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
MathSciNet review:
3008858
Fulltext PDF
Abstract 
References 
Similar Articles 
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.
 1.
M. N. Bleicher, On Prielipp's problem on signed sums of th powers, J. Number Theory 56 (1996), 3651. MR 1370195 (96j:11011)
 2.
Jacques
Boulanger and JeanLuc
Chabert, On the representation of integers as
linear combinations of consecutive values of a polynomial, Trans. Amer. Math. Soc. 356 (2004), no. 12, 5071–5088 (electronic). MR 2084411
(2005d:11154), 10.1090/S000299470403569X
 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.
YongGao
Chen, On subset sums of a fixed set, Acta Arith.
106 (2003), no. 3, 207–211. MR 1957105
(2003j:11019), 10.4064/aa10631
 5.
P.
Erdős and R.
L. Graham, Old and new problems and results in combinatorial number
theory, Monographies de L’Enseignement Mathématique
[Monographs of L’Enseignement Mathématique], vol. 28,
Université de Genève, L’Enseignement
Mathématique, Geneva, 1980. MR 592420
(82j:10001)
 6.
Pál
Erdös and János
Surányi, Über ein Problem aus der additiven
Zahlentheorie, Mat. Lapok 10 (1959), 284–290
(Hungarian, with Russian and German summaries). 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.
Norbert
Hegyvári, On the representation of integers as sums of
distinct terms from a fixed set, Acta Arith. 92
(2000), no. 2, 99–104. MR 1750309
(2001c:11014)
 9.
Tomasz
Łuczak and Tomasz
Schoen, On the maximal density of sumfree sets, Acta Arith.
95 (2000), no. 3, 225–229. MR 1793162
(2001k:11018)
 10.
Melvyn
B. Nathanson, Elementary methods in number theory, Graduate
Texts in Mathematics, vol. 195, SpringerVerlag, New York, 2000. MR 1732941
(2001j:11001)
 11.
Hong
Bing Yu, Signed sums of polynomial
values, Proc. Amer. Math. Soc.
130 (2002), no. 6,
1623–1627. MR 1887008
(2002m:11007), 10.1090/S0002993901064619
 1.
 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)
 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), 111124. MR 0130236 (24:A103)
 4.
 Y. G. Chen, On subset sums of a fixed set, Acta Arith. 106 (2003), no. 3, 207211. 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), 284290. MR 0125825 (23:A3122)
 7.
 R. L. Graham, Complete sequences of polynomial values, Duke Math. J. 31 (1964), 275285. 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), 99104. MR 1750309 (2001c:11014)
 9.
 T. Łuczak and T. Schoen, On the maximal density of sumfree sets, Acta Arith. 95 (2000), 225229. MR 1793162 (2001k:11018)
 10.
 M. B. Nathanson, Elementary Methods in Number Theory, SpringerVerlag, New York, 2000. MR 1732941 (2001j:11001)
 11.
 H. B. Yu, Signed sums of polynomial values, Proc. Amer. Math. Soc. 130 (2002), 16231627. MR 1887008 (2002m:11007)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
11A67,
11B50,
11B83,
11P05
Retrieve articles in all journals
with MSC (2010):
11A67,
11B50,
11B83,
11P05
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
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.
