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A Diophantine inequality with four squares and one $$k$$th power of primes. (English) Zbl 07088789
Summary: Let $$k\geq 5$$ be an odd integer and $$\eta$$ be any given real number. We prove that if $$\lambda_1$$, $$\lambda_2$$, $$\lambda_3$$, $$\lambda_4$$, $$\mu$$ are nonzero real numbers, not all of the same sign, and $$\lambda_1/\lambda_2$$ is irrational, then for any real number $$\sigma$$ with $$0<\sigma<1/(8\vartheta(k))$$, the inequality $|\lambda_1p_1^2+\lambda_2p_2^2+\lambda_3p_3^2+\lambda_4p_4^2+\mu p_5^k+\eta|<\Bigl(\max_{1\leq j\leq 5}p_j\Bigr)^{-\sigma}$ has infinitely many solutions in prime variables $$p_1,p_2,\cdots,p_5$$, where $$\vartheta(k)=3\times 2^{(k-5)/2}$$ for $$k=5,7,9$$ and $$\vartheta(k)=[(k^2+2k+5)/8]$$ for odd integer $$k$$ with $$k\geq 11$$. This improves a recent result in W. Ge and T. Wang [Acta Arith. 182, No. 2, 183–199 (2018; Zbl 1422.11205)].
##### MSC:
 11D75 Diophantine inequalities 11P55 Applications of the Hardy-Littlewood method
##### Keywords:
Diophantine inequalities; Davenport-Heilbronn method; prime
Zbl 1422.11205
Full Text:
##### References:
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