinality of some symmetric differences
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- by Po-Yi Huang, Wen-Fong Ke and Günter F. Pilz
- Proc. Amer. Math. Soc. 138 (2010), 787-797
- DOI: https://doi.org/10.1090/S0002-9939-09-10189-2
- Published electronically: October 23, 2009
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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
- The GAP Group, GAP—Groups, Algorithms, and Programming, Version 4.4.12, 2008, http://www.gap-system.org.
- M. Nair, On Chebyshev-type inequalities for primes, Amer. Math. Monthly 89 (1982), no. 2, 126–129. MR 643279, DOI 10.2307/2320934
- Günter Pilz, On polynomial near-ring codes, Contributions to general algebra, 8 (Linz, 1991) Hölder-Pichler-Tempsky, Vienna, 1992, pp. 233–238. MR 1281844
- Guy Robin, Estimation de la fonction de Tchebychef $\theta$ sur le $k$-ième nombre premier et grandes valeurs de la fonction $\omega (n)$ nombre de diviseurs premiers de $n$, Acta Arith. 42 (1983), no. 4, 367–389 (French). MR 736719, DOI 10.4064/aa-42-4-367-389
- J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 137689
Bibliographic 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
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 787-797
- MSC (2010): Primary 05A05; Secondary 11N05, 94B05
- DOI: https://doi.org/10.1090/S0002-9939-09-10189-2
- MathSciNet review: 2566544