The Exceptional Sets Of Waring-Goldbach Problem For Unlike Powers

As one of the main problems of additive prime number theory,the study of the Waring-Goldbach problem is of great significance.With the development of the study on the Waring-Goldbach problem,people are becoming more and more interested in Waring-Goldbach problem for unlike powers.The Waring-Goldbach problem for unlike powers is to study the possibility of representing the integer n as where k1,k2,…,kr are natural numbers satisfying 2≤k1≤k2≤…≤kr,p1,p2,…,pr are prime numbers.In this paper,combining with the circle method created by Hardy and Littlewood,we study the exceptional sets of two Waring-Goldbach problems for unlike powers:(1)the exceptional sets of expressing an even number n as the sum of the square of two prime numbers,the fourth power of a prime number and the κ-th power of a prime number(κ≥4),that is where p1,p2,p3,p4 are prime numbers.(2)the exceptional sets of expressing an even number n as the sum of the square of a prime number,the cube of a prime number,the fourth power of a prime number and the κ-th power of a prime number(κ>5),that is where p1,p2,p3,p4 are prime numbers.In order to obtain the exception sets,we use the Dirichlet approximation theorem to divide the studied interval[0,1]into the main interval and its complementary interval in the process of using the circle method.For the estimation of the generating function on the main interval,the methed of expanding the main interval in Jianya Liu[16]is used to get a lower bound.In addition,while dealing with the complementary interval,the integral on the complementary interval is also divided into two parts to be estimated,and an upper bound of the integral on the complementary interval is obtained by using the estimation of the exponential sum and the estimation of the mean value in Zhenzhen Feng[30].In Chapter 1,we mainly introduce the background and research progress of Waring problem and Waring-Goldbach problem,and gives the basic conclusions of this paper.In Chapter 2,we give the preliminary knowledge and necessary lemmas needed in the proof of the theorems.In Chapter 3,we give the proofs of the conclusions of this paper.