Email this article Slim exceptional sets of Waring–Goldbach problem: two squares, two cubes and two biquadrates
- Authors: Tian S.1
-
Affiliations:
- Department of Mathematics, Tongji University, Shanghai, P. R. China
- Issue: Vol 216, No 1 (2025)
- Pages: 96-108
- Section: Articles
- URL: https://journals.rcsi.science/0368-8666/article/view/306674
- DOI: https://doi.org/10.4213/sm10001
- ID: 306674
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Abstract
Let $N$ be a sufficiently large number. We show that, with at most $O(N^{3/32+\varepsilon})$ exceptions, all even positive integers not exceeding $N$ can be represented 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 is an improvement of the result $O(N^{7/18+\varepsilon})$ due to Zhang and Li.Bibliography: 13 titles.
About the authors
Shuangrui Tian
Department of Mathematics, Tongji University, Shanghai, P. R. China
Author for correspondence.
Email: tianshuangrui@tongji.edu.cn
PhD, no status
References
- J. Brüdern, “Sums of squares and higher powers. I”, J. Lond. Math. Soc. (2), 35:2 (1987), 233–243
- K. Kawada, T. D. Wooley, “Relations between exceptional sets for additive problems”, J. Lond. Math. Soc. (2), 82:2 (2010), 437–458
- A. V. Kumchev, “On Weyl sums over primes and almost primes”, Michigan Math. J., 54:2 (2006), 243–268
- Yuhui Liu, “On a Waring–Goldbach problem involving squares, cubes and biquadrates”, Bull. Korean Math. Soc., 55:6 (2018), 1659–1666
- Xiao Dong Lü, “Waring–Goldbach problem: two squares, two cubes and two biquadrates”, Chinese Ann. Math. Ser. A, 36:2 (2015), 161–174 (in Chinese)
- Reviews in number theory, Reviews reprinted in Mathematical reviews, 1940 through 1972, volumes 1–44 inclusive, v. 1–6, ed. W. J. Leveque, Amer. Math. Soc., Providence, RI, 1974, v+420 pp., v+672 pp., vi+377 pp., vi+582 pp., v+470 pp., ix+410 pp.
- R. C. Vaughan, On the representation of numbers as sums of squares, cubes and fourth powers and on the representation of numbers as sums of powers of primes, Ph.D. thesis, Univ. London, 1970
- Р. Вон, Метод Харди–Литтлвуда, Мир, М., 1985, 184 с.
- T. D. Wooley, “Slim exceptional sets for sums of four squares”, Proc. Lond. Math. Soc. (3), 85:1 (2002), 1–21
- Min Zhang, Jinjiang Li, “Exceptional set for sums of unlike powers of primes”, Taiwanese J. Math., 22:4 (2018), 779–811
- Min Zhang, Jinjiang Li, “Exceptional set for sums of unlike powers of primes (II)”, Ramanujan J., 55:1 (2021), 131–140
- Lilu Zhao, “On the Waring–Goldbach problem for fourth and sixth powers”, Proc. Lond. Math. Soc. (3), 108:6 (2014), 1593–1622
- Li Zhu, “Goldbach–Linnik type problems with unequal powers of primes”, J. Korean Math. Soc., 59:2 (2022), 407–420
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