happy number
Let an agent explain happy numbers and the digit-square process, test whether numbers are happy, describe generalisations and known results, and support recreational and educational use.
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 happy numbers and the digit-square process, test whether numbers are happy, describe generalisations and known results, and support recreational and educational use.
A positive integer that, when the squares of its decimal digits are repeatedly summed, eventually reaches one, such as 1, 7, 10, 13, 19, 23, 28, 31 and 32, while numbers that instead fall into the cycle beginning with 4 are called unhappy or sad; happy numbers are a topic of recreational mathematics and number theory, generalised to other bases and powers, and studied for their density and patterns.
What it is for: Integers whose digit-square iteration reaches one.
It can be explain the definition; test numbers; describe generalisations; support education.
Distinguishing features
Digit-based iteration
Base-dependent
Two possible fates
Recreational interest
What it looks like
Not physical; a set of integers.
How it is recognised
Digit-square iteration reaches one
Otherwise falls into the cycle containing four
Perfect and amicable numbers are other named integer classes
Related models
is a kind of - category
is related to - another named integer class
is related to - numbers in general
is related to - proofs about happy numbers
In practice
Families and kinds
happy numbers in base ten
happy primes
happy numbers in other bases
generalisations with other powers
consecutive happy numbers
Standards and regulation
Mathematical conventions
No regulation
Failure modes and hazards
Confusing base dependence
Assuming patterns hold without proof
Treating the term as having practical significance
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 happy numbers are.
Mathematics.
Definition
Definition and process.
Definition
Definition.
- How is a happy number defined, and why does every number end at one or in the cycle containing four? definition
- Which base is meant? boundary
Examples
Examples.
Examples
Examples.
- Which small numbers are happy, and how are happy primes defined? definition
- Which entry fits prime numbers? action
Compute Testing and computing.
Practice.
Test
Testing a number.
Test
Testing.
- Is this number happy, and what is its iteration sequence? action
- Which entry fits integer algorithms? action
Program
Programming.
Program
Programming.
- How can happy numbers be computed efficiently with cycle detection? action
- Which entry fits algorithms? action
Theory Results and generalisations.
Number theory.
Results
Known results.
Results
Results.
- What is known about the density of happy numbers and runs of consecutive happy numbers? provenance
- Which questions remain open? boundary
Generalise
Generalisations.
Generalise
Generalisations.
- How do happy numbers generalise to other bases and powers? definition
- Which entry fits recreational mathematics? action
Learn History and teaching.
Education.
History
History.
History
History.
- Where did the concept of happy numbers originate? provenance
- Which references are standard? provenance
Teach
Teaching.
Teach
Teaching.
- How can happy numbers be used to teach iteration and proof? action
- Which misconceptions arise? provenance
What the second pass must settle
- Should happy primes be a separate entry?
- How should integer sequence databases be linked?
- How should educational resources be linked?