## References

SHOWING 1-10 OF 11 REFERENCES

On a question of Mordell

- MathematicsProceedings of the National Academy of Sciences
- 2021

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

- MathematicsProceedings 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

- Mathematics
- 2015

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

- Mathematics
- 2015

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

- Mathematics
- 2015

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

- Mathematics
- 2012

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

- MathematicsPhilosophical 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

- MathematicsGraduate texts in mathematics
- 1982

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.