Kaprekar number
Let an agent define Kaprekar numbers precisely, test whether a number is one, list the small examples, relate them to the Kaprekar routine and constant, and point to sequence references.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
Researched by: Claude
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Define The definition.
Definition
Definition.
Definition
Definition.
- What exactly makes an integer a Kaprekar number, and how are zero parts and the number 1 handled? definition
- Is the question about Kaprekar numbers or about the Kaprekar routine? boundary
Base
Base dependence.
Base
Base.
- How does the property depend on the base? definition
- Which base is assumed? boundary
Compute Testing and listing.
Test
Testing.
Test
Test.
- How is a given integer tested, and what is the result for the number in question? action
- What are the examples below a given bound? measurement
Sequence
The sequence.
Sequence
Sequence.
- Where is the sequence recorded, and what is known about its growth? provenance
- Which entry fits integer sequences? action
Theory Properties.
Properties
Known results.
Properties
Properties.
- What is known about the structure of Kaprekar numbers, such as their relation to unitary divisors of powers of ten? provenance
- Which references are standard? provenance
Generalise
Generalisations.
Generalise
Generalise.
- What generalisations exist to other powers and splittings? provenance
- Should a generalisation be a separate entry? boundary
Context History and teaching.
History
Kaprekar.
History
History.
- Who was Kaprekar, and how did the numbers get their name? provenance
- Which entry fits the person? action
Teach
Teaching.
Teach
Teaching.
- How are Kaprekar numbers used in teaching and puzzles? action
- Which misconceptions arise? provenance
Classifiers Filled
- Family
- Thing Registry
- Category
- Cross-cutting context
- Entry kind
- thing
- Plane
- XCT
- Domain
- XCT.QTY
What it is Filled
A non-negative integer whose square, written in a given base, can be split into two parts that add up to the original number, such as 45, since 45 squared is 2025 and 20 plus 25 is 45; named after the Indian mathematician D. R. Kaprekar and studied in recreational number theory.
Why it exists Filled
Let an agent define Kaprekar numbers precisely, test whether a number is one, list the small examples, relate them to the Kaprekar routine and constant, and point to sequence references.
Distinguishing features Filled
- Defined by a splitting of the square
- Depends on the base
- Conventions differ on zero parts and on 1
- Named after Kaprekar, like the unrelated routine
What robots and AI may and may not do Filled
Must not
- Present a number as a Kaprekar number without checking the definition in the stated base.
- Confuse Kaprekar numbers with Kaprekar's routine or constant.
- Attach mystical or predictive meaning to these numbers as fact.
Only with a human decision
- Publishing new results about them.
May
- Explain Kaprekar numbers with examples and check whether a number is one.
- Generate lists in a given base and show the working.
Moral aspects Filled
- Correct attribution credits a self-taught mathematician.
- Numerology can mislead people into harmful decisions.
Who is affected
- Students
- Mathematicians
Owners Filled
Steward
Nobody: a mathematical concept held in common.
Links to other meta-models Filled
parent
- Q28920052 - registry parent class
related
- non-negative integer - a property of integers
- Kaprekar constant - a different object named after the same person
- On-Line Encyclopedia of Integer Sequences - sequence A006886
- recreational mathematics - the field
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.kaprekar-number
- Wikidata: Q1478274 (https://www.wikidata.org/wiki/Q1478274)
- OEIS: A006886 sequence of Kaprekar numbers in base ten
Direct properties not applicable Not applicable
- smallest examples in base ten: 1, 9, 45, 55, 99 integers - with the convention that the right part is nonzero
Plane XCT: no invented physical properties.
Recognition optional Filled
- Square splits into two parts summing to the number
- Examples 1, 9, 45, 55, 99, 297, 703, 999
- The Kaprekar constant 6174 comes from a different procedure
- Not visible; a property of integers.
Capabilities and actions required Filled
- test whether an integer is a Kaprekar number
- list examples in base ten and other bases
- explain the definition and edge cases
- distinguish it from the Kaprekar routine and constant 6174
Hazards and failure modes required Filled
- Inconsistent conventions about zero parts
- Confusing the number with the routine
- Base-dependent claims stated as absolute
Standards and interfaces required Filled
- No regulation; definitions follow the literature and the OEIS
Context of use required Filled
- Not for anything; a curiosity in number theory.
- Kaprekar numbers in base ten
- Kaprekar numbers in other bases
- generalised Kaprekar numbers for higher powers
- related: the Kaprekar routine and constant
Sources Filled
- Wikidata item Q1478274: Kaprekar number - identity and sense of the item
- Wikipedia: Kaprekar number - general description of the item
Open questions
- Which convention on zero parts should the entry adopt?
- How should OEIS entries be linked?
- Should generalisations be separate entries?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q1478274/spec.json