Base scaling in four-digit Kaprekar dynamics, and an infinite family of bases with no unique terminating cycle
Abstract
The four-digit Kaprekar routine in base b — sort the digits downward, subtract the upward arrangement, repeat — depends only on the two differences d₁=a₁-a₄ and d₂=a₂-a₃, and so acts on a set X_b of size about b²/2. Chen, Ono, Schwartz and Thakur classified its terminal cycles for every odd base B>3 and closed with a remark, supported only by numerical evidence, predicting that the even bases with a unique terminating cycle are exactly 2²ⁿ⁺¹ · 3, 2²ⁿ⁺¹ · 5 and 2^(3n ± 1) · 7, with cycle lengths 6(n+1), 1 and 3. We work with the scaling law K_(mb)(md₁,md₂)=m K_b(d₁,d₂), valid at every base on the region where one step is {lvert 2d₁-brvert,lvert 2d₂-brvert} — the case m=2 is due to Devlin and Zeng — and with the transport of terminal cycles that it yields. Our main result is that every base with an odd divisor in [9,51] carries two disjoint terminal cycles and hence has no unique terminating cycle; the covered set contains more than half of the even bases below 10⁶, and no previously published result applies to most of those. We also decide the predicted classification, in both directions, for every even b ≤ 200 and for b=224 and b=384. Every statement is machine-checked in Lean 4.
Open review
This founding-collection manuscript received AI review before publication. Independent human review is open. Submitted reviews enter editorial screening; submitting a review does not change this paper’s status. Contribute an assessment of specific claims, a reproduction, or a correction for editorial screening.
Archived files
- Version 1 · current (opens in a new tab)
Source snapshot 2026-08-30 15:34 UTC
File fingerprint
31f10c60565efdaaab6d94077ff0f5a034f31ee7e216b29978b2bea76924a8eb
Claim ledger
Stated results
kap-01routine2026-08-22
The four-digit routine reduces to d1(b³-1) + d2(b²-b) at EVERY base, and the difference coordinates carry the dynamics
kap-02routine2026-08-22
One Kaprekar step is |2d1-b|, |2d2-b| whenever d1 > d2 > 0 and d1 + d2!= b, at EVERY base
kap-03routine2026-08-22
b - 1 divides every Kaprekar difference, at EVERY digit count
kap-04candidate2026-08-23
Chen-Ono-Schwartz-Thakur's even-base Remark, DECIDED for every even b <= 200, plus the members 224 and 384
kap-05routine2026-08-22
The source's odd-base classification reproduced at cells, and cₘax(B) = lambda(B) for every odd B >= 7 in range
kap-06routine2026-08-22
Negative controls: the source's Table 1 is not exhaustive at even bases, and the oddness hypothesis of Corollary 4.5(ii) is load-bearing
kap-07routine2026-08-28
The base-scaling law K_(m·b)(m·d₁, m·d₂) = m·K_b(d₁,d₂) on the doubling region, at EVERY base, and the cycle transport it gives
kap-08routine2026-08-28
Each member of two of the three families of the Remark carries a cycle of exactly the conjectured length, for every n — and the region explains why the third does not
kap-09candidate2026-08-28
Every base with an odd divisor in [9,51] has two disjoint terminal cycles, hence NO unique terminating four-digit cycle — an infinite family, 53.5% of all even bases
kap-10routine2026-08-28
Every four-digit Kaprekar fixed point, at EVERY base, and the resulting Kaprekar constants as integers
kap-11routine2026-08-28
Negative controls for the scaling law: each region hypothesis is load-bearing, and the 2²ⁿ⁺¹·3 family provably does not scale
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- Source. Evan Chen, Ken Ono, Richard E. Schwartz, Dinesh S. Thakur, *Four-digit Kaprekar dynamics in odd bases*, arXiv:2606.20439 (v1 June 2026; v3 2026-08-11, accepted in Journal of Integer Sequences).
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7