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On the Waring–Goldbach problem for squares, cubes and higher powers

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Abstract

Let \(\mathcal {P}_r\) denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, we generalize the result of Vaughan (Bull Lond Math Soc 17(1):17–20, 1985) for ternary ‘admissible exponent’. Moreover, we use the refined ‘admissible exponent’ to prove that, for \(3\leqslant k\leqslant 14\) and for every sufficiently large even integer n, the following equation

$$\begin{aligned} n=x^2+p_1^2+p_2^3+p_3^3+p_4^3+p_5^k \end{aligned}$$

is solvable with x being an almost-prime \(\mathcal {P}_{r(k)}\) and the other variables primes, where r(k) is defined in Theorem 1.1. This result constitutes a deepening of previous results.

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Acknowledgements

The authors would like to express their most sincere gratitude to the referee for his/her patience in refereeing this paper.

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  1. School of Applied Science, Bei**g Information Science and Technology University, Bei**g, 100192, People’s Republic of China

    Min Zhang

  2. Department of Mathematics, China University of Mining and Technology, Bei**g, 100083, People’s Republic of China

    **jiang Li

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  1. Min Zhang
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  2. **jiang Li
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Correspondence to **jiang Li.

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This work is supported by the National Natural Science Foundation of China (Grant Nos. 11901566, 11971476, 12001047, 12071238), the Fundamental Research Funds for the Central Universities (Grant No. 2019QS02), and the Scientific Research Funds of Bei**g Information Science and Technology University (Grant No. 2025035).

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Zhang, M., Li, J. On the Waring–Goldbach problem for squares, cubes and higher powers. Ramanujan J 56, 1123–1150 (2021). https://doi.org/10.1007/s11139-020-00334-2

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