# Vestnik КRAUNC. Fiz.-Mat. nauki. 2024. vol. 46. no. 1. P. 9-21. ISSN 2079-6641

**MATHEMATICS**

https://doi.org/10.26117/2079-6641-2024-46-1-9-21

Research Article

Full text in Russian

MSC 34A99

**On One Way to Solve Linear Equations Over a Euclidean Ring**

**U. M. Pachev^\ast, A. Kh. Kodzokov, A. G. Ezaova, A. A. Tokbaeva, Z. Kh. Guchaeva**

Kabardino-Balkarian State University named after H.M. Berbekov, 360004, Nalchick, Chernyshevskogo str., 173, Russia

Abstract. Linear equations, i.e. Equations of the first degree, as well as systems of such equations, receive much attention both in algebra and in number theory. Of greatest interest is the case of such equations with integer coefficients, and in this case they need to be solved in integers. Such equations with the specified conditions are called linear Diophantine equations. Euler also considered ways to solve linear Diophantine equations with two unknowns, and one of these methods was based on the use of the Euclid algorithm. Another method for solving such equations, based on continued fractions, was also used by Lagrange. Euler’s method turned out to be more convenient and promising than the method of continued fractions. In this paper, we consider one new method for solving linear equations over a Euclidean ring, based on comparisons over suitable moduli. The previously known matrix method for solving such equations with an increasing number of unknowns is quite cumbersome due to the fact that it is associated with finding the inverses of unimodular integer matrices. Essential in our method of solving linear equations over a Euclidean ring is the use of the Euclidean algorithm and the linear GCD representation of elements in the Euclidean ring. The theorem proved in the work is applied to finding a solution to a linear equation in three unknowns over a ring of Gaussian integers, which, as is known, is a Euclidean ring. In conclusion, comments are made on possible ways of further development of the presented research.

*Key words: linear equation, Euclidean ring, Euclidean norm, Gaussian integers, module congruences.*

Received: 30.11.2023; Revised: 10.01.2024; Accepted: 18.01.2024; First online: 07.03.2024

**For citation.** Pachev U. M. et al. On one way to solve linear equations over a Euclidean ring. Vestnik KRAUNC. Fiz.-mat. nauki. 2024, 46: 1, 9-21. EDN: CZKZBA. https://doi.org/10.26117/2079-6641-2024-46-1-9-21.

**Funding.** The work was not carried out within the framework of funds

**Competing interests.** There are no conflicts of interest regarding authorship and publication.

**Contribution and Responsibility.** All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

^\astCorrespondence: E-mail: urusbi@rambler.ru

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Pachev U. M. et al., 2024

© Institute of Cosmophysical Research and Radio Wave Propagation, 2024 (original layout, design, compilation)

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**Information about authors**

**Pachev Urusbi Muhamedovich** – D. Sci. (Phys. & Math.), Professor, Department of Algebra and Differential Equations, Institute of Physics and Mathematics, Kabardino-Balkarian State University, Nalchik, Russia, ORCID 0009-0002-8362-6174.

**Kodzokov Azamat Khasanovich **– Senior Lecturer, Department of Algebra and Differential Equations, Institute of Physics and Mathematics, Kabardino-Balkarian State University, Nalchik, Russia, ORCID 0009-0007-3431-1228.

**Ezaova Alena Georgievna** – Ph.D. (Phys. & Math. Sci.), Department of Algebra and Differential Equations, Institute of Physics and Mathematics, Kabardino-Balkarian State University, Nalchik, Russia, ORCID 0009-0004-8691-0706.

**Tokbaeva Al’bina Aniuarovna** – Ph.D. (Phys. & Math. Sci.), Department of Algebra and Differential Equations, Institute of Physics and Mathematics, Kabardino-Balkarian State University, Nalchik, Russia, ORCID 0009-0007-4926-4452.

**Guchaeva Zera Hamidbievna** – Senior Lecturer, Department of Algebra and Differential Equations, Institute of Physics and Mathematics, Kabardino-Balkarian State University, Nalchik, Russia, ORCID 0009-0000-9777-4018.