# Slim exceptional set for sums of two squares, two cubes, and two biquadrates of primes

Research Article

First Online:

- 11 Downloads

## Abstract

We prove that, with at most \(O(N^{\frac{17}{192}+\varepsilon})\) exceptions, all even positive integers up to *N* are expressible in the form *p* _{1} ^{2} + *p* _{2} ^{2} + *p* _{3} ^{3} + *p* _{4} ^{3} + *p* _{5} ^{4} + *p* _{6} ^{4} , where *p*_{1}, *p*_{2}, …, *p*_{6} are prime numbers. This gives large improvement of a recent result \(O(N^{\frac{13}{16}+\varepsilon})\) due to M. Zhang and J. J. Li.

## Keywords

Waring-Goldbach problem circle method exceptional set## MSC

11P05 11P32 11P55## Preview

Unable to display preview. Download preview PDF.

## Notes

### Acknowledgements

The author would like to express the most sincere gratitude to Prof. Zhixin Liu for his valuable advice and constant encouragement. This work was supported by the National Natural Science Foundation of China (Grant No. 11871367).

## References

- 1.Hua L K. Some results in the additive prime number theory. Quart J Math, 1938, 9: 68–80MathSciNetCrossRefGoogle Scholar
- 2.Hua L K. Additive Theory of Prime Numbers. Providence: Amer Math Soc, 1965zbMATHGoogle Scholar
- 3.Kawada K, Wooley T D. Relations between exceptional sets for additive problems. J Lond Math Soc, 2010, 82: 437–458MathSciNetCrossRefGoogle Scholar
- 4.Kumchev A V. On Weyl sums over primes and almost primes. Michigan Math J, 2006, 54: 243–268MathSciNetCrossRefGoogle Scholar
- 5.Li T Y. Enlarged major arcs in the Waring-Goldbach problem. Int J Number Theory, 2016, 12: 205–217MathSciNetCrossRefGoogle Scholar
- 6.Liu J Y. Enlarged major arcs in additive problems. Math Notes, 2010, 88: 395–401MathSciNetCrossRefGoogle Scholar
- 7.Liu Y H. On a Waring-Goldbach problem involving squares, cubes and biquadrates. Bull Korean Math Soc, 2018, 55: 1659–1666MathSciNetzbMATHGoogle Scholar
- 8.Liu Y H. Exceptional set for sums of unlike powers of primes. Int J Number Theory, 2019, 15(2): 339–352MathSciNetCrossRefGoogle Scholar
- 9.Liu Z X. Goldbach-Linnik type problems with unequal powers of primes. J Number Theory, 2017, 176: 439–448MathSciNetCrossRefGoogle Scholar
- 10.Lü X D. Waring-Goldbach problem: two squares, two cubes and two biquadrates. Chinese Ann Math Ser A, 2015, 36: 161–174 (in Chinese)MathSciNetCrossRefGoogle Scholar
- 11.Ren X M. On exponential sums over primes and application in Waring-Goldbach problem. Sci China Ser A, 2005, 48: 785–797MathSciNetCrossRefGoogle Scholar
- 12.Vaughan R C. On the representation of numbers as sums of powers of natural numbers. Proc Lond Math Soc, 1970, 21: 160–180MathSciNetCrossRefGoogle Scholar
- 13.Vaughan R C. The Hardy-Littlewood Method. 2nd ed. Cambridge: Cambridge Univ Press, 1997CrossRefGoogle Scholar
- 14.Vinogradov I M. Elements of Number Theory. New York: Dover Publications, 1954zbMATHGoogle Scholar
- 15.Zhang M, Li J J. Exceptional set for sums of unlike powers of primes. Taiwanese J Math, 2018, 22: 779–811MathSciNetCrossRefGoogle Scholar
- 16.Zhao L L. The additive problem with one cube and three cubes of primes. Michigan Math J, 2014, 63: 763–779MathSciNetCrossRefGoogle Scholar
- 17.Zhao L L. On the Waring-Goldbach problem for fourth and sixth powers. Proc Lond Math Soc, 2014, 108: 1593–1622MathSciNetCrossRefGoogle Scholar

## Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019