Geodesic
On solving of some classes of diophantine equations by the method of identities
News of the Kabardin-Balkar scientific center of RAS, no. 2 (2019), pp. 37-45 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Solving algebraic equations with integer coefficients with more than one unknown integer coefficient is one of the most important problems in number theory. It is known that in the general formulation the problem of describing the set of solutions of Diophantine equations in integers is algorithmically insoluble. Despite the efforts of many generations of mathematicians, there are still no general effective methods for their solving in this area. In this paper, we consider some classes of Diophantine equations, the solution in integers of which is of interest both by itself and, for example, in the process of studying complex discrete systems, in searching for optimal structures in organic chemistry and molecular physics, in decoding of computer algorithms, in cryptography, economics and probability theory. Using the identities and their modifications known in algebra, as well as the method of mathematical induction and the Lagrange method, the work shows an effective method for solving both classical Diophantine equations and some of their generalized variants. As a result, we can get formulas expressing general solutions of Diophantine equations. This method can also be applied to other classes of Diophantine equations.
Mots-clés : Diophantine equations, Pell equation.
Keywords: algebraic identities, mathematical induction method, Lagrange identities 
@article{IZKAB_2019_2_a3,
     author = {F. Kh. Uvizheva and M. Kh. Kalazhokov{\cyra}},
     title = {On solving of some classes of diophantine equations by the method of identities},
     journal = {News of the Kabardin-Balkar scientific center of RAS},
     pages = {37--45},
     year = {2019},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IZKAB_2019_2_a3/}
}
TY  - JOUR
AU  - F. Kh. Uvizheva
AU  - M. Kh. Kalazhokovа
TI  - On solving of some classes of diophantine equations by the method of identities
JO  - News of the Kabardin-Balkar scientific center of RAS
PY  - 2019
SP  - 37
EP  - 45
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/IZKAB_2019_2_a3/
LA  - ru
ID  - IZKAB_2019_2_a3
ER  - 
%0 Journal Article
%A F. Kh. Uvizheva
%A M. Kh. Kalazhokovа
%T On solving of some classes of diophantine equations by the method of identities
%J News of the Kabardin-Balkar scientific center of RAS
%D 2019
%P 37-45
%N 2
%U http://geodesic.mathdoc.fr/item/IZKAB_2019_2_a3/
%G ru
%F IZKAB_2019_2_a3
F. Kh. Uvizheva; M. Kh. Kalazhokovа. On solving of some classes of diophantine equations by the method of identities. News of the Kabardin-Balkar scientific center of RAS, no. 2 (2019), pp. 37-45. http://geodesic.mathdoc.fr/item/IZKAB_2019_2_a3/

[1] I. M. Vinogradov, Osnovy teorii chisel [Fundamentals of number theory], Nauka Publishing House. Main editorial office of physical and mathematical literature, M, 1965 | MR

[2] I. V. Arnold, Teoriya chisel [Number theory], State Pedagogical Publishing House of the People's Commissariat of Education of the RSFSR, M., 1989

[3] A. K. Sushkevich, “Teoriya chisel [Number theory]”, Kharkov: Publishing House of Kharkov University n.a. A.M. Gorky, 1954 | MR

[4] A. O. Gelfond, “Reshenie uravnenij v celyh chislah [Solving equations in integers]”, M, 1952, State Publishing House of technical and theoretical literature, L. | MR

[5] V. Serpinsky, O reshenii uravnenij v celyh chislah [On the solution of equations in integers], State Publishing House of physical and mathematical literature, M., 1961

[6] B. L. Minukhin, Ob odnom metode resheniya diofantovyh uravnenij [On a method for solving diophantine equations] (Mathematical Enlightenment), Issue three, State Publishing House for physical and mathematical lit, M, 1958

[7] S. I. Novoselov, Special'nyj kurs elementarnoj algebry [Special course of elementary algebra], «Higher School» Publishing House, M, 1965

[8] G. Davenport, “Vysshaya arifmetika [Higher arithmetic]”, Nauka Publishing House. Main editorial office of physical and mathematical literature, M, 1965