On the asymptotics of representations by a sum of a pair of integers by a sum and a linear form with a congruential condition of a special form
https://doi.org/10.22405/2226-8383-2025-26-1-62-75
Abstract
In the work, asymptotic formulas with a remainder term are obtained for the number of representations of a pair of integers 𝑚 and 𝑛, respectively, as a sum of 𝑠 ⩾ 5 variables, and each solution of such a Diophantine system satisfies the congruential a condition of a special type, associated in a certain way with a linear form.
Asymptotic formulas with a remainder term for the number of solutions of such a Diophantine system are derived for 𝑁 → ∞, where 𝑁 = Δ𝑚 − 𝑛2 and Δ equals the sum of the squares on the coefficients on the linear form. In addition, two-sided lower and upper bounds are obtained for a special series of Diophantine system under study based on the upper bound based on formulas for the number of solutions of
a congruence of the second degree modulo the power 𝑥21+ . . . + 𝑥2𝑠 ≡ 𝑎 (mod 𝑝𝑘) of the prime number, where 𝑎 is natural number.
This work is a continuation of a previous study, relating to the case of an even number of variables.
About the Authors
Urusbi Mukhamedovich Pachev
Berbekov Kabardino-Balkarian State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
Azamat Khasanovich Kodzokov
Berbekov Kabardino-Balkarian State University
Russian Federation
Russian Federation
Mariana Malilovna Isakova
Berbekov Kabardino-Balkarian State University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
Marina Sefovna Nirova
Berbekov Kabardino-Balkarian State University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
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Review
For citations:
Pachev U.M., Kodzokov A.Kh., Isakova M.M., Nirova M.S. On the asymptotics of representations by a sum of a pair of integers by a sum and a linear form with a congruential condition of a special form. Chebyshevskii Sbornik. 2025;26(1):62-75. (In Russ.) https://doi.org/10.22405/2226-8383-2025-26-1-62-75
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