Research On Goldbach-Linnik Type Problem

Number theory is an ancient discipline with a history of more than two thousand years.There are many conjectures in it.Every conjecture shines with the wisdom of predecessors and inspires generations of researchers on number theory to explore and pursue.In this paper,we consider the following famous conjectures in number theory by circle method,Conjecture 1:Every sufficiently large even integer is a sum of eight cubes of primes;Conjecture 2:Every sufficiently large even integer is a sum of two prime squares and four prime cubes.Based on Linnik’s idea and method of dealing with Goldbach’s conjecture,the above two conjectures are studied,which are the following two types of Goldbach-Linnik problems:(?)Obviously,as for(A),when k=0,the conjecture 1 holds;as for(B),when k=0,the conjecture 1 is true.Hence,we will focus on the value of k.We will do whatever we can to sharpen the value of k so that k=0.And our final goal is to prove these two conjectures.As for(A),the best result so far is k=341.In this paper,we improve the value of k to 204;As for(B),the best result so far is k=211.In this paper,we improve the value of k to 43.