prime power
Let an agent define prime powers, relay their properties and roles in number theory, algebra and finite fields, and distinguish them from primes, perfect powers and composite numbers generally.
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Define Definition.
Definition
Definition.
Definition
Definition.
- What is a prime power, and does the definition include primes and 1? definition
- Is the number a prime power, a perfect power or a general composite? boundary
Test
Testing.
Test
Test.
- How is a number tested for being a prime power? action
- Which entry fits integer factorisation? action
Properties Properties.
Arithmetic
Arithmetic properties.
Arithmetic
Arithmetic.
- What properties do prime powers have, such as totient values and divisor counts? provenance
- Which references are standard? provenance
Distribution
Distribution.
Distribution
Distribution.
- How are prime powers distributed among the integers? provenance
- Which sources are cited? provenance
Algebra Algebraic roles.
Fields
Finite fields.
Fields
Fields.
- Why does a finite field exist exactly for each prime power order? provenance
- Which entry fits finite field? action
Groups
Groups.
Groups
Groups.
- What are p-groups and Sylow theorems, and how do prime powers enter group theory? provenance
- Which entry fits p-group? action
Context Applications and history.
Applications
Applications.
Applications
Applications.
- Where do prime powers appear in coding theory, cryptography and combinatorics? provenance
- Which entry fits the specific application? action
History
History.
History
History.
- How did the study of prime powers develop with the fundamental theorem of arithmetic and Galois fields? provenance
- Which entry fits the history of algebra? action
Classifiers Filled
- Family
- Thing Registry
- Category
- Cross-cutting context
- Entry kind
- thing
- Plane
- XCT
- Domain
- XCT.QTY
What it is Filled
A positive integer that is a power of a single prime number, of the form p to the k for a prime p and a positive integer k, such as 2, 4, 8, 3, 9, 27, 5 and 25, including the primes themselves as the case k equals 1; prime powers are the building blocks of integers through the fundamental theorem of arithmetic, are exactly the orders of finite fields and appear in group theory, cyclotomic polynomials and many number-theoretic results.
Why it exists Filled
Let an agent define prime powers, relay their properties and roles in number theory, algebra and finite fields, and distinguish them from primes, perfect powers and composite numbers generally.
Distinguishing features Filled
- Single prime factor
- Orders of finite fields
- Cyclic unit groups for odd primes
- Convention on including 1
What robots and AI may and may not do Filled
Must not
- Treat a composite number with several prime factors as a prime power.
- Present a probabilistic primality test result as a proof.
- Rely on untested prime powers in cryptographic or field construction work.
- Omit the method and bound when reporting a computed result.
Only with a human decision
- Choosing parameters for a cryptographic system.
May
- State the definition of a prime power and give correct examples.
- Test primality and prime power form within stated limits and methods.
Moral aspects Filled
- Weak or wrongly tested parameters break cryptography that people depend on.
- Teaching material is widely copied, so errors propagate.
Who is affected
- Users of cryptographic systems
- Students
- Software developers
Owners Filled
Steward
Nobody owns a mathematical fact.
Links to other meta-models Filled
parent
- Q16317911 - registry parent class
related
- positive integer - in registry terms
- prime number - as the exponent-one case
- finite field - for every prime power
- perfect power - which allows composite bases
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.prime-power
- Wikidata: Q1667469 (https://www.wikidata.org/wiki/Q1667469)
- OEIS: A000961 with 1
- OEIS: A246655 without 1
Direct properties not applicable Not applicable
- count below 100: 35 numbers - including 1 by some conventions
Plane XCT: no invented physical properties.
Recognition optional Filled
- Powers of a single prime
- Sequence OEIS A000961 including primes
- Perfect powers may have composite bases; semiprimes are products of two primes
- Numbers such as 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32.
Capabilities and actions required Filled
- define and characterise
- relay properties
- explain roles in algebra
- distinguish related classes
Hazards and failure modes required Filled
- Confusing with perfect powers
- Convention disputes on including 1
- Errors in factorisation
Standards and interfaces required Filled
- No regulation; mathematical conventions
Context of use required Filled
- Not applicable; a class of integers.
- primes as prime powers with exponent 1
- proper prime powers with exponent at least 2
- powers of 2
- prime powers as finite field orders
- prime power groups in group theory
Sources Filled
- Wikidata item Q1667469: prime power - identity and sense of the item
- Wikipedia: Prime power - general description of the item
Open questions
- Should the convention on 1 be recorded explicitly?
- How should reference sequences be linked?
- Which algebraic applications should be primary links?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q1667469/spec.json