square-free integer
Let an agent explain square-free integers by definition, tests, properties and links to number theory functions.
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 explain square-free integers by definition, tests, properties and links to number theory functions.
A positive integer that is not divisible by any perfect square other than 1, so that in its prime factorisation no prime appears more than once; examples are 1, 2, 3, 5, 6, 7, 10, and the natural density of square-free integers is 6/pi^2, about 0.6079.
What it is for: Number theory and algebra.
It can be test whether a number is square-free; explain the Mobius function connection; compute the density of square-free numbers; find sequences in OEIS.
Distinguishing features
No square divisor above 1
Distinct prime factors
Density 6/pi^2
Linked to the Mobius function
What it looks like
Not physical; integers and their factorisations.
Physical character
natural density: 6/pi^2, about 0.6079 - share of integers that are square-free
How it is recognised
No repeated prime factor
Mobius function nonzero
A square number is the opposite extreme
Related models
is a kind of - category
is characterised by - function
is related to - concept
is listed in - reference
In practice
Families and kinds
square-free integers
primes (a subset)
sphenic numbers
square-free elements in rings
Identifiers
OEIS A005117 sequence of square-free numbers
Standards and regulation
No specific regulation
Failure modes and hazards
Forgetting that 1 is square-free
Slow tests for very large 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.
Definition What it means.
Definition is simple.
Test
Checking.
Test
Testing.
- Is this number square-free? boundary
- What is its prime factorisation? measurement
Examples
Examples.
Examples
Examples.
- Which numbers below a given bound are square-free? measurement
- Why is 1 included? definition
Properties Theory.
Properties link to functions.
Mobius
Mobius function.
Mobius
Mobius function.
- How does the Mobius function detect square-free numbers? definition
- What is its value here? measurement
Density
Proportion.
Density
Density.
- Why is the density of square-free numbers 6/pi^2? definition
- How is it related to the zeta function? definition
Computation Algorithms.
Large numbers are harder.
Algorithm
Tests.
Algorithm
Algorithms.
- How can square-freeness be tested efficiently? definition
- Is it as hard as factoring? boundary
Sequence
OEIS.
Sequence
Sequence data.
- What does the OEIS entry for square-free numbers contain? provenance
- Which related sequences exist? provenance
Generalisation Beyond integers.
Concept extends.
Rings
Algebra.
Rings
Square-free elements.
- What does square-free mean for polynomials or ring elements? definition
- Which textbooks treat it? provenance
Teaching
Explaining.
Teaching
Teaching.
- How can square-free numbers be explained with examples? action
- Which exercises help? action
What the second pass must settle
- Should related number classes be separate entries?
- How should OEIS be linked?
- How should algebraic generalisations be linked?