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Chebyshevskii Sbornik, 2023, Volume 24, Issue 4, Pages 325–334
DOI: https://doi.org/10.22405/2226-8383-2023-24-4-325-334
(Mi cheb1361)
 

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On the diophantine inequalities with prime numbers

D. V. Goryashin, S. A. Gritsenko

Lomonosov Moscow State University, Department of mathematics and mechanics (Moscow)
References:
Abstract: The article deals with two problems of approximating a given positive number $N$ by the sum of two primes, and by the sum of a prime and two squares of primes.
In 2001, R. Baker, G. Harman, and J. Pintz proved for the number of solutions of the inequality $|p-N|\leqslant H$ in primes $p$ a lower bound for $H\geqslant N^{21/40+\varepsilon}$, where $\varepsilon$ is an arbitrarily small positive number. Using this result and the density technique, in this paper we prove a lower bound for the number of solutions of the inequality $|p_1+p_2-N| \leqslant H$ in prime numbers $p_1$, $p_2$ for $H\geqslant N^{7/80+\varepsilon}$.
Also based on the density technique, we prove a lower bound for the number of solutions of the inequality $\left|p_1^2+p_2^2+p_3-N\right| \leqslant H$ in prime numbers $p_1$, $p_2$ and $p_3$ for $H\geqslant N^{7/72+\varepsilon}$.
Keywords: diophantine inequalities, prime numbers, density theorems.
Received: 18.08.2023
Accepted: 11.12.2023
Document Type: Article
UDC: 511.3
Language: Russian
Citation: D. V. Goryashin, S. A. Gritsenko, “On the diophantine inequalities with prime numbers”, Chebyshevskii Sb., 24:4 (2023), 325–334
Citation in format AMSBIB
\Bibitem{GorGri23}
\by D.~V.~Goryashin, S.~A.~Gritsenko
\paper On the diophantine inequalities with prime numbers
\jour Chebyshevskii Sb.
\yr 2023
\vol 24
\issue 4
\pages 325--334
\mathnet{http://mi.mathnet.ru/cheb1361}
\crossref{https://doi.org/10.22405/2226-8383-2023-24-4-325-334}
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