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On Diophantine problems with mixed powers of primes. (English) Zbl 1422.11205
Summary: Let \(k\) be an integer with \(k\geq 3\) and \(\varepsilon > 0\). Let \(s(k)=[{(k+1)}/{2}]\) and
\[ \sigma(k)=\min\bigl(2^{s(k)-1},\tfrac{1}{2}s(k)(s(k)+1)\bigr). \]
Suppose that \(\lambda_1,\lambda_2,\lambda_3\) are non-zero real numbers, not all negative, and \(\lambda_1/\lambda_2\) is irrational and algebraic. Let \(\mathcal{V}\) be a well-spaced sequence and \(\delta > 0\). We prove that
\[ E(\mathcal{V},X,\delta) \ll X^{1-1/(8\sigma(k))+2\delta+\varepsilon}, \]
where \(E(\mathcal{V},X,\delta)\) denotes the number of \(v\in \mathcal{V}\) with \(1\le v\le X\) such that the inequality
\[ |\lambda_1p_1^2+\lambda_2p_2^2+\lambda_3p_3^k-v| < v^{-\delta}\]
has no solution in primes \(p_1,p_2,p_3\). Furthermore, suppose that \(\lambda_1,\lambda_2,\lambda_3,\lambda_4,\lambda_5\) are non-zero real numbers, not all of the same sign, \(\lambda_1/\lambda_2\) is irrational and \(\varpi\) is a real number. We prove that there are infinitely many solutions in primes \(p_j\) to the inequality
\[ |\lambda_1p_1^2+\lambda_2p_2^2+\lambda_3p_3^2+\lambda_4p_4^2+\lambda_5p_5^k+\varpi| < (\max p_j)^{-{1}/{(8\sigma(k))}+\varepsilon}. \]
This gives an improvement of an earlier result. (18 Refs.)

MSC:
11P32 Goldbach-type theorems; other additive questions involving primes
11D75 Diophantine inequalities
11P55 Applications of the Hardy-Littlewood method
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