The connection between the linear Diophantine equations solutions under the actions of the symmetric group and the automoprhism group of the integers
https://doi.org/10.22405/2226-8383-2025-26-5-259-279
Abstract
This article shows that there is a direct connection between the actions of the symmetric
group 𝑆𝑛 on the set of linear Diophantine equations and on the set of their solutions.
Thus, it was found that for coefficients rearrangement in a linear Diophantine equation the
coordinates of its general solution vector are rearranged in the same order, and for variables rearrangement we get the reverse order of the vector coordinates.
There is a similar connection between the actions of the group of the automorphisms of the
integers on the set of linear Diophantine equations and their solutions: if on changes the signs of some coefficients in the equation, then the signs of the corresponding coordinates of the general solution vector are changed too.
We also obtained results on the connection of different actions of the symmetric group on
the set of linear Diophantine equations. For example, rearrangement of the equation coefficients is equivalent to rearrangement of its variables in the reverse order.
Established connections between the actions make it possible to quickly find solutions of an
entire class of linear diophantine equations from a solution of only one of its elements. In turn, studying the actions of other groups on the given set and the connections generated by them would expand this class.
About the Authors
Ivan Sergeevich Chistov
Moscow Pedagogical State University
Russian Federation
Russian Federation
student
Liliya Mikhailovna Tsybulya
Moscow Pedagogical State University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
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Review
For citations:
Chistov I.S., Tsybulya L.M. The connection between the linear Diophantine equations solutions under the actions of the symmetric group and the automoprhism group of the integers. Chebyshevskii Sbornik. 2025;26(5):259-279. (In Russ.) https://doi.org/10.22405/2226-8383-2025-26-5-259-279
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