On an additive problem of unlike powers in short intervals

Abstract

We prove that almost all positive even integers n can be represented as p ²₂ + p ³₃ + p ⁴₄ + p ⁵₅ with $\left| {p_k^k - {1 \over 4}N} \right| \leqslant {N^{1 - 1/54 + \varepsilon }}$ for 2 ⩽ k ⩽ 5. As a consequence, we show that each sufficiently large odd integer N can be written as p₁ + p ²₂ + p ³₃ + p ⁴₄ + p ⁵₅ with $\left| {p_k^k - {1 \over 5}N} \right| \leqslant {N^{1 - 1/54 + \varepsilon }}$ for 1 ⩽ k ⩽ 5.