Fermat number
Let an agent explain Fermat numbers and their properties, describe known results on Fermat primes and factorisations, relate them to constructible polygons, and support number theory learning.
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 Fermat numbers and their properties, describe known results on Fermat primes and factorisations, relate them to constructible polygons, and support number theory learning.
A positive integer of the form two raised to the power of two to the n, plus one, named after Pierre de Fermat, who conjectured that all such numbers are prime; the first five, 3, 5, 17, 257 and 65537, are prime, but Euler showed the sixth is composite, and no further Fermat primes are known; Fermat numbers are a class of Cunningham numbers, connect to constructible polygons, and are studied in number theory and computational primality testing.
What it is for: Integers of the form 2^(2^n) + 1.
It can be explain definition and properties; describe known results; relate to geometry; support learning.
Distinguishing features
Double exponential form
Pairwise coprime
Few known primes
Constructible polygon link
What it looks like
Not physical; a sequence of integers.
How it is recognised
Form 2^(2^n) + 1
Only five known primes
Mersenne numbers have the form 2^p - 1
Related models
is a kind of - category
is a kind of - category
is related to - Fermat primes
is related to - another geometry-related entry
In practice
Families and kinds
Fermat primes
composite Fermat numbers with known factors
generalised Fermat numbers
Fermat numbers in primality testing
Standards and regulation
Mathematical conventions and notation
Distributed computing project conventions for factor searches
Failure modes and hazards
Misstating which Fermat numbers are prime
Confusing Fermat and Mersenne numbers
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.
Understand What Fermat numbers are.
Mathematics.
Definition
Definition and properties.
Definition
Definition.
- What are Fermat numbers, and what properties such as pairwise coprimality and recurrence relations do they satisfy? definition
- How do they differ from Mersenne numbers? definition
Primes
Fermat primes.
Primes
Primes.
- Which Fermat numbers are known to be prime or composite, and what did Fermat conjecture and Euler show? provenance
- What is known about the factors of larger Fermat numbers? provenance
Connect Connections.
Theory.
Geometry
Constructible polygons.
Geometry
Geometry.
- How do Fermat primes determine which regular polygons are constructible with compass and straightedge, according to the Gauss-Wantzel theorem? definition
- Which entry fits constructible polygons? action
Tests
Primality testing.
Tests
Testing.
- How is the primality of Fermat numbers tested with the Pepin test, and how are factors searched? provenance
- Which entry fits primality testing? action
Open Open problems.
Research.
Problems
Open questions.
Problems
Open questions.
- Are there infinitely many Fermat primes or composites, and what is the status of these open problems? boundary
- Which references are standard? provenance
Computing
Computational searches.
Computing
Computing.
- Which projects search for Fermat number factors, and what records exist? provenance
- Which entry fits distributed computing? action
Learn History and teaching.
Education.
History
History.
History
History.
- How did Fermat, Euler and later mathematicians study these numbers? provenance
- Which entry fits Fermat? action
Teach
Teaching.
Teach
Teaching.
- How can Fermat numbers be used to teach conjecture and proof? action
- Which misconceptions arise? provenance
What the second pass must settle
- Should Fermat primes be a separate entry?
- How should factor tables be linked?
- How should open problems be documented?