Proth number
Let an agent define Proth numbers and Proth primes, explain Proth theorem and its use as a primality test, relate them to Fermat and Cullen numbers, and help test or list examples.
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Define What Proth numbers are.
Definition
Definition and examples.
Definition
Definition.
- What is a Proth number, and what are the first examples? definition
- Does the number in question satisfy the form and the condition on k? boundary
Primes
Proth primes.
Primes
Primes.
- What are Proth primes, and which large primes have this form? definition
- Which entry fits Proth prime? action
Test Primality testing.
Theorem
Proth theorem.
Theorem
Theorem.
- What does Proth theorem state, and how is it applied as a test? action
- Is the Proth number in question prime? measurement
Algorithms
Algorithms.
Algorithms
Algorithms.
- How is the test implemented efficiently for large numbers? provenance
- Which references are standard? provenance
Theory Related forms.
Fermat
Fermat numbers.
Fermat
Fermat.
- How are Fermat numbers Proth numbers, and what does Pepin test have to do with Proth theorem? provenance
- Which entry fits Fermat number? action
Sierpinski
Sierpinski and related problems.
Sierpinski
Sierpinski.
- How do Proth numbers relate to Sierpinski numbers and the Seventeen or Bust project? provenance
- Which sources are cited? provenance
Context History and computing.
History
History.
History
History.
- Who was Francois Proth, and how was his theorem received? provenance
- Which entry fits the history of number theory? action
Computing
Distributed prime searches.
Computing
Computing.
- How do distributed projects search for Proth primes? provenance
- Which entry fits distributed computing? 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 of the form k times 2 to the power n plus 1, where k is odd and k is less than 2 to the power n, such as 3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81 and 97; Proth numbers are named after Francois Proth, whose 1878 theorem gives a simple primality test for them, and Proth primes include many of the largest known primes found by distributed computing.
Why it exists Filled
Let an agent define Proth numbers and Proth primes, explain Proth theorem and its use as a primality test, relate them to Fermat and Cullen numbers, and help test or list examples.
Distinguishing features Filled
- Special form
- Simple primality test
- Source of large primes
- Connection to Fermat numbers
What robots and AI may and may not do Filled
Must not
- Omit the condition that k is odd and less than 2 to the n.
- Confuse Proth numbers with Proth primes.
- Report unverified results for large cases as confirmed.
- Claim new prime discoveries without verification.
Only with a human decision
- Announcing a newly found large Proth prime.
May
- State the definition of a Proth number including the condition that k is less than 2 to the n.
- Check whether a number is a Proth number and show the working.
- Explain Proth's theorem and its use for testing primes.
Moral aspects Filled
- Correct credit for discoveries matters to mathematicians and volunteer projects.
- Accurate definitions keep students from learning errors.
Who is affected
- Students
- Mathematicians
- Volunteer computing participants
Owners Filled
Steward
Nobody: a mathematical concept held in common.
Links to other meta-models Filled
parent
- Q16317911 - registry parent class
related
- positive integer - in registry terms
- Proth theorem - the primality test
- Fermat number - as a special case
- Francois Proth - the mathematician
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.proth-number
- Wikidata: Q593418 (https://www.wikidata.org/wiki/Q593418)
- OEIS: A080075 Proth numbers
Direct properties not applicable Not applicable
- first terms: 3, 5, 9, 13, 17, 25, 33, 41, 49, 57 sequence
- Proth theorem: 1878 year
Plane XCT: no invented physical properties.
Recognition optional Filled
- k times 2 to the n plus 1 with odd k less than 2 to the n
- Sequence 3, 5, 9, 13, 17, 25, 33
- Fermat numbers are a special case; Sierpinski numbers are related
- Not a visible object; integers of a special form.
Capabilities and actions required Filled
- define the numbers and primes
- explain Proth theorem
- relate to other special forms
- help test or list examples
Hazards and failure modes required Filled
- Omitting the condition k less than 2 to the n
- Confusing Proth numbers with Proth primes
- Computation errors for large cases
Standards and interfaces required Filled
- No regulation; the OEIS and prime databases are references
Context of use required Filled
- Primality testing and prime searching.
- Proth numbers
- Proth primes
- Fermat numbers as Proth numbers with k equal to 1
- related forms such as Cullen and Sierpinski numbers
Sources Filled
- Wikidata item Q593418: Proth number - identity and sense of the item
- Wikipedia: Proth number - general description of the item
Open questions
- Should Proth prime be a separate entry?
- How should prime databases be linked?
- How should records of largest Proth primes be kept current?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q593418/spec.json