A new variant of the HilbertWaring problem
Authors:
Tianxin Cai and Deyi Chen
Journal:
Math. Comp. 82 (2013), 23332341
MSC (2010):
Primary 11P05; Secondary 11D85, 11D72
Published electronically:
March 5, 2013
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Abstract 
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Additional Information
Abstract: In this paper, we propose a new variant of Waring's problem: to express a positive integer as a sum of positive integers whose product is a th power. We define, in a similar way to that done for and in Waring's problem, numbers and . We obtain , for primes , for odd primes . Moreover, we obtain several interesting results and make two conjectures about and .
 1.
Şaban
Alaca and Kenneth
S. Williams, Introductory algebraic number theory, Cambridge
University Press, Cambridge, 2004. MR 2031707
(2005d:11152)
 2.
W.
J. Ellison, Waring’s problem, Amer. Math. Monthly
78 (1971), no. 1, 10–36. MR 0414510
(54 #2611)
 3.
Trygve
Nagell, Introduction to Number Theory, John Wiley & Sons,
Inc., New York; Almqvist & Wiksell, Stockholm, 1951. MR 0043111
(13,207b)
 4.
Charles
Small, Waring’s problem 𝑚𝑜𝑑
𝑛, Amer. Math. Monthly 84 (1977), no. 1,
12–25. MR
0424734 (54 #12693)
Charles
Small, Solution of Waring’s problem
𝑚𝑜𝑑\𝑛, Amer. Math. Monthly
84 (1977), no. 5, 356–359. MR 0439787
(55 #12671)
 5.
Vaughan R. C. and Wooley T. D., Waring's Problem: Surveys in Number Theory, A. K. Peters, pp. 285324, edited by M. A. Bennett et al. (2003).
 6.
Weisstein E. W., Prime representation. From MathWorld, A Wolfram Web Resource.
http://mathworld.wolfram.com/PrimeRepresentation.html
 1.
 Alaca S. and Williams K., Introductory algebraic number theory. CUP, Cambridge (2004). MR 2031707 (2005d:11152)
 2.
 Ellison W. J., Waring's problem, Amer. Math. Monthly. 78, 1036 (1971). MR 0414510 (54:2611)
 3.
 Nagell T., Introduction to Number Theory. pp. 205206. Wiley, New York (1951). MR 0043111 (13:207b)
 4.
 Small C., Waring's problem mod . Amer. Math. Monthly. 84, 1225 (1977); Solution of Waring's problem mod . Amer. Math. Monthly. 84, 356359 (1977). MR 0424734; MR 0439787
 5.
 Vaughan R. C. and Wooley T. D., Waring's Problem: Surveys in Number Theory, A. K. Peters, pp. 285324, edited by M. A. Bennett et al. (2003).
 6.
 Weisstein E. W., Prime representation. From MathWorld, A Wolfram Web Resource.
http://mathworld.wolfram.com/PrimeRepresentation.html
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Additional Information
Tianxin Cai
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou, 310027, China
Email:
txcai@zju.edu.cn
Deyi Chen
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou, 310027, China
Email:
chendeyi1986@126.com
DOI:
http://dx.doi.org/10.1090/S002557182013026853
PII:
S 00255718(2013)026853
Received by editor(s):
September 4, 2011
Received by editor(s) in revised form:
January 22, 2012, January 27, 2012, and February 1, 2012
Published electronically:
March 5, 2013
Additional Notes:
Project supported by China National Science Foundation Grant No.10871169.
Article copyright:
© Copyright 2013
American Mathematical Society
