Dudeney number¶

A base-dependent natural number that is a perfect cube whose digit sum equals its cube root.

Version: v2 · 2026-09-19 · History
Domain-specific #: 4278
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Recreational Number Theory → Mathematics

Core Idea¶

In the classical decimal case, m is a Dudeney number when m equals n cubed and the sum of the base-ten digits of m equals n; generalized versions vary the base and exponent. Exponentiation produces the digit string, digit summation maps it back to a candidate root, and solutions are fixed points of the resulting digit-power transform. 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 Cube Round Trip

Take 8 and multiply it by itself three times: 8 times 8 times 8 makes 512. Now add up the digits of 512: 5 plus 1 plus 2 makes 8, the number you started with! Numbers like 512 that lead you back to where you started this way are called Dudeney numbers.

Cube, Add Digits, Come Back

A Dudeney number is a number that is a cube, some number n times itself three times, and whose digits add up to that same n. For example, 512 is 8 times 8 times 8, and 5 + 1 + 2 = 8. So cubing takes you from 8 to 512, and adding the digits takes you right back to 8. Mathematicians also look at versions using other powers or other number systems besides our usual base ten.

Digit-Sum Fixed Points of Cubes

In the classic version, a number m is a Dudeney number if m = n³ for some natural number n and the sum of m's decimal digits equals n. For example, 512 = 8³ and 5 + 1 + 2 = 8. You can think of it as a round trip through two operations: cubing produces a string of digits, and summing those digits maps back to a possible root. A Dudeney number is a fixed point of that round trip. Generalized versions change the base (instead of base ten) or the exponent (instead of cubes), but in each case the base and exponent must be fixed and the digit sum must match the root exactly.

A Dudeney number, in the classical decimal case, is a natural number m such that m = n^3 for some natural n and the sum of the base-ten digits of m equals n; 512 = 8^3 with digit sum 8 is an example. The structure is a composition of two maps: exponentiation sends n to the digit string of n^3, and digit summation sends that string back to a candidate root. Dudeney numbers are then precisely the fixed points of the combined transform n to digit-sum(n^k). Generalizations vary the base b and exponent k, requiring m = n^k with base-b digit sum equal to n. The identity depends on all three conditions holding for a declared base and exponent; a number that is a power but checked in a different base, or has a matching digit sum without being the stated power, is not an instance.

Scope of Application¶

Dudeney number belongs to recreational number theory and is useful where the analyst can specify the typed recreational number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the base and exponent are fixed, the number is the stated power of a natural root, and its digit sum in that base equals the root. The scope is broad within that domain but bounded by the need for the base and exponent are fixed, the number is the stated power of a natural root, and its digit sum in that base equals the root. 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 the base and exponent are fixed, the number is the stated power of a natural root, and its digit sum in that base equals the root 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 Dudeney 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 Dudeney number. Dudeney 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: the typed recreational number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express the base and exponent are fixed, the number is the stated power of a natural root, and its digit sum in that base equals the root independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of recreational number theory because they reuse the typed recreational number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Exponentiation produces the digit string, digit summation maps it back to a candidate root, and solutions are fixed points of the resulting digit-power transform., and type the carrier, state every parameter and convention in the definition, test that the base and exponent are fixed, the number is the stated power of a natural root, and its digit sum in that base equals the root, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Dudeney number Domain-specific

Parents (1) — outgoing typed relations

Dudeney number is a kind of Fixed Point Prime

What makes it its own entry: the domain-specific identity determined by the base and exponent are fixed, the number is the stated power of a natural root, and its digit sum in that base equals the root.

Hierarchy path (1) — routes to 1 parentless root

Dudeney number → is a kind of Fixed Point

Neighborhood in Abstraction Space¶

Dudeney number sits in a crowded region of the domain-specific corpus (26^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Numeral Systems & Integer Sequences (22 abstractions)

Nearest neighbors

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