Hostname: page-component-5cf477f64f-zrtmk Total loading time: 0 Render date: 2025-04-02T07:00:30.846Z Has data issue: false hasContentIssue false
  • English
  • Français

On the sum of four cubes

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  26 February 2010

Koichi Kawada
Koichi Kawada
Affiliation:
Department of Mathematics, Faculty of Education, Iwate University, Morioka 020, Japan.
Get access

Extract

On Waring's problem for cubes, it is conjectured that every sufficiently large natural number can be represented as a sum of four cubes of natural numbers. Denoting by E(N) the number of the natural numbers up to N that cannot be written as a sum of four cubes, we may express the conjecture as E(N)≪1.

MSC classification

Secondary: 11P05: Waring's problem and variants
Type
Research Article
Information
Mathematika , Volume 43 , Issue 2 , December 1996 , pp. 323 - 348
Copyright
Copyright © University College London 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Brüdern, J.. A problem in additive number theory. Math. Proc. Cambridge Phil. Soc., 103 (1988), 27–33.CrossRefGoogle Scholar
2.Brudern, J.. Sums of four cubes. Monatsh. Math., 107 (1989), 179–188.CrossRefGoogle Scholar
3.Brüdern, J.. On Waring's problem for cubes. Math. Proc. Cambridge Phil. Soc. 109 (1991), 229–256.CrossRefGoogle Scholar
4.Brudern, J. and Watt, N.. On Waring's problem for four cubes. Duke Math. J., 11 (1995), 583–606.Google Scholar
5.Davenport, H.. On Waring's problem for cubes. Acta Math., 71 (1939), 123–143.CrossRefGoogle Scholar
6.Hooley, C.. On Waring's problem. Acta Math., 157 (1986), 49–97.CrossRefGoogle Scholar
7.Loxton, J. H. and Vaughan, R. C.. The estimation of complete exponential sums. Canad. Math. Bull., 28 (1985), 440–454.CrossRefGoogle Scholar
8.Titchmarsh, E. C.. The Theory of the Riemann Zeta Function (The Clarendon Press, 1951).Google Scholar
9.Vaughan, R. C.. The Hardy- Littlewood Method (Cambridge University Press, 1981).Google Scholar
10.Vaughan, R. C.. Some remarks on Weyl sums. In Topics in classical number theory, Colloq. Math. Soc. János Bolyai, 34 (North-Holland, Amsterdam, 1981), 1585–1602.Google Scholar
11.Vaughan, R. C.. On Waring's problem for cubes. J. Reine. Angew. Math., 365 (1986), 122–170.Google Scholar
12.Vaughan, R. C.. A new iterative method in Waring's problem. Acta Math., 162 (1989), 1–71.CrossRefGoogle Scholar
13.Wooley, T. D.. Large improvements in Waring's problem. Ann. Math., 135 (1992), 131–164.CrossRefGoogle Scholar