A new variant of the Hilbert-Waring problem

Authors:
Tianxin Cai and Deyi Chen

Journal:
Math. Comp. **82** (2013), 2333-2341

MSC (2010):
Primary 11P05; Secondary 11D85, 11D72

DOI:
https://doi.org/10.1090/S0025-5718-2013-02685-3

Published electronically:
March 5, 2013

MathSciNet review:
3073204

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Abstract | References | Similar Articles | 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.**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, 10-36 (1971). MR**0414510 (54:2611)****3.**Nagell T., Introduction to Number Theory. pp. 205-206. Wiley, New York (1951). MR**0043111 (13:207b)****4.**Small C., Waring's problem mod . Amer. Math. Monthly. 84, 12-25 (1977); Solution of Waring's problem mod . Amer. Math. Monthly. 84, 356-359 (1977). MR**0424734**; MR**0439787****5.**Vaughan R. C. and Wooley T. D., Waring's Problem: Surveys in Number Theory, A. K. Peters, pp. 285-324, 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:
https://doi.org/10.1090/S0025-5718-2013-02685-3

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