Waring’s Problem for Cubes

  title={Waring’s Problem for Cubes},
  author={Tim Browning},
  journal={Cubic Forms and the Circle Method},
  • Tim Browning
  • Published 2021
  • Mathematics
  • Cubic Forms and the Circle Method


On a question of Mordell
This paper concludes a 65-y search with an affirmative answer to Mordell’s question and strongly supports a related conjecture of Heath-Brown and makes several improvements to methods for finding integer solutions to x3+y3+z3=k for small values of k.
Nested efficient congruencing and relatives of Vinogradov's mean value theorem
  • T. Wooley
  • Mathematics
    Proceedings of the London Mathematical Society
  • 2018
We apply a nested variant of multigrade efficient congruencing to estimate mean values related to that of Vinogradov. We show that when φj∈Z[t] (1⩽j⩽k) is a system of polynomials with non‐vanishing
Every integer greater than 454 is the sum of at most seven positive cubes
A long-standing conjecture states that every positive integer other than 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, 303, 364, 420, 428, 454 is a sum of at most seven positive cubes. This
Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three
We prove the main conjecture in Vinogradov's Mean Value Theorem for degrees higher than three. This will be a consequence of a sharp decoupling inequality for curves
Sums of three cubes, II
Estimates are provided for $s$th moments of cubic smooth Weyl sums, when $4\le s\le 8$, by enhancing the author's iterative method that delivers estimates beyond classical convexity. As a
Vinogradov's mean value theorem via efficient congruencing
We obtain estimates for Vinogradov’s integral that for the rst time approach those conjectured to be the best possible. Several applications of these new bounds are provided. In particular, the
The circle method and diagonal cubic forms
  • D. R. Heath-Brown
  • Mathematics
    Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
  • 1998
The Hardy–Littlewood circle method is used to investigate the number of integer zeros of diagonal cubic forms, and it is shown that there are O(P3+ϵ) zeros up to height P, for any ϵ>0.
A classical introduction to modern number theory
This second edition has been corrected and contains two new chapters which provide a complete proof of the Mordell-Weil theorem for elliptic curves over the rational numbers, and an overview of recent progress on the arithmetic of elliptic curve.