amicable numbers
Let an agent define amicable numbers, verify and find pairs, explain known rules and results, and present open questions and history.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify
Researched by: Claude
Purpose and description
Let an agent define amicable numbers, verify and find pairs, explain known rules and results, and present open questions and history.
Two different natural numbers such that the sum of the proper divisors of each equals the other, the smallest pair being 220 and 284, known since antiquity, with many pairs found by Thabit ibn Qurra rule, Euler and computer searches; amicable numbers are a case of sociable numbers of period two and remain a topic of open questions in number theory.
What it is for: A classical topic in number theory.
It can be define and verify amicable pairs; explain rules for generating pairs; present known results and open problems; describe history.
Distinguishing features
Divisor sum relation
Pairs
Sociable numbers of period two
Open problems
What it looks like
Not physical; pairs of integers.
How it is recognised
Each is the sum of the proper divisors of the other
Smallest pair 220 and 284
A perfect number equals its own divisor sum
Related models
is a kind of - category
is related to - related concept
is related to - contrast with fixed values
is related to - another paired concept in mathematics
In practice
Families and kinds
amicable pairs
regular and irregular pairs
pairs from the Thabit rule
sociable numbers of longer periods (related)
quasi-amicable and other variants
Standards and regulation
Mathematical notation conventions
Failure modes and hazards
Confusing with friendly or perfect numbers
Miscounting divisors
Overstating what is proven
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.
Define What they are.
Definition.
Definition
Definition and examples.
Definition
Definition.
- What are amicable numbers, and why are 220 and 284 amicable? definition
- How do they differ from perfect and friendly numbers? definition
Verify
Checking a pair.
Verify
Verification.
- Is this pair of numbers amicable, computed from their divisor sums? action
- How can pairs be found by computer? action
Theory Rules and results.
Mathematics.
Rules
Generating rules.
Rules
Rules.
- How do the rules of Thabit ibn Qurra and Euler generate amicable pairs? definition
- Why do they not produce all pairs? definition
Open
Open problems.
Open
Open problems.
- Are there infinitely many amicable pairs, and is there a pair with one odd and one even number? What is known? provenance
- How many pairs are known? provenance
History History.
Attribution.
Ancient
Antiquity and medieval work.
Ancient
History.
- How were amicable numbers known to the Pythagoreans and studied by medieval Arabic mathematicians? provenance
- What did Fermat, Descartes and Euler contribute? provenance
Modern
Computer era.
Modern
Modern era.
- How have computer searches expanded the known pairs? provenance
- Which databases list them? provenance
Learn Teaching and culture.
Education.
Teach
Teaching.
Teach
Teaching.
- How can amicable numbers be used to teach divisors and programming? action
- Which misconceptions arise? provenance
Culture
Cultural references.
Culture
Culture.
- How have amicable numbers appeared in folklore, talismans and literature? provenance
- Which entry fits sociable numbers? action
What the second pass must settle
- Should sociable numbers be a single entry?
- How should databases of pairs be linked?
- How should open problems be tracked?