Acknowledgement
The author would like to thank the anonymous referee for his/her patience and time in refereeing this manuscript and express the most sincere gratitude to Professor Yingchun Cai for his valuable advice and constant encouragement.
References
- Y. Cai, The Waring-Goldbach problem: one square and five cubes, Ramanujan J. 34 (2014), no. 1, 57-72. https://doi.org/10.1007/s11139-013-9486-y
- L. K. Hua, Additive theory of prime numbers, Translations of Mathematical Monographs, Vol. 13, American Mathematical Society, Providence, RI, 1965.
- A. V. Kumchev, On Weyl sums over primes and almost primes, Michigan Math. J. 54 (2006), no. 2, 243-268. https://doi.org/10.1307/mmj/1156345592
- Y. Liu, Exceptional set for sums of unlike powers of primes, Int. J. Number Theory 15 (2019), no. 2, 339-352. https://doi.org/10.1142/S1793042119500155
- R. C. Vaughan, The Hardy-Littlewood method, second edition, Cambridge Tracts in Mathematics, 125, Cambridge University Press, Cambridge, 1997. https://doi.org/10.1017/CBO9780511470929
- I. M. Vinogradov, Elements of Number Theory, translated by S. Kravetz, Dover Publications, Inc., New York, 1954.
- T. D. Wooley, Slim exceptional sets in Waring's problem: one square and five cubes, Q. J. Math. 53 (2002), no. 1, 111-118. https://doi.org/10.1093/qjmath/53.1.111
- T. D. Wooley, Slim exceptional sets and the asymptotic formula in Waring's problem, Math. Proc. Cambridge Philos. Soc. 134 (2003), no. 2, 193-206. https://doi.org/10.1017/S030500410200628X
- F. Xue, M. Zhang, and J. Li, On the Waring-Goldbach problem for one square and five cubes in short intervals, Czechoslovak Math. J. 71(146) (2021), no. 2, 563-589. https://doi.org/10.21136/CMJ.2020.0013-20
- L. Zhao, On the Waring-Goldbach problem for fourth and sixth powers, Proc. Lond. Math. Soc. (3) 108 (2014), no. 6, 1593-1622. https://doi.org/10.1112/plms/pdt072