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On gaps between numbers that are sums of three cubes

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  26 February 2010

Stephan Daniel
Stephan Daniel
Affiliation:
Mathematisches Institut A, Pfaffenwaldring 57, D-70550 Stuttgart, Germany. e-mail: sdaniel@mathematik.uni-stuttgart.de
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Extract

Let (Sn)n = 1,2,… be the strictly increasing sequence of those natural numbers that can be represented as the sum of three cubes of positive integers. The estimate

is easily proved as follows: Let x1 be the largest natural number with Then This procedure is iterated by choosing x2 and then x3 as the largest natural numbers satisfying and Thus Since this implies (1).

MSC classification

Secondary: 11P05: Waring's problem and variants
Type
Research Article
Information
Mathematika , Volume 44 , Issue 1 , June 1997 , pp. 1 - 13
Copyright
Copyright © University College London 1997

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References

1.Davenport, H.. On Waring's problem for cubes. Acta Math., 71 (1939), 123–143.CrossRefGoogle Scholar
2.Hooley, C.. On the intervals between numbers that are sums of two squares. Acta Math., 127 (1971), 279–297.CrossRefGoogle Scholar
3.Vaughan, R. C.. The Hardy Littlewood Method (Cambridge University Press, London, 1981).Google Scholar
4.Vaughan, R. C.. Some remarks on Weyl sums. In: Topics in classical number theory, Coll. Math. Soc Janos Bolyui 34 (Budapest, 1984).Google Scholar
5.Vaughan, R. C.. Sums of three cubes. Bull. London Math. Soc., 17 (1985), 17–20.CrossRefGoogle Scholar
6.Vaughan, R. C.. On Waring's problem for sixth powers. J. London Math. Soc., 33 (2) (1986), 227–236.CrossRefGoogle Scholar