arXiv subjects
Publications and source records attributed to Fernando Nuez.
In general terms, we establish algebraic relations that numbers must satisfy in order for their images to match after one or several transformations. Some groups associated with these relationships are identified, such as the Klein group. Such equivalences are applied to numbers of 2, 3, 4 or 5 digits. The relationship between cycles and the transformation trees structure are analyzed.
The Kaprekar transformation is uniquely determined by parameters based on differences between symmetric values in the ordered numeric sequence. The parametric analysis of the iteration of the process requires developing Ki functions that provide the parameters of the transformed number. This is the objective of this work. As application examples we use these functions to deduce constants and cycles. But its main utility is to study the algebraic architecture of the transformation trees that we develop in the second part of this work.
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.