Abdullah N. Arslan

Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132

Volume 21, 2015, Number 3, Pages 22–26

**Full paper (PDF, 155 Kb)**

## Details

### Authors and affiliations

Abdullah N. Arslan

*Department of Computer Science, Texas A&M University
Commerce, TX 75428, USA
*

### Abstract

We introduce a variant of Waring’s problem. For a given positive integer k, consider the problem of writing any given positive integer N as the sum of the kth powers of consecutive integers starting at 1 using each of these kth powers (summands) exactly once, and repeating some of these summands as necessary. Let Ck denote the total number of such repeats. Determine minimum Ck for positive integers k required to write all positive integers using the kth powers of consecutive integers as described. We show that Ck ≤ g(k), where g(k) is the usual notation in Waring’s problem, the least number of non-negative kth powers sufficient to represent all positive integers. This result implies that for any given positive integer k, every positive integer N can be expressed as a linear combination of the kth powers of consecutive integers with positive integer coefficients that satisfy certain inequalities. Another implication is that for all positive integers N; n′; and k, the equation N = Σ_{i = 1}^{n′}i^{k}x_{i} has at least one solution (x_{1}, x_{2}, …, x_{n′}) in nonnegative integers if Σ_{i = 1}^{n′}i^{k} ≥ N.

### Keywords

- Partitions
- Lagrange’s four square theorem
- Hilbert–Waring theorem
- function
*g*

### AMS Classification

- 11P05
- 11Y99
- 11D85
- 11B83

### References

- Balasubramanian, R., Deshouillers, J.-M., Dress, F. (1986) Problème de Waring pour les bicarrés. I. Schéma de la solution. (French. English summary) [Waring’s problem for biquadrates. I. Sketch of the solution] C. R. Acad. Sci. Paris Sér. I Math. 303, no. 4, pp. 85-88
- Balasubramanian, R., Deshouillers, J.-M., Dress, F. (1986) Problème de Waring pour les bicarrés. II. Résultats auxiliaires pour le théorème asymptotique. (French. English summary)[Waring’s problem for biquadrates. II. Auxiliary results for the asymptotic theorem] C. R. Acad. Sci. Paris Sér. I Math. 303, no. 5, pp. 161-163
- Cooper, S. B. (2003) Computability Theory. Chapman Hall/CRC Mathematics Series, p. 98
- Hilbert, D. (1909) Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem). Mathematische Annalen 67: 281-300
- Jingrun, C. (1964) Waring’s problem g(5) = 37, Acta Math. Sin 14, 715-734.
- Kempner, A. (1912) Bemerkungen zum Waringschen Problem. Mathematische Annalen 72(3): 387-399. doi:10.1007/BF01456723
- Pillai, S. S. (1940) On Waring’s problem g(6) = 73. Proc. Indian Acad. Sci. 12A: 30-40
- Rabin, M. O. and Shallit, J. O. (1986) Randomized Algorithms in Number Theory,

Communications on Pure and Applied Mathematics 39, no. S1, pp. S239-S256.

doi:10.1002/cpa.3160390713 - Sloane, N. J. A. (2015) Sequence A002804. The On-Line Encyclopedia of Integer Sequences.

http://oeis.org/A002804 (retrieved on May 2, 2015) - Sums of Consecutive Powers Project (2015) http://www.math.rutgers.edu/~erowland/

sumsofpowers-project.html (retrieved on May 2, 2015) - Wieferich, A. (1909) Beweis des Satzes, daß sich eine jede ganze Zahl als Summe von

höchstens neun positiven Kuben darstellen läßt. Mathematische Annalen 66 (1): 95-101.

doi:10.1007/BF01450913

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## Cite this paper

Arslan, A. (2015). A variant of Waring’s problem. *Notes on Number Theory and Discrete Mathematics*, 21(3), 22-26.