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 ²₂ + p ³₃ + p ⁴₄ + p ⁵₅ with $\left| {p_k^k - \tfrac{N} {4}} \right| \leqslant N^{\tfrac{{321}} {{325}} + \varepsilon }$ for 2 ⩽ k ⩽ 5. Moreover, it is proved that each sufficiently large odd integer N can be represented as N = p _p + p ²₂ + p ³₃ + p ⁴₄ + p ⁵₅ with $\left| {p_k^k - \tfrac{N} {5}} \right| \leqslant N^{\tfrac{{321}} {{325}} + \varepsilon }$ for 1 ⩽ k ⩽ 5.