Kaprekar´s transformations. Part I - theoretical discussion

The paper is devoted to discussion of the minimal cycles of the so called Kaprekar´s transformations and some of its generalizations. The considered transformations are the self-maps of the sets of natural numbers possessing n digits in their decimal expansions. In the paper there are introduced several new characteristics of such maps, among others, the ones connected with the Sharkovsky´s theorem and with the Erdös-Szekeres theorem concerning the monotonic subsequences. Because of the size the study is divided into two parts. Part I includes the considerations of strictly theoretical nature resulting from the definition of Kaprekar´s transformations. We find here all the minimal orbits of Kaprekar´s transformations T_n, for n = 3,..., 7. Moreover, we define many different generalizations of the Kaprekar´s transformations and we discuss their minimal orbits for the selected cases. In Part II (ibidem), which is a continuation of the current paper, the theoretical discussion will be supported by the numerical observations. For example, we notice there that each fixed point, familiar to us, of any Kaprekar´s transformation generates an infinite sequence of fixed points of the other Kaprekar´s transformations. The observed facts concern also several generalizations of the Kaprekar´s transformations defined in Part I.