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Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

The cardinality of some symmetric differences

Author(s): Po-Yi Huang; Wen-Fong Ke; Günter F. Pilz
Journal: Proc. Amer. Math. Soc. 138 (2010), 787-797.
MSC (2010): Primary 05A05; Secondary 11N05, 94B05
Posted: October 23, 2009
MathSciNet review: 2566544
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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:

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), 126-129. MR 643279 (83f:10043)

3.
G. Pilz, On polynomial near-ring codes, in Contributions to General Algebra, 8, Hölder-Pichler-Tempsky, Vienna, 1992, 233-238. MR 1281844 (95e:11131)

4.
G. Robin, Estimate of the Chebyshev function $ \theta $ on the $ k$th prime number and large values of the number of prime divisors function $ \omega (n)$ of $ n$. Acta Arith. 42 (1983), 367-389. MR 736719 (85j:11109)

5.
J. B. Rosser and L. Schoenfeld. Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962), 64-94. MR 0137689 (25:1139)

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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: 10.1090/S0002-9939-09-10189-2
PII: S 0002-9939(09)10189-2
Received by editor(s): June 1, 2009
Posted: 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 of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




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