Abstract
Let 1<c<37∕18,c≠2 and N be a sufficiently large real number. In this paper, we prove that, for almost all R∈(N,2N], the Diophantine inequality |p1c+p2c+p3c−R|<log−1N is solvable in primes p1,p2,p3. Moreover, we also investigate the problem of six primes and prove that the Diophantine inequality |p1c+p2c+p3c+p4c+p5c+p6c−N|<log−1N is solvable in primes p1,p2,p3,p4,p5,p6 for sufficiently large real number N.
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