Leyland number
Let an agent define Leyland numbers precisely, compute and test them, relay known Leyland primes and their role in primality testing, and point to sequence references.
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Define The definition.
Definition
Definition.
Definition
Definition.
- What is a Leyland number, and why are x and y required to exceed 1? definition
- Is the question about the first or second kind? boundary
Examples
Examples.
Examples
Examples.
- What are the Leyland numbers below a given bound, and which pairs produce them? measurement
- Which entry fits integer sequences? action
Primes Leyland primes.
Known
Known primes.
Known
Known.
- Which Leyland primes are known, and where are records kept? provenance
- Is the list current? boundary
Testing
Role in primality proving.
Testing
Testing.
- Why are Leyland numbers useful test cases for general primality proving algorithms? provenance
- Which entry fits primality proving? action
Compute Computation.
Generate
Generating.
Generate
Generate.
- How are Leyland numbers generated in order without duplicates? action
- What are the computational limits? measurement
Second kind
Second kind.
Second
Second.
- What are Leyland numbers of the second kind, and what is known about them? provenance
- Which references are standard? provenance
Context History and teaching.
History
Origin.
History
History.
- Who is Paul Leyland, and how did the numbers get their name? provenance
- Which entry fits the person? action
Teach
Teaching.
Teach
Teaching.
- How are Leyland numbers used in teaching and computation exercises? action
- Which misconceptions arise? provenance
Classifiers Filled
- Family
- Thing Registry
- Category
- Cross-cutting context
- Entry kind
- thing
- Plane
- XCT
- Domain
- XCT.QTY
What it is Filled
A number of the form x to the power y plus y to the power x, where x and y are integers greater than 1, named after Paul Leyland; the first are 8, 17, 32, 54, 57, 100 and 145, and Leyland primes such as 17, 593 and 32993 are used as test cases for primality proving because they have no special algebraic form.
Why it exists Filled
Let an agent define Leyland numbers precisely, compute and test them, relay known Leyland primes and their role in primality testing, and point to sequence references.
Distinguishing features Filled
- Symmetric exponential form
- No special algebraic structure aiding primality proofs
- Named after Paul Leyland
- Second kind with subtraction
What robots and AI may and may not do Filled
Must not
- Include cases with x or y equal to 1 as Leyland numbers.
- Quote outdated lists of the largest known Leyland primes as current.
- Claim a probable prime is a proven prime without a primality certificate.
- Confuse Leyland numbers of the first kind with those of the second kind.
Only with a human decision
- Announcing a new record Leyland prime.
May
- Define Leyland numbers and compute them for given values of x and y.
- Relay known Leyland primes and their use as test cases for primality proving, with sources.
- Test whether a given number is a Leyland number.
Moral aspects Filled
- Credit for record primes belongs to the people and projects that found and proved them.
- Errors in published primes can propagate into software tests and benchmarks.
Who is affected
- Mathematicians
- Prime search volunteers
- Students and learners
Owners Filled
Steward
Nobody owns the concept; the mathematical community and prime record lists keep the record.
Links to other meta-models Filled
parent
- Q16317911 - registry parent class
related
- positive integer - a property of integers
- primality test - as test cases
- On-Line Encyclopedia of Integer Sequences - sequence A076980
- prime number - Leyland primes
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.leyland-number
- Wikidata: Q2661866 (https://www.wikidata.org/wiki/Q2661866)
- OEIS: A076980 Leyland numbers
- OEIS: A094133 Leyland primes
Direct properties not applicable Not applicable
- first terms: 8, 17, 32, 54, 57, 100, 145, 177, 320 integers
- condition: x and y greater than 1 condition - to exclude trivial cases
Plane XCT: no invented physical properties.
Recognition optional Filled
- Form x to the y plus y to the x with x, y greater than 1
- First terms 8, 17, 32, 54, 57, 100
- Leyland numbers of the second kind subtract instead of add
- Not visible; a property of integers.
Capabilities and actions required Filled
- compute Leyland numbers for given x and y
- list examples and known primes
- explain their use in primality proving
- point to sequence references
Hazards and failure modes required Filled
- Including trivial cases with x or y equal to 1
- Confusing the two kinds
- Quoting outdated lists of known primes
Standards and interfaces required Filled
- No regulation; definitions follow the literature and the OEIS
Context of use required Filled
- Not for anything; test cases in computational number theory.
- Leyland numbers
- Leyland primes
- Leyland numbers of the second kind
- generalisations with other bases
Sources Filled
- Wikidata item Q2661866: Leyland number - identity and sense of the item
- Wikipedia: Leyland number - general description of the item
Open questions
- Should Leyland primes be a separate entry?
- How should prime records be linked?
- Which conjectures should be recorded?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q2661866/spec.json