Two Diophantine Inequalities over Primes with Fractional Power

Let 1 < c < (256) /(119), c not equal 2 and N be a sufficiently large real number. In this paper, we first prove that the Diophantine inequality |p (c)(1)+p (c)(2)+<middle dot> <middle dot> <middle dot>+p (c)(6)-N| < log(-1) N is solvable in primes p1, p2, ... , p6. Moreover, we prove that for almost all R is an element of (N, 2N], the Diophantine inequality |p (c)(1)+ p (c)(2 )+ p (c)(3) - R| < log(-1) N is solvable in primes p1, p2, p3. These results constitute further improvements upon previous results.