## Abstract

Let *R*(*n*) denote the number of representations of an even integer *n* as the sum of two squares, two cubes and two sixth powers of primes, and by \(\mathcal {E}(N)\) we denote the number of even integers \(n \leqslant N\) such that the expected asymptotic formula for *R*(*n*) fails to hold. In this paper, it is proved that \(\mathcal {E}(N) \ll N^{\frac{127}{288} + \varepsilon }\) for any \(\varepsilon >0\).

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## Acknowledgements

This project was supported by the National Natural Science Foundation of China (Grant No. 11771333). The author would like to thank the anonymous referee for his/her patience, time and valuable suggestions in refereeing this paper.

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Communicating Editor: Sanoli Gun

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Liu, Y. Exceptional set in Waring–Goldbach problem: Two squares, two cubes and two sixth powers.
*Proc Math Sci* **130, **8 (2020). https://doi.org/10.1007/s12044-019-0540-6

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### Keywords

- Waring–Goldbach problem
- exceptional set
- Hardy–Littlewood method

### 2010 Mathematics Subject Classification

- 11P05
- 11P55