Waring-Goldbach problems for unlike powers with almost equal variables

Abstract

We prove that almost all positive even integers n can be written as n=p_2^2+p_3^3+p_4^4+p_5^5 with \left|p{\displaystyle \frac{k}{k}}-{\displaystyle \frac{N}{\mathrm{4}}}\right|\le N^{\frac{\mathrm{321}}{\mathrm{325}}+ϵ} for 2≤k≤5. Moreover, it is proved that each sufficiently large odd integer N can be represented as N=p_1+p_2^2+p_3^3+p_4^4+p_5^5 with \left|p{\displaystyle \frac{k}{k}}-{\displaystyle \frac{N}{\mathrm{5}}}\right|\le N^{\frac{\mathrm{321}}{\mathrm{325}}+ϵ}for 1≤k≤5.

Key words： Exceptional set ofWaring-Goldbach problem; exponential sum over primes in short intervals; circle method

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