Sums Of A Prime And Two Squares Of Primes In Short Intervals

Studying the representations of positive integers by powers of primes is the main topic of additive number theory. The circle method of Hardy and Littlewood in combination with the estimations of Vinogradov for exponen-tial sums over primes give an affirmative answer to the problem. In resent years, new ideas of circle method, sieving methods, and exponential sums are incorporated into the aspect, and lots of remarkable advances is appearing.The representation of positive integers by powers of primes in short in-tervals are also studied by many number theorist. For example, the represen-tations of positive integers by three squares of primes in short intervals have been studied by J. Y. Liu, T. Zhan and A. V. Kumchev etc. And in this paper we firstly consider the the representations of positive integers by a prime with two squares of primes in short intervals. Let (?)={n≥1:n≡1,3(mod 6)}, E(N)=#{n∈(?): n≤N,n≠p1+p22+p23 for any primes P1,P2,P3}.We prove that when H≥X0.252, almost all integers n∈(?)∩(X,X+H] can be represented as the sum of a prime and two squares of primes, and this is the meaning of the following theorem:Theorem 1. Let A> 0 be given, if X0.252≤H≤X, then we haveNext We consider the representations of positive integers by the sum of three primes powers respectively with 1,2, k(k≥3). Let Ek(N)=#{n≤N n≠P1+P22+Pk3,k≥3, for any primes p1,p2,p3}. We give an upper bound of Ek(N) by using the circle method. Improving the estimations of exponential sums on the minor arcs is a key problem, then a better result is following. For this purpose we use the method of dividing the minor arcs into two parts, which was developed by X. M. Ren [5], then we can use A. V. Kumchev's estimations of exponential sums and Ren's estimations respectively, and give the estimations on the whole minor arcs. We have the following results:Theorem 2. If k(?)3 is given, Ek(N) is defined as above, then we have whereσ(3)=1/14;σ(k)=2/3×2-k, while k(?)4.