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Diophantine approximation with four squares and one $$k$$th power of primes. (English) Zbl 1396.11059
Summary: Let $$k$$ be an integer with $$k\geq 3$$ and $$\eta$$ be any real number. Suppose that $$\lambda_1,\lambda_2,\lambda_3,\lambda_4,\mu$$ are non-zero real numbers, not all of the same sign and $$\lambda_1/\lambda_2$$ is irrational. It is proved that the inequality $|\lambda_1p_1^2+\lambda_2p_2^2+\lambda_3p_3^2+\lambda_4p_4^2+\mu p_5^k+\eta|<(\max p_j)^{-\sigma}$ has infinitely many solutions in prime variables $$p_1,p_2,\dots,p_5$$, where $$0<\sigma<\frac{1}{16}$$ for $$k=3$$, $$0<\sigma<\frac{5}{3k2^k}$$ for $$4\leq k\leq 5$$ and $$0<\sigma<\frac{40}{21k2^k}$$ for $$k\geq 6$$. This gives an improvement of an earlier result [W. Li and W. Wang, J. Math. Sci. Adv. Appl. 6, No. 1, 1–16 (2010; Zbl 1238.11047)].

##### MSC:
 11D75 Diophantine inequalities 11P32 Goldbach-type theorems; other additive questions involving primes
##### Keywords:
Diophantine inequalities; Davenport-Heilbronn method; prime
Zbl 1238.11047
Full Text:
##### References:
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