On The Sum Of A Prime And A Kth Power Of Prime In Short Intervals

In this paper, we are concerned with representation of integers by a prime and a kth power of prime, i.e. Define Let Ek(X) denote the number of integers n∈Hk∩[1,X] which cannot be represented as the sum of a prime and a kth power of prime. We will introduce the sieve method to prove the following result.Let k≥3, then for any A> 0, and anyε>0, for X11/20(1-1/k+ε≤H≤X. The implied constant depends at most on k, A andε.This implies that all but O(HL-A) integers n∈Hk∩(X,X+H]can be represented as the sum of a prime and a kth. power of prime forκ> 3.Moreover, we consider representation of integers by a prime and aκ-th power of integer, i.e.Let E'k(X) denote the number of integers n∈[1.X] which cannot be repre-sented as the sum of a prime and aκ-th power of integer. We will obtain a similar result. Let k≥3,then for any A>0,and anyε>0, Ek'(X+H)-Ek'(X)《H(logX)-A for X11/20(1-1/k)+ε≤H≤X.The implied constant depends at most on k,A andε.This implies that all but O(HL-A)integers n∈(X,X+H]can be rep-resented as the sum of a prime and a kth power of integer for k≥3.