On the sum of a prime and a k-th power of prime in short intervals

摘要

Let H _k denote the set {n{pipe}2{pipe}n,n ? 1(mod p)~? p > 2 with p-1{pipe}k}. We prove that when X ~(11/20(1-1/2k)+ε) ≦ H ≦ X,almost all integers n∈ H _k ∩(X,X+H] can be represented as the sum of a prime and a k-th power of prime for k≧3. Moreover, when X ~(11/20(1-1/k)+ε) ≦ H ≦ X almost all integers n∈(X,X+H] can be represented as the sum of a prime and a k-th power of integer for k≧3.