On Diophantine Inequalities Concerning Primes

In 1952,Piatetski-Shapiro[13] studied the solvability of the following Dio-phantine inequality|P1c+P2c+…+Prc-N|<ε(0.4)for sufficiently large N,where c>1 is not an integer,ε is a fixed small positive number and p1,…,pr are primes.He established the existence of a number H(c),depending only on c,such that for all sufficiently large real N,(0.4)has solution whenever r≥H(c).Piatetski-Shapiro[13] showed thatand he also proved that H(c)≤5 holds for 1<c<3/2.Later,Zhai and Cao[20],Garaev[7],Zhai and Cao[22],Shi and Liu[16]also investigated this problem and gave some improvements.The best result at present was obtained by Baker and Weingartner[1],who proved that H(c)≤5 holds for 1<c<2.041 and c ≠2.When c is close to 1,by Vinogradov-Goldbach theorem[19],one should expect H(c)≤3.The first result in this direction was obtained by D.I.Tolev[18],who showed that the inequality|p1c+p2c+p3c-N|<ε(0.5)is solvable for 1<c<15/14,where ε=N-(/c)(15/14-c)log9 N.Later,many researchers studied this problem(See[3][4][11][10][2]).The optimal result at present was given by Baker and Weingartner[2],who established that(0.4)was solvable for H(c)≤3 with 1<c<10/9.For the solvability of the Diophantine inequalities(0.4),one expects to get a larger c for fixed r.But it can not always be realised.Thus,in order to obtain larger range of c,we sometimes consider the exceptional set problem.Actually,many mathematicians have investigated this problem.Let Br denote R ∈(N,2N] such that(0.4)is unsolvable.In 1999,Laporta[12]studied the exceptional set of(0.4)with r=2.Laporta proved that,if 1<c<15/14,the inequality|p1c+p2c-R|<ε(0.6)is solvable for all R ∈(N,2N]\B2,whereIn 2003,Zhai and Cao[21]improved Laporta’s result[12],and showed that,for 1<c<43/36,the inequality(0.6)is solvable for all R E(N,2N]\B2,where B2 is defined as above and ε=N1-43/(36c).In this paper,we study the exceptional set of(0.4)with r=3.We will establish the following theorem.Theorem 0.1 Let 1<c<33/16 and c≠2.Then there exists a subset(?)withsuch that for all R ∈(N,2N]\B3,the Diophantine inequality|p1c+p2c+p3c-R|<log-1 Nis solvable in primes p1,p2,p3 for N>No(c)>0.Note that 33/16=2.0625,this range of c can be compared with the result obtained by Baker and Weingartner[2]:1<c<10/9=1.11….We have enlarged the range of c.In this paper,the second problem that we are going to investigate is the solvability of(0.4)with r=6.Our result is the following.Theorem 0.2 Suppose that 1<c<33/16 and c≠2.Then the Diophantine inequality|p1c+p2c+p3c+p4c+p5c+p6c-N|<log-1 Nis solvable in primes p1,p2,p3,p4,p5,p6 for N>N0(c)>0.Theorem 0.2 can be compared with the result of Baker and Weingartner[1],who showed that H(c)≤5 with 1<c<2.041,c≠2.On the base of increasing a prime variable,we get a larger c.