The Kaprekar transformation \(T_{b,n}\) (named after the Indian mathematician D. R. Kaprekar) takes an integer \(x\) in base \(b\) with \(n\) digits as an input, rearranges the digits of \(x\) in descending and ascending orders, and outputs the difference of the newly generated integers. A. Yamagami [J. Number Theory 185, 257–280 (2018; Zbl 1431.11016)] obtained a complete description of the Kaprekar transformation in base \(2\) (fixed points, cycles etc.), and A. Yamagami and Y. Matsui [JP J. Algebra Number Theory Appl. 40, No. 6, 957–1028 (2018; Zbl 1429.11019)] treated the much more involved case of the transformation in base \(3\).
The aim of the paper under review is to extend their result to higher odd bases. The structure of the cycles (primitive proper, primitive non-proper, and general cycles) has a similar appeal in all odd bases, but new features arise depending on whether the base is congruent to \(1\) or \(3\) modulo \(4\). The authors examine in detail the cycles and fixed points in bases \(5\) and \(7\), and comment on several results for higher bases where the transformation becomes even more complex, such that the existence of arbitrarily long cycles is related to the Artin’s conjecture on primitive roots.