On Sum Of Four Prime Squares And K-TH Powr Of Prime In Short Intervals

In the additive theory of prime numbers, one studies the representation of positive integers by powers of primes. In 1937, Vinogradov [1] proved that each sufficiently large integer odd N can be written as the sum of three primes, which is known as the famous three prime theorem. For the nonlinear case, Hua [2] proved that each large integer congruent to 5 modulo 24 can be written as the sum of five squares of primes and each large odd integer can be written as the sum of nine cubes of primes. And for higher or hybrid power case, there are many results referring to [3], [4], [5].On the other hand, the additive theory of prime numbers with certain conditions appeals to many researchers to study, in which to restrict the variables in short intervals is frequently considered. The problems are called the additive theory of prime numbers in the short intervals. For example, basing on the work of Pan and Pan([7], [8], [26]), Zhan [9] proved that every large odd integer N can be represented aswhere i = 1,2,3. Baker and Harman [10] had reduced the exponent 5/8 to 4/7. There exist many results about the sum of five square primes in the short intervals.First of all, assuming the Generalized Riemann Hypothesis, Liu and Zhan [11] considered short interval version of this problem. They showed the following result for any sufficiently large integer N satisfying N≡5 (mod 24) can be written aswhere U=(?).Later on, in 1998, Liu and Zhan [12] found a new approach to treat the enlarged major arcs in the Waring-Glodbach problem, in which the possible existence of the Siegel zero does not have a special influence, and hence the Deuring-Heilbronn phenomenon can be avoided. Due to this approach, they obtained that (1.2) is true for U = (?). This approach has been successfully applied to a number of additive problems concerning primes. Recently Bauer [13] used the approach mentioned above and showed that U = (?). In 2006, Bauer and Wang [14] obtained U =(?). Liu Lu and Zhan in [17] obtain the U = (?) .In this paper , we study the generalization of the problem. Hua in [2] proved that every sufficiently large integer N satisfying the congruence conditioncan be written asOur goal is to study this problem in short intervals. The case k = 1 was studied by L(?) in [21]. Many results about k = 2 have been worked out by different authors.We shall prove the following theorem,Theorem For each sufficiently large integer N as in (0.1), k≥4 and K = 2^k-1, the equationhas solutions.