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Waring–Goldbach problem: two squares and some unlike powers

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Abstract

Let P r denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper it is proved that for every sufficiently large odd integer N, the equation

$$N = x^2+ p_{2}^{2}+ p_{3}^{3}+ p_{4}^{4}+ p_{5}^{5}+ p_{6}^{6}+ p_{7}^{7}$$

is solvable with x being an almost-prime P 6 and the other terms powers of primes. This result constitutes a refinement upon that of C. Hooley [7].

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Authors and Affiliations

  1. Department of Mathematics, Tongji University, Shanghai, 200092, P. R. China

    Y. C. Cai

  2. School of Science, Xi’an Polytechnic University, Xi’an, Shaanxi Province, 710048, P. R. China

    Q. W. Mu

Authors
  1. Y. C. Cai
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  2. Q. W. Mu
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Correspondence to Y. C. Cai.

Additional information

This work is supported by the National Natural Science Foundation of China (grant Nos. 11201107, 11271283) and the Natural Science Foundation of Anhui Province in China (grant No. 1208085QA01).

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Cai, Y.C., Mu, Q.W. Waring–Goldbach problem: two squares and some unlike powers. Acta Math. Hungar. 145, 46–67 (2015). https://doi.org/10.1007/s10474-014-0472-7

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  • DOI: https://doi.org/10.1007/s10474-014-0472-7

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