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)].


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


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