Signed sums of terms of a sequence
Authors:
FengJuan Chen and YongGao Chen
Journal:
Proc. Amer. Math. Soc. 141 (2013), 11051111
MSC (2010):
Primary 11A67, 11B50, 11B83, 11P05
Published electronically:
August 9, 2012
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Additional Information
Abstract: We give a sufficient and necessary condition on the sequence of integers that for any integer , every integer can be represented in the form , where . This generalizes the known result on integralvalued 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.
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 M. N. Bleicher, On Prielipp's problem on signed sums of th powers, J. Number Theory 56 (1996), 3651. 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), 50715088. MR 2084411 (2005d:11154)
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 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), 111124. MR 0130236 (24:A103)
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 Y. G. Chen, On subset sums of a fixed set, Acta Arith. 106 (2003), no. 3, 207211. MR 1957105 (2003j:11019)
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 P. Erdős and J. Surányi, Über ein Problem aus der additiven Zahlentheorie, Mat. Lapok 10 (1959), 284290. MR 0125825 (23:A3122)
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 H. B. Yu, Signed sums of polynomial values, Proc. Amer. Math. Soc. 130 (2002), 16231627. MR 1887008 (2002m:11007)
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Additional Information
FengJuan 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
YongGao 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:
http://dx.doi.org/10.1090/S000299392012113978
PII:
S 00029939(2012)113978
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 (CXZZ110868).
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.
