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arXiv · 2608.22564

There and back again with Kaprekar 2-cycles

Rebekah Mayne·Quinn Shapiro·Cornelia A. Van Cott
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Abstract

A curious property of integers, first observed by D.~R. Kaprekar in 1949, is as follows: write the digits of a 4-digit number in both descending and ascending order, and then find the positive difference between these integers. Iterating this process on any 4-digit number - except for multiples of 1111 - eventually produces the number 6174. The process ends here, since 6174 is a fixed point of the procedure. In general, if we start with any integer and iterate this process, it either ends at a fixed point (as with 4 digits) or enters a cycle of numbers, called a Kaprekar cycle. In 2011, Dolan classified all fixed points of this process. We study Kaprekar cycles of length 2. We find general properties of numbers in a Kaprekar 2-cycle, and we classify all 2-cycles such that the two integers have the same smallest digit.

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Rebekah Mayne, Quinn Shapiro, Cornelia A. Van Cott. 2026-08-23. There and back again with Kaprekar 2-cycles. https://arxiv.org/abs/2608.22564

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