automorphic number
Let an agent explain automorphic numbers, relay their definition, examples and construction from mathematical references, describe the connection to p-adic numbers, and distinguish them from related recreational number classes.
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Understand What an automorphic number is.
Definition
Definition.
Definition
Definition.
- What is an automorphic number, and what are the first examples in base ten? definition
- Is the question about automorphic numbers or a related digit-based class? boundary
Construction
Construction.
Construction
Construction.
- How are longer automorphic numbers built digit by digit, and why are there exactly two sequences? action
- Which entry fits the specific construction? action
Theory Theory.
p-adic
p-adic view.
p-adic
p-adic.
- How do automorphic numbers correspond to idempotents in the 10-adic integers? provenance
- Which references are standard? provenance
Bases
Other bases.
Bases
Bases.
- How do automorphic numbers behave in bases that are prime powers or products of primes? provenance
- Which entry fits number base? action
Explore Related classes.
Trimorphic
Trimorphic numbers.
Trimorphic
Trimorphic.
- What are trimorphic numbers, and how do they relate? provenance
- Which sources are cited? provenance
Recreational
Recreational mathematics.
Recreational
Recreational.
- How do automorphic numbers fit into recreational number theory? provenance
- Which entry fits recreational mathematics? action
Context History and computation.
History
History.
History
History.
- When were automorphic numbers studied, and by whom? provenance
- Which entry fits the history of number theory? action
Computation
Computation.
Computation
Computation.
- How can automorphic numbers be computed programmatically? action
- Which entry fits computational number theory? action
Classifiers Filled
- Family
- Thing Registry
- Category
- Cross-cutting context
- Entry kind
- thing
- Plane
- XCT
- Domain
- XCT.QTY
What it is Filled
A natural number whose square ends in the same digits as the number itself in a given base, such as 5, 6, 25, 76, 376 and 625 in base ten, arising from the solutions of n^2 = n modulo powers of the base and related to the 10-adic idempotents; automorphic numbers form two infinite sequences in base ten extending the digits ...90625 and ...09376, and generalise to trimorphic numbers whose cubes end in the same digits.
Why it exists Filled
Let an agent explain automorphic numbers, relay their definition, examples and construction from mathematical references, describe the connection to p-adic numbers, and distinguish them from related recreational number classes.
Distinguishing features Filled
- Self-reproducing square
- Base dependence
- Two infinite sequences
- p-adic idempotents
What robots and AI may and may not do Filled
Must not
- State that a number is automorphic without saying in which base.
- Present a computational search result as a proof of a general pattern.
- Claim significance for these numbers in numerology, prediction or finance.
Only with a human decision
- Publishing a claimed new result about these numbers under a person's name.
May
- Explain what automorphic numbers are and list the known small ones in a stated base.
- Check or generate automorphic numbers in a given base and show the modular condition used.
Moral aspects Filled
- Number curiosities are often dressed up as mystical or predictive, which misleads people.
- Credit for computational results should go to those who produced them.
Who is affected
- Students and learners
- Recreational mathematicians
Owners Filled
Steward
Nobody owns the concept; integer sequence databases record the known values.
Master systems
- Published integer sequence databases
Links to other meta-models Filled
parent
- Q28920052 - registry parent class
related
- non-negative integer - in registry terms
- p-adic number - through 10-adic idempotents
- trimorphic number - using cubes
- OEIS - as A003226
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.automorphic-number
- Wikidata: Q2313686 (https://www.wikidata.org/wiki/Q2313686)
- OEIS: A003226 automorphic numbers
Direct properties not applicable Not applicable
- first base ten examples: 0, 1, 5, 6, 25, 76, 376, 625 sequence - OEIS A003226
- base ten sequences: 2 count - for n > 1 digits
Plane XCT: no invented physical properties.
Recognition optional Filled
- Numbers whose square ends in the number
- Base ten sequences ending 90625 and 09376
- Trimorphic numbers use cubes; narcissistic numbers use digit powers
- Not a visible object; integers with a self-reproducing square.
Capabilities and actions required Filled
- explain the definition
- relay examples and construction
- describe the p-adic connection
- distinguish related classes
Hazards and failure modes required Filled
- Assuming base independence
- Confusing with other digit-based classes
- Miscounting sequences
Standards and interfaces required Filled
- No regulation; standard mathematical definitions
Context of use required Filled
- Not applicable; a mathematical concept.
- automorphic numbers in base ten
- automorphic numbers in other bases
- trimorphic numbers
- circular and other digit-based number classes
Sources Filled
- Wikidata item Q2313686: automorphic number - identity and sense of the item
- Wikipedia: Automorphic number - general description of the item
Open questions
- Should trimorphic number be a separate entry?
- How should OEIS and references be linked?
- How should the p-adic interpretation be recorded?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q2313686/spec.json