Waring-Goldbach Problem For One Prime Power And Four Prime Cubes

Goldbach’s problem is one of the most important problem in the whole number theory circle.Odd Goldbach’s conjecture has been proved to be successful,and the proof of even Goldbach’s conjecture is still working hard.In addition,it is necessary to mention the Waring-Goldbach problem,which is about that natural numbers can be expressed as the sum of the powers of several prime numbers.In 2019,Feng and Ma’s article Waring-Goldbach Problem For Unlike Powers gave the asymptotic formula and estimation results of Waring-Goldbach problem with power 3 and power k.Based on this understanding,Recently,Feng and Ma considered the exception set E（k,N） of the Waring-Goldbach problem as follows.n=p₁^k+p₂³+p₃³+p₄³+p₅³ The corresponding estimate was E（k,N）<<N^{1-θ（k）+ε}.Here （?）This paper starts from the perspective of the short interval,using the knowledge of the circle method in analytic number theory and some of the methods employed in the work of A.Languasco and A.Zaccagnini to improve the above results.In addition,we also study the above problem when the Riemann conjecture holds in this paper and get a better asymptotic formula.