## Nagoya Mathematical Journal

### On Waring's problem: three cubes and a sixth power

#### Abstract

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.

#### Article information

Source
Nagoya Math. J., Volume 163 (2001), 13-53.

Dates
First available in Project Euclid: 27 April 2005

https://projecteuclid.org/euclid.nmj/1114631619

Mathematical Reviews number (MathSciNet)
MR1854387

Zentralblatt MATH identifier
0989.11047

#### Citation

Brüdern, Jörg; Wooley, Trevor D. On Waring's problem: three cubes and a sixth power. Nagoya Math. J. 163 (2001), 13--53. https://projecteuclid.org/euclid.nmj/1114631619

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