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Signed sums of terms of a sequence


Authors: Feng-Juan Chen and Yong-Gao Chen
Journal: Proc. Amer. Math. Soc. 141 (2013), 1105-1111
MSC (2010): Primary 11A67, 11B50, 11B83, 11P05
DOI: https://doi.org/10.1090/S0002-9939-2012-11397-8
Published electronically: August 9, 2012
MathSciNet review: 3008858
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Abstract | References | Similar Articles | Additional Information

Abstract: We give a sufficient and necessary condition on the sequence $ \{a_n\} $ of integers that for any integer $ l\ge 1$, every integer can be represented in the form $ \varepsilon _l a_l+\varepsilon _{l+1} a_{l+1}+\cdots + \varepsilon _ka_k$, where $ \varepsilon _i\in \{-1, 1\}\ (i=l,l+1,\ldots , k)$. This generalizes the known result on integral-valued polynomial values. Moreover, we show that such sequences exist with any growth rate. This answers two problems posed by Bleicher. We also pose several problems for further research.


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

  • 1. M. N. Bleicher, On Prielipp's problem on signed sums of $ k$th powers, J. Number Theory 56 (1996), 36-51. MR 1370195 (96j:11011)
  • 2. J. Boulanger and J. L. Chabert, On the representation of integers as linear combinations of consecutive values of a polynomial, Trans. Amer. Math. Soc. 356 (2004), 5071-5088. MR 2084411 (2005d:11154)
  • 3. J. W. S. Cassels, On the representation of integers as the sums of distinct summands taken from a fixed set, Acta Sci. Math. Szeged 21 (1960), 111-124. MR 0130236 (24:A103)
  • 4. Y. G. Chen, On subset sums of a fixed set, Acta Arith. 106 (2003), no. 3, 207-211. MR 1957105 (2003j:11019)
  • 5. P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory, Monographie 28 de L'enseignement mathématique, Geneva, 1980. MR 592420 (82j:10001)
  • 6. P. Erdős and J. Surányi, Über ein Problem aus der additiven Zahlentheorie, Mat. Lapok 10 (1959), 284-290. MR 0125825 (23:A3122)
  • 7. R. L. Graham, Complete sequences of polynomial values, Duke Math. J. 31 (1964), 275-285. MR 0162759 (29:63)
  • 8. N. Hegyvári, On the representation of integers as sums of distinct terms from a fixed set, Acta Arith. 92 (2000), 99-104. MR 1750309 (2001c:11014)
  • 9. T. Łuczak and T. Schoen, On the maximal density of sum-free sets, Acta Arith. 95 (2000), 225-229. MR 1793162 (2001k:11018)
  • 10. M. B. Nathanson, Elementary Methods in Number Theory, Springer-Verlag, New York, 2000. MR 1732941 (2001j:11001)
  • 11. H. B. Yu, Signed sums of polynomial values, Proc. Amer. Math. Soc. 130 (2002), 1623-1627. MR 1887008 (2002m:11007)

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Additional Information

Feng-Juan Chen
Affiliation: School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210046, People’s Republic of China – and – Department of Mathematics, Suzhou University, Suzhou 215006, People’s Republic of China
Email: cfjsz@126.com

Yong-Gao Chen
Affiliation: School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210046, People’s Republic of China
Email: ygchen@njnu.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-2012-11397-8
Keywords: Signed sums, sequences, polynomials
Received by editor(s): March 14, 2011
Received by editor(s) in revised form: August 4, 2011
Published electronically: August 9, 2012
Additional Notes: This work was supported by the National Natural Science Foundation of China, Grant No. 11071121 and the Project of Graduate Education Innovation of Jiangsu Province (CXZZ11-0868).
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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