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Four-digit Kaprekar dynamics in odd bases

This paper establishes that in every odd base greater than three, the four-digit Kaprekar routine exhibits a rigid structure where all nonconstant orbits quickly enter a specific triangular region and behave like projective doubling, thereby providing a complete finite classification of all terminal cycles, their lengths, and their counts.

Original authors: Evan Chen, Ken Ono, Richard E. Schwartz, Dinesh S. Thakur

Published 2026-06-19

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CC0 1.0

Original authors: Evan Chen, Ken Ono, Richard E. Schwartz, Dinesh S. Thakur

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a magic number machine. You feed it a four-digit number, and it performs a very specific, repetitive trick:

Sort: It rearranges the digits from biggest to smallest.
Reverse: It rearranges the same digits from smallest to biggest.
Subtract: It subtracts the second number from the first.
Repeat: It takes the result and does the whole process again.

In our everyday world (Base 10), if you start with almost any four-digit number, this machine eventually spits out the number 6174 and then just keeps spitting out 6174 forever. It's like a magnet pulling everything to a single spot.

But what happens if we change the rules of the game? What if we aren't counting in tens, but in sevens, elevens, or any other "odd" number system? This paper, written by a team of mathematicians (including some famous names like Ken Ono), explores exactly that. They discovered that while the behavior in other bases is more complex than in Base 10, it follows a surprisingly rigid and beautiful pattern.

Here is the story of their discovery, explained simply.

The Secret Language of Differences

The authors realized that looking at the actual numbers (like 5100 or 9650) is like trying to understand a storm by looking at individual raindrops. It's too messy. Instead, they decided to look at the distance between the numbers.

Imagine your four digits are standing in a line.

The Outer Difference is the gap between the tallest person and the shortest person.
The Inner Difference is the gap between the two people in the middle.

The magic of this paper is that the entire future of the machine depends only on these two gaps. Once you know the two gaps, you don't need to know the original digits anymore. The machine's behavior is entirely determined by how these two gaps change.

The "Warm-Up" Phase

When you first start the machine, the gaps might be weird. Sometimes they are zero, sometimes they are even numbers, sometimes they are equal. The authors proved that no matter what you start with, the machine is very efficient at cleaning up this mess.

Within three steps (at most), the machine forces the gaps into a special, tidy zone. In this zone:

The gaps are never zero.
The gaps are never equal.
The gaps are always "odd" numbers.

Think of this like a chaotic dance floor. At first, people are bumping into each other, standing in circles, and moving randomly. But within three songs, everyone has found their perfect spot in a neat, orderly line. Once they are in this line, the dance becomes incredibly predictable.

The "Doubling" Dance

Once the machine is in this tidy zone, the authors found a hidden code. They realized that if you translate the gaps into a special "projective" language (a way of looking at numbers where positive and negative are treated as the same), the machine stops doing subtraction and sorting.

Instead, it simply doubles the numbers.

Imagine you have a pair of numbers, say 3 and 2.

Step 1: Double them $\rightarrow$ 6 and 4.
Step 2: Double them again $\rightarrow$ 12 and 8.
Step 3: Double them again $\rightarrow$ 24 and 16.

In the world of this machine, "doubling" works like a clock. If you keep doubling, eventually the numbers wrap around and come back to where they started. This creates a cycle.

The paper proves that the complex, messy process of sorting and subtracting digits is actually just a fancy disguise for this simple "doubling" game.

How Long is the Cycle?

Because the machine is just doubling numbers, the length of the cycle (how many steps it takes to return to the start) depends entirely on the base you are using.

The Limit: The longest possible cycle can never be longer than half the base minus one. For example, in Base 11, the longest cycle is 5 steps. In Base 17, it's at most 8 steps.
The Prime Rule: The machine only reaches this maximum length if the base is a prime number (like 7, 11, 13, 17) and the number 2 has a special property in that base. It's like a lock and key: the base must be a prime, and the "key" (the number 2) must take exactly the right amount of steps to turn the lock.

If the base is a composite number (like 9 or 15), the cycle is always shorter than the maximum possible.

The AI Connection

One of the most unique parts of this paper is how it was written. The authors mention that the core ideas were first proposed by two of the co-authors, but they used an AI tool called AxiomProver to help write the formal mathematical proof.

Think of the AI as a very strict editor. The human authors wrote the story and the main ideas, and the AI checked every single logical step to ensure there were no mistakes, translating the math into a computer language (Lean) that a machine could verify. This paper is a test case for how humans and AI can work together to solve deep mathematical mysteries.

Summary

In short, this paper takes a fun number trick (Kaprekar's routine) and reveals its hidden skeleton.

Messy Start: The process starts chaotic.
Quick Cleanup: It settles into a stable pattern within three steps.
Simple Core: Once stable, the process is just "doubling" numbers in a special mathematical world.
Predictable End: This means we can now calculate exactly how long the cycles will be and how many of them exist for any odd base, simply by looking at the properties of the number 2 in that base.

It turns a "magic trick" into a predictable, elegant dance governed by simple arithmetic rules.

Technical Summary: Four-Digit Kaprekar Dynamics in Odd Bases

Problem Statement
The paper investigates the dynamics of the Kaprekar routine applied to four-digit strings in odd bases $B > 3$ . The routine involves taking a four-digit number, arranging its digits in descending and ascending order, subtracting the latter from the former, and iterating the process. While the base-10 case is famous for converging to the fixed point 6174, the behavior in general bases is more complex, often involving multiple terminal cycles rather than a single attractor. The authors aim to provide a complete, explicit description of the terminal cycles (attractors) and their lengths for all odd bases $B > 3$ .

Methodology
The authors reduce the four-digit digit dynamics to a simpler system based on the differences between the sorted digits.

Difference Coordinates: For a sorted string $a_1 \ge a_2 \ge a_3 \ge a_4$ , the authors define two key variables: the outer difference $d_1 = a_1 - a_4$ and the inner difference $d_2 = a_2 - a_3$ . They demonstrate that the next state in the Kaprekar routine depends solely on the pair $(d_1, d_2)$ , effectively reducing the state space from digit strings to a two-dimensional integer lattice $X_B$ .
Pre-periodic Analysis: The authors prove that for any odd base $B > 3$ , every orbit enters a specific "stable odd region" $T_B$ (where $d_1 > d_2 > 0$ and both are odd) within at most three iterations.
Projective Transformation: Within the stable region $T_B$ , the authors introduce a change of variables to projective coordinates: $r = (d_1 + d_2)/2$ and $s = (d_1 - d_2)/2$ . They map these pairs to unordered two-element subsets of the projective residue classes modulo $B$ , denoted as $\binom{P_B}{2}$ , where $P_B = ((\mathbb{Z}/B\mathbb{Z}) \setminus \{0\}) / \{\pm 1\}$ .
Conjugacy to Doubling: The core technical result is proving that, under this transformation, the Kaprekar map restricted to $T_B$ is conjugate to the "projective doubling" map: $\{[r], [s]\} \mapsto \{[2r], [2s]\}$ . This transforms the digit subtraction problem into a problem of the multiplicative order of 2 in the group of units modulo $B$ (modulo sign).

Key Contributions and Results

Structural Classification (Theorem 1.1): The paper establishes that the Kaprekar dynamics in odd bases are governed by projective doubling after a bounded pre-period. This provides a finite, explicit description of the system's behavior.
Cycle Length Bounds (Corollary 1.2): The maximum length of any terminal cycle, $c_{\max}(B)$ , is bounded by $(B-1)/2$ . The paper proves that equality $c_{\max}(B) = (B-1)/2$ holds if and only if $B$ is a prime $p \ge 7$ and the least positive integer $m$ satisfying $2^m \equiv \pm 1 \pmod p$ is exactly $m = (p-1)/2$ .
Cycle Counting (Theorem 4.4): The authors derive an explicit formula for the number of terminal cycles of a given length $L$ . This formula depends on the projective order of 2 modulo the divisors of $B$ and involves combinatorial terms accounting for pairs of cycles and antipodal points.
Prime Base Specifics (Corollary 1.3): For prime bases where the maximal cycle length is achieved, the number of such maximal cycles is given by $\lfloor (c_{\max}(B) - 1)/2 \rfloor$ .
Formal Verification: The results were first formulated by Schwartz and Thakur and subsequently formalized in Lean/mathlib using the AI-assisted system AxiomProver, providing a machine-verified proof of the structural and arithmetic claims.

Significance and Claims
The paper claims to reveal a "rigid structure" underlying the Kaprekar routine in odd bases, demystifying the process by showing it is governed by simple modular arithmetic (projective doubling) rather than complex digit interactions.

Simplification: The authors emphasize that the reduction to projective doubling offers a "transparent route" to understanding cycle lengths and counts, which were previously studied via more intricate subgroup/coset methods or specific base analyses.
Completeness: Unlike previous works that may have focused on fixed points or specific bases, this paper provides a unified, complete classification for all odd bases $B > 3$ with four digits.
Modesty: The authors position this work as a "low-dimensional model" that makes the dynamics explicit for the four-digit case. They acknowledge that their results complement broader theories (such as those by Yamagami, Matsui, Kay, and Downes-Ward) but do not claim to solve the generalized problem for arbitrary digit lengths or even bases, noting that the even base case is "more intricate."

The paper concludes that the behavior of the Kaprekar routine is not a "numerical magic trick" but a direct consequence of the order of 2 modulo the base, with signs ignored.