Cycle Structure of Kaprekar Transformations on 5-Digit Integers

This paper studies the Kaprekar transformation on 5-digit integers. Unlike the classical 4-digit case (which converges to 6174), the 5-digit system exhibits periodic cycles rather than a fixed point. We prove that all iterates are divisible by 9 and analyze the system as a finite dynamical process. Computational experiments reveal stable cycles of lengths 2 and 4, and we conjecture that all valid initial values eventually enter these cycles. We also explore structural properties such as permutation invariance and reduction to digit differences.