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ON THE WARING–GOLDBACH PROBLEM FOR ONE SQUARE, FOUR CUBES AND ONE BIQUADRATE

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  15 September 2022

JINJIANG LI Open the ORCID record for JINJIANG LI [Opens in a new window] ,
FEI XUE Open the ORCID record for FEI XUE [Opens in a new window]  and
MIN ZHANG Open the ORCID record for MIN ZHANG [Opens in a new window]
JINJIANG LI
Affiliation:
Department of Mathematics, China University of Mining and Technology, Beijing 100083, PR China e-mail: jinjiang.li.math@gmail.com
FEI XUE
Affiliation:
Department of Mathematics, China University of Mining and Technology, Beijing 100083, PR China e-mail: fei.xue.math@gmail.com
MIN ZHANG*
Affiliation:
School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, PR China
*
e-mail: min.zhang.math@gmail.com
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Abstract

Let N be a sufficiently large integer. We prove that, with at most $O(N^{23/48+\varepsilon })$ exceptions, all even positive integers up to N can be represented in the form $p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^4$ , where $p_1,p_2,p_3,p_4,p_5,p_6$ are prime numbers.

Keywords

Waring–Goldbach problemHardy–Littlewood methodexceptional set

MSC classification

Primary: 11P05: Waring's problem and variants
Secondary: 11P32: Goldbach-type theorems; other additive questions involving primes 11P55: Applications of the Hardy-Littlewood method
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 107 , Issue 3 , June 2023 , pp. 416 - 431
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11901566, 12001047, 11971476 and 12071238) and the Fundamental Research Funds for the Central Universities (Grant No. 2022YQLX05).

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