On the diophantine inequalities with prime numbers
https://doi.org/10.22405/2226-8383-2023-24-4-325-334
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| ⩽ 𝐻 in primes 𝑝 a lower bound for 𝐻 ⩾ 𝑁^(21/40+𝜀), where 𝜀 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| ⩽ 𝐻 in prime numbers 𝑝_1, 𝑝_2 for 𝐻 ⩾ 𝑁^(7/80+𝜀.)
Also based on the density technique, we prove a lower bound for the number of solutions of the inequality |𝑝^2_1+ 𝑝^2_2+ 𝑝_3 − 𝑁|⩽ 𝐻 in prime numbers 𝑝_1, 𝑝_2 and 𝑝_3 for 𝐻 ⩾ 𝑁^(7/72+𝜀).
About the Authors
Dmitry Victorovich Goryashin
Lomonosov Moscow State University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences, associate professor
Sergei Alexandrovich Gritsenko
Lomonosov Moscow State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
References
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Review
For citations:
Goryashin D.V., Gritsenko S.A. On the diophantine inequalities with prime numbers. Chebyshevskii Sbornik. 2023;24(4):325-334. (In Russ.) https://doi.org/10.22405/2226-8383-2023-24-4-325-334
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