Symmetries in generalized Kaprekar routine

TL;DR

This paper explores the algebraic and group-theoretic properties of the generalized Kaprekar routine, analyzing invariance, cycles, and equivalence classes through parametric functions and binary relations.

Contribution

It introduces a new methodology for analyzing the structure of the generalized Kaprekar routine using algebraic relations, group theory, and parametric transformations.

Findings

Identifies algebraic invariance conditions in the generalized Kaprekar routine

Analyzes the group structure, including Klein group relations

Develops a parametric framework for understanding cycles and equivalence classes

Abstract

Algebraic relations are established that determine the invariance of the transformed number after several transformations. The restrictions that determine the group structure of these relationships are analyzed, as is the case of the Klein group. Parametric Kr functions associated with the existence of cycles are presented, as well as the role of the number of their links in the grouping of numbers in higher order equivalence classes. For this we have developed a methodology based on binary equivalence relations and the complete parameterization of the Kaprekar routine using Ki functions of parametric transformation.