### 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.

Translated title of the contribution | On Waring's problem: Three cubes and a sixth power |
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Original language | English |

Pages (from-to) | 13 - 53 |

Number of pages | 41 |

Journal | Nagoya Mathematical Journal |

Volume | 163 |

DOIs | |

Publication status | Published - Sep 2001 |

### Bibliographical note

Publisher: Nagoya Univ, Dept Math - Faculty Sci### Keywords

- IMPROVEMENTS
- SUMS

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## Cite this

Brudern, J., & Wooley, TD. (2001). On Waring's problem: Three cubes and a sixth power.

*Nagoya Mathematical Journal*,*163*, 13 - 53. https://doi.org/10.1007/BF01231451