## An improvement on Waring-Goldbach problem for unlike powers.(English)Zbl 1274.11154

Summary: Let $$p_i$$ be prime numbers. In this paper, it is proved that for any integer $$k\geqq 5$$, with at most $$O\big(N^{1-\frac{1}{3k\times 2^{k-2}}+\varepsilon}\big)$$ exceptions, all positive even integers up to $$N$$ can be expressed in the form $$p^2_2+ p^3_3+p^5_5+p^k_k$$. This improves the result $$O\big(\frac{N}{\log^c N}\big)$$ for some $$c > 0$$ due to M.-G. Lu and Z. Shan [Kexue Tongbao, Foreign. Lang. Ed. 27, 246–250 (1982; Zbl 0497.10037); J. China Univ. Sci. Tech., Suppl. I, 1–8 (Chinese)(1982)], and it is a generalization for a series of results of X.-M. Ren and K.-M. Tsang [Acta Math. Sin., Engl. Ser. 23, No. 2, 265–280 (2007; Zbl 1128.11043), Acta Math. Sin., Chin. Ser. 50, No. 1, 175–182 (2007; Zbl 1121.11312)], and C. Bauer [Rocky Mt. J. Math. 38, No. 4, 1073–1090 (2008; Zbl 1232.11101), a.o.] for the problem in the form $$p^2_2 + p^3_3+p^4_4 +p^5_5$$. This method can also be used for some other similar forms.

### MSC:

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

### Keywords:

Waring-Goldbach problem; circle method; exceptional set

### Citations:

Zbl 0497.10037; Zbl 1128.11043; Zbl 1121.11312; Zbl 1232.11101
Full Text:

### References:

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