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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
Citations:
Zbl 1422.11205
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References:
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