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Published

Achilles number

vr.tr.achilles-number · thing-q3232444

Let an agent define Achilles numbers, distinguish them from powerful numbers and perfect powers, give examples, and help test whether a number qualifies.

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

Define What Achilles numbers are.

Definition

Definition and examples.

Definition

Definition.

  1. What is an Achilles number, and what are the first examples? definition
  2. Is the number powerful, and is it a perfect power? boundary

Relations

Related classes.

Relations

Relations.

  1. How do Achilles numbers relate to powerful numbers and perfect powers? definition
  2. Which entry fits powerful number? action
Compute Testing and generating.

Test

Testing a number.

Test

Test.

  1. How is a number tested for the Achilles property by factorisation and exponent gcd? action
  2. Is the number in question an Achilles number? measurement

Strong

Strong Achilles numbers.

Strong

Strong.

  1. What are strong Achilles numbers, and how are they found? provenance
  2. Which references are standard? provenance
Theory Properties.

Density

Density and structure.

Density

Density.

  1. How are Achilles numbers distributed, and what structure do they have? provenance
  2. Which sources are cited? provenance

Related

Related problems.

Related

Related.

  1. How do powerful numbers figure in problems such as consecutive powerful numbers? provenance
  2. Which entry fits the specific problem? action
Context Naming and status.

Naming

Naming.

Naming

Naming.

  1. Why are they named after Achilles, and who introduced the term? provenance
  2. Which entry fits recreational mathematics? action

Status

Mathematical status.

Status

Status.

  1. How do mathematicians regard the class, with positions attributed? provenance
  2. Is the presentation neutral? boundary

Classifiers Filled

Family
Thing Registry
Category
Cross-cutting context
Entry kind
thing
Plane
XCT
Domain
XCT.QTY

What it is Filled

A positive integer that is powerful, meaning every prime factor appears at least squared, but is not a perfect power, such as 72, 108, 200, 288, 392, 432, 500, 648, 675 and 800; Achilles numbers are named for the hero who was powerful but imperfect, and they include strong Achilles numbers whose Euler totient is also an Achilles number.

Why it exists Filled

Let an agent define Achilles numbers, distinguish them from powerful numbers and perfect powers, give examples, and help test whether a number qualifies.

Distinguishing features Filled

  • Every prime factor at least squared
  • Not a perfect power
  • Named after Achilles
  • Strong subclass via totient

What robots and AI may and may not do Filled

Must not

  • Forget to exclude perfect powers.
  • Misidentify powerful numbers.
  • Overstate the mathematical significance of Achilles numbers.
  • Confuse the mathematical term with the mythological figure.

Only with a human decision

  • Publishing new claimed results.

May

  • Define Achilles numbers and check whether a number is one, showing why.
  • Explain the link to powerful numbers and perfect powers.

Moral aspects Filled

  • Accurate definitions matter for students.
  • Mathematical play deserves honesty about significance.

Who is affected

  • Students
  • Puzzle enthusiasts

Owners Filled

Steward

Nobody: a mathematical concept held in common.

Links to other meta-models Filled

parent

  • Q3191171 - registry parent class

related

  • powerful number - in registry terms
  • perfect power - the excluded case
  • Euler totient function - in the strong subclass
  • recreational mathematics - the field

What else AI and robots need to interact with it Filled

Identity and identifiers required Filled

  • Vercy registry: vr.tr.achilles-number
  • Wikidata: Q3232444 (https://www.wikidata.org/wiki/Q3232444)
  • OEIS: A052486 Achilles numbers

Direct properties not applicable Not applicable

  • first terms: 72, 108, 200, 288, 392, 432, 500, 648, 675, 800 sequence

Plane XCT: no invented physical properties.

Recognition optional Filled

  • Powerful but not a perfect power
  • Sequence 72, 108, 200, 288, 392
  • Perfect powers such as 64 are powerful but excluded; squarefree numbers are not powerful
  • Not a visible object; integers of a special form.

Capabilities and actions required Filled

  • define the class and its conditions
  • distinguish from powerful numbers and perfect powers
  • give examples
  • test numbers

Hazards and failure modes required Filled

  • Forgetting to exclude perfect powers
  • Misidentifying powerful numbers
  • Overstating mathematical significance

Standards and interfaces required Filled

  • No regulation; the OEIS is the reference database

Context of use required Filled

  • A recreational and number-theoretic classification.
  • Achilles numbers
  • strong Achilles numbers
  • powerful numbers as the parent class
  • perfect powers as the excluded class

Sources Filled

  1. Wikidata item Q3232444: Achilles number - identity and sense of the item
  2. Wikipedia: Achilles number - general description of the item

Open questions

  • Should powerful number be the primary entry?
  • How should the OEIS be linked?
  • How should the naming story be sourced?

Machine files

Card as JSON Specification Card backlog vercy.card/1

Provenance

thing registry research (pass 2) · unreviewed

Built from: models/things/publications/thing-q3232444/spec.json