On the diophantine inequalities with prime numbers
Čebyševskij sbornik, Tome 24 (2023) no. 4, pp. 325-334
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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.
@article{CHEB_2023_24_4_a18,
author = {D. V. Goryashin and S. A. Gritsenko},
title = {On the diophantine inequalities with prime numbers},
journal = {\v{C}eby\v{s}evskij sbornik},
pages = {325--334},
publisher = {mathdoc},
volume = {24},
number = {4},
year = {2023},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/CHEB_2023_24_4_a18/}
}
D. V. Goryashin; S. A. Gritsenko. On the diophantine inequalities with prime numbers. Čebyševskij sbornik, Tome 24 (2023) no. 4, pp. 325-334. http://geodesic.mathdoc.fr/item/CHEB_2023_24_4_a18/