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perfect power

vr.tr.perfect-power · thing-q1972803

Let an agent handle perfect powers by definition, testing, examples and notable theorems.

Thing Registry Cross-cutting context XCT.QTY
Bundle → Layer → Finding → QuestionsClassifiersWhat it isWhy it existsDistinguishing featuresWhat robots and AI may and may not doMoral aspectsOwnersLinks to other meta-modelsIdentity and identifiersDirect propertiesRecognitionCapabilities and actionsHazards and failure modesStandards and interfacesContext of useSources

Bundle → Layer → Finding → Questions Filled

4 bundles · 8 layers · 8 findings · 16 questions

Test Is it a perfect power.

Check

Roots.

Check

Perfect power test.

  1. Is this number a perfect power, and of which base and exponent? boundary
  2. How is it tested exactly? action

Code

Implementation.

Code

Implementation.

  1. How can integer roots be computed without floating point errors? action
  2. Which libraries help? provenance
Examples Sequences.

List

Early terms.

List

List.

  1. What are the first perfect powers? provenance
  2. How dense are they among integers? measurement

Kinds

Squares and cubes.

Kinds

Kinds.

  1. Is it a square, cube or higher power? definition
  2. Can it be several at once, such as 64? definition
Theorems Results.

Catalan

Consecutive powers.

Catalan

Catalan conjecture.

  1. What does the Catalan conjecture, now a theorem, state? provenance
  2. Who proved it and when? provenance

Open

Open problems.

Open

Open problems.

  1. Which related problems remain open? provenance
  2. Is their status stated accurately? boundary
Learning Teaching.

Patterns

Visual models.

Patterns

Patterns.

  1. How can squares and cubes be shown with arrays and blocks? action
  2. Which exercises help? action

Notation

Exponents.

Notation

Exponent notation.

  1. How is exponent notation written and read? definition
  2. What common mistakes occur? boundary

Classifiers Filled

Family
Thing Registry
Category
Cross-cutting context
Entry kind
thing
Plane
XCT
Domain
XCT.QTY
Other names and narrower kinds
power of five, power of four, power of six, cube number, power of three

What it is Filled

A positive integer that can be written as m to the power k with integers m greater than 1 and k at least 2, such as squares (4, 9), cubes (8, 27) and higher powers; Catalan conjecture, proved by Mihailescu, shows 8 and 9 are the only consecutive perfect powers.

Why it exists Filled

Let an agent handle perfect powers by definition, testing, examples and notable theorems.

Distinguishing features Filled

  • Integer power of an integer
  • Includes squares and cubes
  • Rare among large numbers
  • Subject of deep theorems

What robots and AI may and may not do Filled

Must not

  • Rely on floating point roots alone to decide that a large number is a perfect power.
  • Confuse perfect powers with powers of a fixed base.
  • Present an unproved conjecture about powers as a theorem.

Only with a human decision

  • Publishing a claimed new result about perfect powers as established.

May

  • Test whether an integer is a perfect power using exact integer arithmetic.
  • List perfect powers and their bases and exponents for teaching or research.

Moral aspects Filled

  • Correct mathematics matters where perfect power tests feed into cryptographic and number theory software.

Who is affected

  • Students and teachers
  • Users of number theory software

Owners Filled

Steward

Nobody owns the concept; it belongs to mathematics.

Links to other meta-models Filled

parent

  • Q28920052 - registry parent class

related

  • non-negative integer - category
  • square number - kind
  • Catalan conjecture - theorem
  • number theory - field

What else AI and robots need to interact with it Filled

Identity and identifiers required Filled

  • Vercy registry: vr.tr.perfect-power
  • Wikidata: Q1972803 (https://www.wikidata.org/wiki/Q1972803)
  • OEIS: A001597 integer sequence

Direct properties not applicable Not applicable

Not applicable

Plane XCT: no invented physical properties.

Recognition optional Filled

  • Forms such as m squared or m cubed
  • Squares and cubes are special cases
  • Powers of a fixed base such as 2 form a different sequence
  • Integers such as 16, 27, 32 and 64.

Capabilities and actions required Filled

  • test whether a number is a perfect power
  • list squares, cubes and higher powers
  • explain related theorems
  • implement exact tests in code

Hazards and failure modes required Filled

  • Floating point errors in root tests
  • Confusing perfect powers with powers of a base

Standards and interfaces not applicable Not applicable

Not applicable

No specific standard, law or official classification applies to perfect powers as an abstract mathematical concept.

Context of use required Filled

  • Number theory and computation.
  • square numbers
  • cube numbers
  • fourth and higher powers
  • powers of two, three and five

Sources Filled

  1. Wikidata item Q1972803: perfect power - identity and sense of the item
  2. Wikipedia: Perfect power - general description of the item

Open questions

  • Should cubes be a separate entry?
  • How should OEIS be linked?
  • How should related conjectures be tracked?

Machine files

Card as JSON Specification Card backlog vercy.card/1

Provenance

thing registry research (pass 2) · unreviewed

Built from: models/things/publications/thing-q1972803/spec.json