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.

Version: v2 · 2026-09-19 · History
Domain-specific #: 5176
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Number Theory, Digit Defined Integer Sequences → Mathematics

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.

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The Square-Split-Add Trick

Take a number and multiply it by itself. Cut the answer into two pieces, then add the pieces together. If you get your starting number back, it's a Kaprekar number! For 45: 45 times 45 is 2025, and 20 plus 25 makes 45 again.

Square, Split, Add Back

A Kaprekar number is one that comes back to itself after a square-split-add trick. You square the number, chop the answer into a left part and a right part with a chosen number of digits, and add the two parts. If the sum equals the number you started with, it's a Kaprekar number. For example, 9 × 9 = 81, and 8 + 1 = 9. Likewise 45 × 45 = 2025, and 20 + 25 = 45. You have to decide in advance where to cut. You can't try every cut until one happens to work.

Square–Split–Sum Fixed Points

A p-Kaprekar number in base b is defined by a square–split–sum test. Square n, then divide n² by b^p to get a quotient α and a remainder β, so that n² = α·b^p + β with 0 ≤ β < b^p. In other words, β is the last p digits and α is the rest. If α + β = n, then n is a p-Kaprekar number; it is a fixed point of the 'split and add' function. The base and the split length must be stated in advance, and leading zeros in the low part count where permitted. In base 10 with p = 2, 45 works: 2025 → 20 + 25 = 45. Don't confuse it with Kaprekar's constant, which comes from a different digit routine.

A p-Kaprekar number in base b is a fixed point of the Kaprekar function K(n) = floor(n²/b^p) + (n² mod b^p). Euclidean division writes n² = αb^p + β with 0 ≤ β < b^p, and n qualifies when α + β = n. The defining structure is this square–split–sum fixed-point equation with the radix and split length stated explicitly.

Testing a candidate correctly means: - declaring b and p, - computing quotient and remainder exactly, - keeping any permitted leading zeros in the low part, - checking equality rather than hunting for a convenient partition.

The concept should not be confused with Kaprekar's constant or other digit routines. Once established, it supports studying digit-defined fixed points and cycles, base-dependent integer sequences, and links between numeral structure and modular constraints.

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.
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) — outgoing typed relations

Kaprekar number is a kind of Function (Mapping) Prime

The identity is literally a fixed point of a declared mapping F_{p,b}.

Hierarchy path (1) — routes to 1 parentless root

Kaprekar number → is a kind of Function (Mapping)

Neighborhood in Abstraction Space¶

Kaprekar number sits in a crowded region of the domain-specific corpus (24^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Integer Classifications & Arithmetic Functions (45 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25