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On Waring’s problem: Three cubes and a sixth power

Published online by Cambridge University Press:  22 January 2016

Jörg Brüdern  and
Trevor D. Wooley
Jörg Brüdern
Affiliation:
Mathematisches Institut A, Universität Stuttgart, Postfach 80 11 40, D-70511 Stuttgart, Germany, bruedern@mathematik.uni-stuttgart.de
Trevor D. Wooley
Affiliation:
Department of Mathematics, University of Michigan, East Hall, 525 East University Avenue, Ann Arbor, Michigan 48109-1109, U.S.A., wooley@math.lsa.umich.edu
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Abstract

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We establish that almost all natural numbers not congruent to 5 modulo 9 are the sum of three cubes and a sixth power of natural numbers, and show, moreover, that the number of such representations is almost always of the expected order of magnitude. As a corollary, the number of representations of a large integer as the sum of six cubes and two sixth powers has the expected order of magnitude. Our results depend on a certain seventh moment of cubic Weyl sums restricted to minor arcs, the latest developments in the theory of exponential sums over smooth numbers, and recent technology for controlling the major arcs in the Hardy-Littlewood method, together with the use of a novel quasi-smooth set of integers.

Keywords

11P0511L1511P55
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 163 , September 2001 , pp. 13 - 53
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2001

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