Kaprekar number¶
Classify a base-b natural number whose square can be split at a declared digit position into two parts that sum back to the number.
Core Idea¶
A p-Kaprekar number in base b is a fixed point of the function that splits n² into high and p-digit low parts and adds them. Euclidean division writes n²=αb^p+β with 0≤β<b^p; the Kaprekar function returns α+β, and equality with n defines membership. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Kaprekar number belongs to number theory and is useful where the analyst can specify a natural number n, radix b greater than one, split length p, and the base-b numeral for n squared, then evaluate n equals floor(n²/b^p) plus n² modulo b^p under the stated base and split convention. The scope is broad within that domain but bounded by the need for n equals floor(n²/b^p) plus n² modulo b^p under the stated base and split convention. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
Clarity¶
The abstraction clarifies a crowded vocabulary by making n equals floor(n²/b^p) plus n² modulo b^p under the stated base and split convention the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Kaprekar number can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Kaprekar number. Kaprekar number compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: a natural number n, radix b greater than one, split length p, and the base-b numeral for n squared. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express n equals floor(n²/b^p) plus n² modulo b^p under the stated base and split convention independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of number theory because they reuse a natural number n, radix b greater than one, split length p, and the base-b numeral for n squared, Euclidean division writes n²=αb^p+β with 0≤β<b^p; the Kaprekar function returns α+β, and equality with n defines membership., and declare b and p, square n, compute quotient and remainder exactly, preserve permitted leading zeros, and test equality rather than one visually convenient split. A theorem, diagnostic, or modeling warning can travel when those roles remain literal.
Relationships to Other Abstractions¶
Current abstraction Kaprekar number Domain-specific
Parents (1) — more general patterns this builds on
-
Kaprekar number is a kind of Function (Mapping) Prime
The proposed strict upward parent is
prime:function_mapping.
Hierarchy path (1) — routes to 1 parentless root
- Kaprekar number → Function (Mapping)
Neighborhood in Abstraction Space¶
Kaprekar number sits in a moderately populated region (42nd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Numeral Bases & Arithmetic Functions (8 abstractions)
Nearest neighbors
- Euclidean domain — 0.90
- Division (mathematics) — 0.90
- Sum of squares function — 0.89
- Sublime number — 0.89
- Square number — 0.89
Computed from structural-signature embeddings · 2026-09-08