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A ternary Diophantine inequality by primes with one of the form \(p=x^2+y^2+1\)

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Abstract

In this paper we solve the ternary Piatetski-Shapiro inequality with prime numbers of a special form. More precisely we show that, for any fixed \(1<c<\frac{427}{400}\), every sufficiently large positive number N and a small constant \(\varepsilon >0\), the Diophantine inequality

$$\begin{aligned} \mid p_1^c+p_2^c+p_3^c-N\mid <\varepsilon \end{aligned}$$

has a solution in prime numbers \(p_1,\,p_2,\,p_3\), such that \(p_1=x^2 + y^2 +1\). For this purpose we establish a new Bombieri–Vinogradov type result for exponential sums over primes.

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  1. Faculty of Applied Mathematics and Informatics, Technical University of Sofia, Blvd. St. Kliment Ohridski 8, 1756, Sofia, Bulgaria

    Stoyan Dimitrov

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  1. Stoyan Dimitrov
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Dimitrov, S. A ternary Diophantine inequality by primes with one of the form \(p=x^2+y^2+1\). Ramanujan J 59, 571–607 (2022). https://doi.org/10.1007/s11139-021-00545-1

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  • DOI: https://doi.org/10.1007/s11139-021-00545-1

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