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A Diophantine inequality with prime variables and mixed power. (Chinese. English summary) Zbl 1340.11055
Summary: Let \(k\) be an integer with \(k\geq 2\) and \(\eta\) be any real number. Suppose that \(\lambda_1,\lambda_2,\lambda_3\) are nonzero real numbers, not all of the same sign and \(\lambda_1/\lambda_2\) is irrational. It is proved that the inequality \(|\lambda_1p_1+\lambda_2p_2+\lambda_3p^k_3+\eta|<(\max p_j)^{-\sigma}\) has infinitely many solutions in prime variables \(p_1,p_2,p_3\), where \(0<\sigma<\frac 1{2(2^{k+1}+1)}\) for \(2\leq k\leq 3\), \(0<\sigma <\frac 5{6k2^k}\) for \(4\leq k\leq 5\), and \(0<\sigma <\frac{20}{21k2^k}\) for \(k\geq 6\).

MSC:
11J25 Diophantine inequalities
11P32 Goldbach-type theorems; other additive questions involving primes
11D75 Diophantine inequalities
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