Four-digit Kaprekar dynamics in odd bases

Abstract: Start with four digits, arrange them in both descending and ascending order, subtract, and repeat. This simple process is known as the Kaprekar routine, famous in base ten for sending every nonconstant four-digit string to $6174$. We show that in every odd base $B>3$, the four-digit Kaprekar map has an unexpectedly rigid structure. After at most three iterations, every nonconstant orbit enters an explicit triangular region $\mathcal{T}_B$, and on this region the map is conjugate to projective doubling: [ {[r],[s]}\longmapsto {[2r],[2s]}. ] This gives a complete finite description of all nonconstant terminal cycles, including an explicit formula for their lengths and counts. In particular, the longest terminal cycle has length at most $(B-1)/2$, and equality can occur only when $B$ is prime. For primes $p>5$, equality occurs precisely when the least positive $m$ with $2^m\equiv\pm1\pmod p$ is $m=(p-1)/2$. The results proved here were first formulated by Schwartz and Thakur. As a test case for AI-assisted formal mathematics, AxiomProver produced Lean/mathlib formalizations of these results.

• The paper establishes a novel arithmetic structure for four-digit Kaprekar dynamics in odd bases, generalizing the base-10 fixed-point phenomenon through difference coordinates and projective doubling.
• It provides explicit formulas to classify terminal cycles by counting their lengths and numbers using modular arithmetic and Euler’s totient function.
• The study enables algorithmic enumeration of periodic orbits, offering practical insights into digit-driven dynamical systems and formal proofs via AI-assisted methods.

Four-Digit Kaprekar Dynamics in Odd Bases: Structure, Cycle Classification, and Formalization

Overview and Motivation

The paper "Four-digit Kaprekar dynamics in odd bases" (2606.20439) establishes a rigorous arithmetic structure for the classical Kaprekar digit-sorting-and-subtracting routine, generalized to four-digit numbers in arbitrary odd bases $B > 3$. Unlike the base-10 case where every nonconstant four-digit integer converges to the unique fixed point 6174, this work demonstrates that for odd bases, the dynamical behavior enters a stable region after at most three iterations, with subsequent dynamics governed by projective doubling on pairs of projective residue classes. Leveraging this structure, the authors precisely enumerate terminal cycles, quantify their lengths, and clarify the underlying arithmetic dependencies.

Main Structural Classification

The crux of the classification relies on difference coordinates: for a given sorted four-digit string $[a_1, a_2, a_3, a_4]$, the outer and inner digit differences $d_1 = a_1 - a_4$ and $d_2 = a_2 - a_3$ parameterize the Kaprekar map. The induced state space $X_B$ admits a forward-invariant triangular region $\mathcal{T}_B$, defined by $d_1 > d_2 > 0$, both odd. After at most three iterations, all nonconstant orbits reside in $\mathcal{T}_B$.

Within $\mathcal{T}_B$, the Kaprekar map $K_B$ is conjugate to projective doubling:

$[a_1, a_2, a_3, a_4]$0

where $[a_1, a_2, a_3, a_4]$1 and $[a_1, a_2, a_3, a_4]$2 are taken modulo $[a_1, a_2, a_3, a_4]$3 and viewed up to sign in the projective residue space $[a_1, a_2, a_3, a_4]$4. This bijection reduces the analysis of terminal cycles to combinatorics and arithmetic within $[a_1, a_2, a_3, a_4]$5.

Cycle Lengths and Explicit Enumeration

The paper proves that the maximal cycle length for Kaprekar dynamics in base $[a_1, a_2, a_3, a_4]$6 is bounded by $[a_1, a_2, a_3, a_4]$7, attaining equality if and only if $[a_1, a_2, a_3, a_4]$8 is prime and the projective order $[a_1, a_2, a_3, a_4]$9 of 2 modulo $d_1 = a_1 - a_4$0 equals $d_1 = a_1 - a_4$1. Specifically,

$d_1 = a_1 - a_4$2

with equality only if $d_1 = a_1 - a_4$3 at $d_1 = a_1 - a_4$4 and $d_1 = a_1 - a_4$5 prime.

The authors give an explicit formula for the number and lengths of terminal cycles as follows. For each odd divisor $d_1 = a_1 - a_4$6 of $d_1 = a_1 - a_4$7, the projective order $d_1 = a_1 - a_4$8 enters the calculation for the cycle counts, denoted $d_1 = a_1 - a_4$9, where $d_2 = a_2 - a_3$0 is Euler's totient. The formula for the number of cycles of length $d_2 = a_2 - a_3$1 is detailed and incorporates both cycles within $d_2 = a_2 - a_3$2 and unordered pairs depending on cyclic distance.

Strong numerical results include:

• In prime bases $d_2 = a_2 - a_3$3 with maximal cycle order, the number of length-$d_2 = a_2 - a_3$4 cycles is $d_2 = a_2 - a_3$5.
• For composite bases or primes where the order condition fails, cycles are strictly shorter than $d_2 = a_2 - a_3$6, precisely determined by the arithmetic of $d_2 = a_2 - a_3$7 modulo $d_2 = a_2 - a_3$8.

Concrete examples illustrate these results, including explicit cycle structures in bases $d_2 = a_2 - a_3$9, $X_B$0, $X_B$1, and $X_B$2.

Practical and Theoretical Implications

Practically, the explicit correspondence between Kaprekar difference dynamics and projective doubling enables algorithmic enumeration of terminal cycles, their representative states, and periodic orbits, facilitating analysis across arbitrary odd bases. The results also generalize the fixed-point phenomenon witnessed in base 10, demonstrating robust group-theoretic and number-theoretic controls over digit dynamics.

Theoretically, the work provides a canonical low-dimensional model of Kaprekar dynamics, emphasizing that, after transient behavior, the core mechanism is arithmetic doubling up to sign on unordered pairs in a finite projective space. This insight is impactful for broader studies of digit dynamics, odd/even base behavior, and finite dynamical systems rooted in algebraic structures.

Importantly, the classification raises further questions regarding even base dynamics, connections with permutation groups, subgroup and coset models (cf. [Kay & Downes-Ward, 2022]), and the structure of fixed points and cycles for higher digit counts (cf. [Prichett, 1978], [Thakur, 2019]). Potential future directions include higher-dimensional analogs, nontrivial subgroup cyclicity for composite bases, and deep connections with modular arithmetic.

Formalization and AI-Assisted Proof

The results were first formalized as conjectures by Schwartz and Thakur, communicated to Ono, and then formalized in Lean/mathlib via AxiomProver. The formal proof files are available and fully verified using Lean 4.28.0, demonstrating the effectiveness of AI-assisted formal mathematics in automating complex arithmetic dynamical proofs. The human-readable account follows these formalizations, ensuring precise mathematical statement and attribution.

Conclusion

The paper provides an explicit, arithmetic-centric classification for four-digit Kaprekar dynamics in arbitrary odd bases, revealing a transparent projective doubling structure on difference coordinates. The resulting complete enumeration of terminal cycles—both their structure and count—clarifies the interaction between digit sorting, subtraction, and modular arithmetic. This work marks a technical advance in understanding digit-driven dynamical systems, with significant implications for algorithmic digit analysis, modular phenomena, and formal mathematics. Further exploration in even bases, higher digits, and algebraic frameworks is warranted.