## The exceptional set for sums of unlike powers of primes.(English)Zbl 1302.11080

Let $$E(N)$$ be the number of positive even integers of size at most $$N$$ which cannot be represented in the form $$p_1^2+p_2^3+p_3^4+p_4^5$$, where $$p_i$$, $$1\leq i\leq 4$$, are prime numbers. The author proves an upper bound for $$E(N)$$ of the form $$E(N)\ll N^{15/16+\varepsilon}$$. This improves on earlier works of Pracher, Ren and Tsang and most recently one of C. Bauer [Rocky Mt. J. Math. 38, No. 4, 1073–1090 (2008; Zbl 1232.11101)].
The proof is based on an application of the Hardy-Littlewood circle method and uses on the minor arcs mean value estimates introduced in earlier work of the author [Proc. London Math. Soc. 108, No. 6, 1593–1622 (2014; Zbl 1370.11116)].

### MSC:

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

### Keywords:

Waring-Goldbach problem; exceptional set; circle method

### Citations:

Zbl 1232.11101; Zbl 1370.11116
Full Text:

### References:

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