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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