On Waring’s problem for larger powers-Reference-Cited by-同舟云学术

On Waring’s problem for larger powers

Published:2023-11-01 Issue:0 Volume:0 Page:
ISSN:0075-4102
Container-title:Journal für die reine und angewandte Mathematik (Crelles Journal)
language:en
Short-container-title:

Author:

Brüdern Jörg¹,Wooley Trevor D.²^ORCID

Affiliation:

1. Mathematisches Institut , Georg-August-Universität Göttingen , Bunsenstr. 3–5, 37073 Göttingen , Germany

2. Department of Mathematics , Purdue University , 150 N. University Street , West Lafayette , IN 47907-2067 , USA

Abstract

Abstract Let G ⁢ ( k )

{G(k)}

denote the least number s having the property that every sufficiently large natural number is the sum of at most s positive integral k-th powers. Then for all k ∈ ℕ

{k\in\mathbb{N}}

, one has

G ⁢ ( k ) ⩽ ⌈ k ⁢ ( log ⁡ k + 4.20032 ) ⌉ .

G(k)\leqslant\lceil k(\log k+4.20032)\rceil.

Our new methods improve on all bounds available hitherto when k ⩾ 14

{k\geqslant 14}

Funder

National Science Foundation

Deutsche Forschungsgemeinschaft

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,General Mathematics

Link

https://www.degruyter.com/document/doi/10.1515/crelle-2023-0072/pdf

Reference24 articles.

1. J. Brüdern, A problem in additive number theory, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 1, 27–33.

2. J. Brüdern and T. D. Wooley, Partitio numerorum: Sums of a prime and a number of k-th powers, preprint (2022), https://arxiv.org/abs/2211.10387.

3. G. H. Hardy and J. E. Littlewood, A new solution of Waring’s problem, Quart. J. Math. Oxford 48 (1920), 272–293.

4. G. H. Hardy and J. E. Littlewood, Some problems of “Partitio Numerorum”: IV. The singular series in Waring’s Problem and the value of the number G ⁢ ( k ) G(k) , Math. Z. 12 (1922), no. 1, 161–188.

5. G. H. Hardy and J. E. Littlewood, Some problems of “Partitio numerorum” (VI): Further researches in Waring’s Problem, Math. Z. 23 (1925), no. 1, 1–37.

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