A System Of Two Diophantine Inequalities With Primes

The Well-known Waring-Goldbach problem is an important subject in analytic number theory.Similar to Waring-Goldbach problem,we consider the solvability of the system of two Diophantine inequalities.The main contents of this paper are as follows.Let 1<d<c<128/119 and 1<?<?<61 d/c.In this paper,we prove that there exist positive real numbers N1(0)and N2(0)depending on c,d,?,? such that for all real numbers N1>N1(0),N2>N2(0)and ??N2/N1d/c ??,the system of two Diophantine inequalities|p1c+p2c+…+p6c-N1|<N1-(1/c)(128/119-c)log109N1,|p1d+p2d+…+p6d-N2|<N2-(1/d)(128/119-d)log109N2 is solvable in prime variables p1,p2,…,p6.In Chapter 1,the research background,current situation and main results of the system of two Diophantine inequalities are described.In Chapter 2,we introduce the outline of the proof of Theorem 1.It is sufficient to prove that|D1|>>?1?2X6-c-d,|D2|<<?1?2X6-c-d/log X,|D3|<<1.In Chapter 3,we introduce some auxiliary lemmas.In Chapter 4,we get the estimate of D1,D2 and D3.