Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On regular polytope numbers

Author: Hyun Kwang Kim
Journal: Proc. Amer. Math. Soc. 131 (2003), 65-75
MSC (2000): Primary 11B13, 11B75, 11P05
Published electronically: June 12, 2002
MathSciNet review: 1929024
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Lagrange proved a theorem which states that every nonnegative integer can be written as a sum of four squares. This result can be generalized as the polygonal number theorem and the Hilbert-Waring problem. In this paper, we shall generalize Lagrange's sum of four squares theorem further. To each regular polytope $V$ in a Euclidean space, we will associate a sequence of nonnegative integers which we shall call regular polytope numbers, and consider the problem of finding the order $g(V)$ of the set of regular polytope numbers associated to $V$.

References [Enhancements On Off] (What's this?)

  • 1. R. Balasubramanian, On Waring’s problem: 𝑔(4)≤20, Hardy-Ramanujan J. 8 (1985), 1–40. MR 876604
  • 2. Jing-run Chen, Waring’s problem for 𝑔(5)=37, Sci. Sinica 13 (1964), 1547–1568. MR 0200236
  • 3. H. S. M. Coxeter, Regular polytopes, 3rd ed., Dover Publications, Inc., New York, 1973. MR 0370327
  • 4. Jean-Marc Deshouillers and François Dress, Sums of 19 biquadrates: on the representation of large integers, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), no. 1, 113–153. MR 1183760
  • 5. A.Kempner, Bemerkungen zum Waringschen Problem, Mat. Annalen $\textbf{72}$ (1912), 387-399.
  • 6. Melvyn B. Nathanson, Additive number theory, Graduate Texts in Mathematics, vol. 164, Springer-Verlag, New York, 1996. The classical bases. MR 1395371
  • 7. Melvyn B. Nathanson, Elementary methods in number theory, Graduate Texts in Mathematics, vol. 195, Springer-Verlag, New York, 2000. MR 1732941
  • 8. S.S.Pillai, On Waring's problem: $g(6)=73$, Proc. Indian Acad. Sci., A $\textbf{12}$(1940), 30-40. MR 2:146c
  • 9. A.Wieferich, Beweis des Satzes, daßsich eine jede ganze Zahl als Summe von höchstens neun positiven Kuben darstellen läßt, Mat. Annalen $\textbf{66}$ (1909), 95-101.
  • 10. J.Worpitzky, Studien über die Bernoullischen und Eulerschen Zahlen, Journal f.d. reine u. angew. Math $\textbf{94}$ (1883), 203-232.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11B13, 11B75, 11P05

Retrieve articles in all journals with MSC (2000): 11B13, 11B75, 11P05

Additional Information

Hyun Kwang Kim
Affiliation: Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784 Korea

Received by editor(s): August 20, 2001
Published electronically: June 12, 2002
Additional Notes: This work was supported by Com$^{2}$MaC-KOSEF
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2002 American Mathematical Society