A Generalized Kaprekar Phenomenon
60714

A five-digit number that is a universal attractor at every digit length d ≥ 5, under an explicit family of generalized Kaprekar rules.

Clay Elmore April 2026 Preprint forthcoming

Kaprekar's classical process fixes 6174 at four digits and 495 at three. The natural question, whether the phenomenon extends to higher digit lengths, has resisted a clean answer because the classical rule, applied at d = 5, has rank deficiency and no full-variable fixed point.

This paper introduces a coefficient-preserving lifting framework that classifies universal full-variable fixed points across digit lengths. At d = 5, exhaustive enumeration over all 5,280 full-variable rules yields 33 such fixed points; at d = 6, 506. Of the 33 candidates at d = 5, exactly one, 60714, survives the lifting test to d = 6 and beyond.

The main theorem establishes that 60714 is a fixed point of an explicit lifting at every d ≥ 5, with strict-universal dynamics at d = 5, 6 and near-universal dynamics at d ≥ 7 on a structurally characterized escape class. The proof closes by induction along two ladders (odd and even d), with the inductive step verified algebraically at large d and exhaustively at low d.

Main Result

Theorem 5.2. The integer 60714 is a fixed point of an explicit coefficient-preserving lifting K(d) at every digit length d ≥ 5. The lifted dynamics are strict-universal at d = 5, 6 (every admissible input reaches 60714) and near-universal at d ≥ 7 (every admissible input outside a structurally characterized escape class Ed reaches 60714; every input in Ed reaches 0).

The escape class Ed has step-1 component of size |Ed(1)| = C(⌊(d−3)/2⌋ + 9, 9) − 10, growing polynomially in d; the basin density 1 − |Ed|/|Ad| is conjectured to approach 1 (Conjecture 7.6, the Basin Density Conjecture).

The Rule at d = 5

Sort the digits of the input in descending order as (a, b, c, d, e). Form two new five-digit numbers by interleaving the digits in fixed patterns, then subtract:

K(n) = | adcbe − deacb |

Both adcbe and deacb are five-digit integers built by reading a, b, c, d, e in the indicated order. The classical Kaprekar process at four digits uses descending minus ascending, abcd − dcba; that simple recipe has no full-variable fixed point at five digits. The interleaved rule adcbe − deacb is the rule the framework selects, and 60714 is its unique cross-dimensional fixed point.

Applied to 60714 itself: digits sorted descending are (7, 6, 4, 1, 0), so adcbe = 71460 and deacb = 10746, and 71460 − 10746 = 60714. ✓

Worked Example

Starting from 10135, the rule reaches 60714 in two iterations.

Step 1
Input: 10135

Sort digits descending and label them:

5a
3b
1c
1d
0e
5 1 1 3 0 = adcbe
− 1 0 5 1 3 = deacb
4 0 6 1 7 = 40617
↓
Step 2
Input: 40617

Sort digits descending and label them:

7a
6b
4c
1d
0e
7 1 4 6 0 = adcbe
− 1 0 7 4 6 = deacb
6 0 7 1 4 = 60714
→ reached 60714 ✓

Notice that the intermediate value 40617 has the same digit multiset as 60714, just permuted. Every five-digit number whose digits are {0, 1, 4, 6, 7} sorts to (7, 6, 4, 1, 0), and the rule maps that descending arrangement directly to 60714. So the second step is the universal "snap to fixed point" move, while the first step is what gets the digit multiset there in the first place.

Read & Try It

Interactive
The 60714 Calculator
Type any number with at least five digits. Watch the lifted Kaprekar process iterate to 60714 (or its padded form) in finitely many steps.
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Interactive
Kaprekar's Classic Routine
The original 1949 routine. Type any 3- or 4-digit number and watch it walk to 495 (at three digits) or 6174 (at four). The result that started it all.
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Paper · PDF
Full Paper (compiled)
76 pages. Sections 1–7, Appendices A–F, References. Compiled from LaTeX; this is the canonical reading version. Updated April 30, 2026. Appendix F (added April 29; §F.5 revised April 30) is a preliminary working note on a strengthened strict-d criterion.
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Plain English
The five-digit cousin of 6174
A 6-minute non-technical read. Kaprekar's classic four-digit routine, a different question you can ask alongside it, and what turned up at five digits.
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Paper · LaTeX
Source (TeX)
Standalone LaTeX source. Compiles to the PDF above with pdflatex. Used for arXiv submission.
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Preprint · arXiv
arXiv:PENDING
Forthcoming. The arXiv preprint will provide the canonical citable version with DOI.
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Sequence · OEIS
|Ed(1)|: 45, 210, 705, 1992, …
Forthcoming OEIS submission for the step-1 escape class size, with closed form C(k+9, 9) − 10.
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Paper Structure

§1
Introduction
Background, the rank-4 reframing, and the framework selection of 60714.
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§2
Framework
Universal full-variable fixed points; admissibility and the rule space.
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§3
Classifications at d ≤ 6
The 0, 4, 33, 506 catalog. Exhaustive enumeration over all full-variable rules.
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§4
Cross-check d = 5 → d = 6
Theorem 4.1: of 33 candidates at d=5, exactly one (60714) lifts to d=6.
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§5
Theorem 5.2
60714 universal at every d ≥ 5. Two-ladder induction; structural closure of Td.
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§6
The {7,6,4,1}-thread
6174, 60714, and 60417 unified through their digit multiset.
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§7
Open Questions
The Basin Density Conjecture and five further open problems.
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App. C
Support Lemmas
The proof's heavy lifting: closure of Td, Lemmas C.3–C.10.
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Reproducibility

Every numerical claim in the paper is reproducible from the verification scripts. The full bundle (paper, sections, scripts, supplementary data) is in the public repository.

Code · Python 3
Verification Scripts
Eight scripts, ~5 to ~30 minutes each on consumer hardware. Includes the d=18 even-ladder exhaustive run (4.69M admissibles, 0 stuck).
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Data
All 506 fixed points at d = 6
Plain text file. The supplementary list referenced in §3.4 and Appendix A.
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Cite

While the arXiv preprint is forthcoming, the paper can be cited as:

@unpublished{elmore2026kaprekar60714, author = {Elmore, Clay}, title = {Universal Full-Variable Fixed Points Under Coefficient-Preserving Liftings: Classification at $d \leq 6$ and the Cross-Dimensional Survivor 60714}, year = {2026}, note = {Preprint forthcoming on arXiv. See https://github.com/clayelmore/Kaprekar-60714} }

This BibTeX entry will be updated to @article with arXiv ID once the preprint is posted.