The Sums Of One Prime And Two Squares Of Primes In Short Intervals

The Waring-Goldbach problem is an important question of additive the-ory of prime numbers,with the efforts of modern mathematicians,a series of significant results were obtained.The Waring-Goldbach problem seeks to find a positive integer s as small as possible,such that every positive integer n satisfying necessary congruences can be represented as n=p1k+p2k+...+psk,where s is dependent on k and pi,p2,...,ps are primes.When k = 1,s = 3,it’s the odd Goldbach conjecture(every sufficiently large positive odd integer can be represented by sums of three primes),which is also called the Three prime number theorem and was proved by Vinogradov[1]in 1937 with the help of analytic methods.When k= 1,s=2,its’s the even Goldbach conjecture which has not been proved so far.The main work of this paper is to study a lower exponent question of the mixed power Waring-Goldbach problem using the method in studying the mixed power Waring-Goldbach problem,which is the solutions of the following equation n=p1+p22+p32.The main aim of this paper is to show that a sufficiently large integer N and n can be almost written as a sum of a prime and two squares of primes in the short intervals[N,N + H]of length H ≥N8/33+∈,where n subject to the necessary congruence conditions n≡1(mod 2),n(?)2(mod 3).