A new variant of the Hilbert-Waring problem
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- by Tianxin Cai and Deyi Chen PDF
- Math. Comp. 82 (2013), 2333-2341 Request permission
Abstract:
In this paper, we propose a new variant of Waring’s problem: to express a positive integer $n$ as a sum of $s$ positive integers whose product is a $k$-th power. We define, in a similar way to that done for $g(k)$ and $G(k)$ in Waring’s problem, numbers $g’(k)$ and $G’(k)$. We obtain $g’(k)=2k-1$, $G’(p)\leq p+1$ for primes $p$, $G’(2p)\leq 2p+2$ for odd primes $p$. Moreover, we obtain several interesting results and make two conjectures about $G’(3)$ and $G’(4)$.References
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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
- 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.
- © Copyright 2013 American Mathematical Society
- 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
- MathSciNet review: 3073204