perfect number
Let an agent handle perfect numbers by definition, known examples, relation to Mersenne primes and open problems.
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 handle perfect numbers by definition, known examples, relation to Mersenne primes and open problems.
A positive integer equal to the sum of its proper divisors, such as 6 = 1 + 2 + 3 and 28; all known perfect numbers are even and correspond to Mersenne primes, and whether any odd perfect number exists is an open problem.
What it is for: Number theory and recreational mathematics.
It can be test whether a number is perfect; list known perfect numbers; relate them to Mersenne primes; describe open problems accurately.
Distinguishing features
Sum of proper divisors equals itself
All known ones are even
Linked to Mersenne primes
Odd perfect numbers unknown
What it looks like
Integers such as 6, 28, 496 and 8128.
How it is recognised
Sum of proper divisors equals the number
Euclid-Euler form 2^(p-1)(2^p - 1)
Abundant and deficient numbers are related
Related models
is a kind of - category
is related to - construction
is defined by - concept
is studied in - field
In practice
Families and kinds
even perfect numbers
hypothetical odd perfect numbers
multiply perfect numbers (related)
Identifiers
OEIS A000396 integer sequence
Failure modes and hazards
Stating that no odd perfect numbers exist as proved
Outdated counts of known perfect numbers
Also called
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.
Test Is it perfect.
Divisors decide.
Check
Sum of divisors.
Check
Perfectness test.
- Is this number perfect, abundant or deficient? boundary
- What is the sum of its proper divisors? measurement
Form
Euclid-Euler.
Form
Euclid-Euler form.
- Can it be written in the Euclid-Euler form? definition
- Which Mersenne prime generates it? definition
Known Examples.
Counts change.
List
Known perfect numbers.
List
Known perfect numbers.
- How many perfect numbers are known, according to current records? provenance
- Which was found most recently? provenance
Search
Mersenne searches.
Search
Searches.
- How are new Mersenne primes and perfect numbers found? definition
- Which project coordinates the search? provenance
Open problems Unknowns.
Accuracy about open problems matters.
Odd
Odd perfect numbers.
Odd
Odd perfect numbers.
- Is the existence of odd perfect numbers still open? provenance
- What lower bounds are known? measurement
Infinitude
Infinitely many.
Infinitude
Infinitely many.
- Is it known whether there are infinitely many perfect numbers? provenance
- What would settle it? definition
Learning Teaching.
Perfect numbers engage learners.
History
Euclid and Euler.
History
History.
- What did Euclid and Euler prove about perfect numbers? provenance
- Which sources explain it? provenance
Activities
Exercises.
Activities
Activities.
- Which exercises help learners explore perfect numbers? action
- How can divisors be listed quickly? action
What the second pass must settle
- Should multiply perfect numbers be separate?
- How should Mersenne prime searches be linked?
- How should the count be kept current?