The Waring-Goldbach Problems On Short Intervals

The Waring-Goldbach problem seeks to represent a natural number N by sums of k-th. powers of primes, i.e. the solvability of the equation: where j depends onκ.Hua [4] proved that every sufficiently large odd integer can be represented as sums of nine cubes of primes. Hua also proved that for 5≤j≤8, if Ej(z) denote the number of positive integers N up to z satisfying some necessary congruence conditions that can not be represented as sums of j cubes of primes, then Ej(z)《z(logz)-A. where A> 0 is an arbitary constant.Hua's result was improved subsequently, the best result was proved by Kumchcv [5], he proved that Ej(z)《zθj,whereIn this paper we will consider the case that pi take values in short intervals, i.e. We writeDenote by Ej(X,U) the number of N∈[X/2,X] which satisfies the above congruence condition but can not be written as (0.5), in this paper we will prove the following results.Theorem 1.1. For j=5,6,7,8, let U=N4(j+1)15j, then we have whereε＞0 is arbitrary.It follows from this theorem that almost all integers N can be represented as sums of j nearly equal cubes of primes.For qradratic Waring-Goldbach problem, Hua proved that every suffi-ciently large integer satisfying N=5(mod4) can be represented as squares of 5 primes. And he also proved that the number E(z) of integers up to z which can not be written as the sum of four squares of primes is《z(logz)-A, for some A>0.Consider this problem in short intervals.Denote by Ej(X, U) the number of integers N∈[X/2,X] satisfying some necessary congruence conditions and can not be represented as (0.8). Lu and Zhai proved that for U=U4=X21/50, In this paper consider the case that j=3.Let E3(X,U)denote the number of integers N satisfying N≡3(mod 24), 5(?)N,N∈[X/2,X] that can not be written as(0.8).We will prove the following result.Theorem 1.2.For U=U3=X13/30,we have whereε>0 is arbitrary.