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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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