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2001 On Waring's problem: three cubes and a sixth power
Jörg Brüdern, Trevor D. Wooley
Nagoya Math. J. 163: 13-53 (2001).


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.


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Jörg Brüdern. Trevor D. Wooley. "On Waring's problem: three cubes and a sixth power." Nagoya Math. J. 163 13 - 53, 2001.


Published: 2001
First available in Project Euclid: 27 April 2005

zbMATH: 0989.11047
MathSciNet: MR1854387

Primary: 11P05
Secondary: 11L15, 11P55

Rights: Copyright © 2001 Editorial Board, Nagoya Mathematical Journal


Vol.163 • 2001
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