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Exceptional sets in Waring-Goldbach problem for fifth powers

Abstract

We consider exceptional sets in the Waring-Goldbach problem for fifth powers. For example, we prove that all but O(N131/132) integers satisfying the necessary local conditions can be represented as the sum of 11 fifth powers of primes, which improves the previous results due to A. V. Kumchev [Canad. J. Math., 2005, 57: 298–327] and Z. X. Liu [Int. J. Number Theory, 2012, 8: 1247–1256].

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Acknowledgements

The authors express their sincere thanks to the referees for valuable suggestions and comments. The first author was supported by the Scientific Research Project of the Education Department of Fujian Province (Grant No. JAT190370) and the Natural Science Foundation of Fujian Province (Grant No. 2020J05162). The second author was supported by the National Natural Science Foundation of China (Grant No. 11871367) and the Natural Science Foundation of Tianjin City (Grant No. 19JCQNJC14200).

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Correspondence to Zhixin Liu.

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Feng, Z., Liu, Z. Exceptional sets in Waring-Goldbach problem for fifth powers. Front. Math. China 16, 49–58 (2021). https://doi.org/10.1007/s11464-021-0899-4

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Keywords

  • Exceptional sets
  • Waring-Goldbach problem
  • circle method

MSC2020

  • 11P32
  • 11P55
  • 11P05