## On the Waring-Goldbach problem for one square and five cubes in short intervals.(English)Zbl 07361086

Summary: Let $$N$$ be a sufficiently large integer. We prove that almost all sufficiently large even integers $$n\in[N-6U,N+6U]$$ can be represented as $\begin{cases}n=p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^3,\\ \Bigl | p_1^2-\dfrac{N}{6}\Bigr | \leq U,\quad\Bigl | p_i^3-\dfrac{N}{6}\Bigr |\leq U,\quad i=2,3,4,5,6,\end{cases}$ where $$U=N^{1-\delta +\varepsilon}$$ with $$\delta\leq 8/225$$.

### MSC:

 11P05 Waring’s problem and variants 11P32 Goldbach-type theorems; other additive questions involving primes 11P55 Applications of the Hardy-Littlewood method
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### References:

 [1] Cai, Y., The Waring-Goldbach problem: One square and five cubes, Ramanujan J. 34 (2014), 57-72 · Zbl 1304.11121 [2] Hua, L. K., Additive Theory of Prime Numbers, Translations of Mathematical Monographs 13. American Mathematical Society, Providence (1965) · Zbl 0192.39304 [3] Kumchev, A. V., On Weyl sums over primes in short intervals, Number Theory: Arithmetic in Shangri-La Series on Number Theory and Its Applications 8. World Scientific, Hackensack (2013), 116-131 · Zbl 1368.11095 [4] Li, J.; Zhang, M., On the Waring-Goldbach problem for one square and five cubes, Int. J. Number Theory 14 (2018), 2425-2440 · Zbl 1445.11111 [5] Liu, J., On Lagrange’s theorem with prime variables, Q. J. Math. 54 (2003), 453-462 · Zbl 1080.11071 [6] Pan, C.; Pan, C., Goldbach Conjecture, Science Press, Beijing (1992) · Zbl 0849.11080 [7] Sinnadurai, J. S.-C. L., Representation of integers as sums of six cubes and one square, Q. J. Math., Oxf. II. Ser. 16 (1965), 289-296 · Zbl 0144.28101 [8] Stanley, G. K., The representation of a number as the sum of one square and a number of $$k$$-th powers, Proc. Lond. Math. Soc. (2) 31 (1930), 512-553 \99999JFM99999 56.0174.01 · JFM 56.0174.01 [9] Stanley, G. K., The representation of a number as a sum of squares and cubes, J. London Math. Soc. 6 (1931), 194-197 · Zbl 0002.18203 [10] Titchmarsh, E. C., The Theory of the Riemann Zeta-Function, Oxford University Press, New York (1986) · Zbl 0601.10026 [11] Vaughan, R. C., On Waring’s problem for smaller exponents, Proc. Lond. Math. Soc., III. Ser. 52 (1986), 445-463 · Zbl 0601.10035 [12] Vaughan, R. C., On Waring’s problem: One square and five cubes, Q. J. Math., Oxf. II. Ser. 37 (1986), 117-127 · Zbl 0589.10047 [13] Vaughan, R. C., The Hardy-Littlewood Method, Cambridge Tracts in Mathematics 125. Cambridge University Press, Cambridge (1997) · Zbl 0868.11046 [14] Vinogradov, I. M., Elements of Number Theory, Dover Publications, New York (1954) · Zbl 0057.28201 [15] Watson, G. L., On sums of a square and five cubes, J. Lond. Math. Soc., II. Ser. 5 (1972), 215-218 · Zbl 0241.10032 [16] Wooley, T. D., Slim exceptional sets in Waring’s problem: One square and five cubes, Q. J. Math. 53 (2002), 111-118 · Zbl 1015.11049 [17] Zhang, M.; Li, J., Waring-Goldbach problem for unlike powers with almost equal variables, Int. J. Number Theory 14 (2018), 2441-2472 · Zbl 1445.11112 [18] Zhao, L., On the Waring-Goldbach problem for fourth and sixth powers, Proc. Lond. Math. Soc. (3) 108 (2014), 1593-1622 · Zbl 1370.11116
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