О разрешимости и специальных решениях параметрических уравнений Пелля

Nikolai Osipov; Nikita Orlov; Irina Orlova

doi:10.32603/2071-2340-2024-4-7-23

Nikolai Osipov Siberian Federal University, 79 Svobodny pr., 660041, Krasnoyarsk, Russian Federation
Nikita Orlov Lomonosov Moscow State University, GSP-1, Leninskie Gory, 119991, Moscow, Russian Federation
Irina Orlova KSPU named after V. P. Astafyev, 82 Ady Lebedevoy str., 660049, Krasnoyarsk, Russian Federation

DOI: https://doi.org/10.32603/2071-2340-2024-4-7-23

Keywords: Diophantine equations, Pell’s equation.

Abstract

The paper investigates a family of solutions of second-order parametric Pell equations of the general form: x² −mx y + y² = B, where m and B are some parameters. An optimal algorithm for solving such equations is found as an alternative to the traditional method. The proposed method allows not only to solve equations of this type for specific values of the parametersm and B, but also to investigate some equations of this class for solvability
as a whole. In the particular case of equations of a special type, sequences of natural numbers are identified —the parameters of the equation, for which it is totally unsolvable.

Author Biographies

Nikolai Osipov, Siberian Federal University, 79 Svobodny pr., 660041, Krasnoyarsk, Russian Federation

Professor of the Department of Applied Mathematics and Data Analysis of the Siberian Federal University, Krasnoyarsk, nosipov@sfu-kras.ru

Nikita Orlov, Lomonosov Moscow State University, GSP-1, Leninskie Gory, 119991, Moscow, Russian Federation

Student of the Faculty of Mechanics and Mathematics of the Lomonosov Moscow State University, Moscow, nikita.orlov@math.msu.ru

Irina Orlova, KSPU named after V. P. Astafyev, 82 Ady Lebedevoy str., 660049, Krasnoyarsk, Russian Federation

Associate Professor of the Department of Physics, Technology and Teaching Methods of the Krasnoyarsk State Pedagogical University named after V. P. Astafyev, Krasnoyarsk, orlhome@rambler.ru

References

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N. Orlov and N. Osipov, “Positive integers k such that the parametric Pell-type equation x2 −mx y + y2 = −m2 −k has no integer solutions (x, y) for all integers m Ê 1, excluding the cases k ≡ 1(mod 4), k ≡ 3(mod 9), and k ≡ 6(mod 9),” in oeis.org, 2024. [Online]. Availble: https://oeis.org/A371957

N. Orlov and N. Osipov, “Positive integers k ≡ 2(mod4) such that the parametric Pell-type equation x2−mx y+y2 =m2+k has no integer solutions (x, y) for all integerm Ê 1,” in oeis.org, 2024. [Online]. Availble: https://oeis.org/A370721

On the Solvability and Special Solutions of Parametric Pell Equations

Abstract

Author Biographies

References