almost prime
Let an agent define almost primes and their classes, give examples, relay their role in sieve theory and results such as Chen theorem, and help classify a number by its count of prime factors.
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Define What almost primes are.
Definition
Definition and classes.
Definition
Definition.
- What is a k-almost prime, and what are the first members of each class? definition
- Is the number classified by factors with or without multiplicity? boundary
Examples
Examples.
Examples
Examples.
- Which numbers are semiprimes and 3-almost primes, and how are they recognised? definition
- Which entry fits semiprime? action
Compute Classification and counting.
Classify
Classifying a number.
Classify
Classify.
- How is a number classified as a k-almost prime by factorisation? action
- What class does the number in question belong to? measurement
Counting
Counting functions.
Counting
Counting.
- How many k-almost primes are there up to x, asymptotically? provenance
- Which references are standard? provenance
Theory Sieve theory.
Sieves
Sieve methods.
Sieves
Sieves.
- How do sieve methods yield results about almost primes? provenance
- Which entry fits sieve theory? action
Chen
Chen theorem.
Chen
Chen.
- What does Chen theorem say, and how does it relate to the Goldbach conjecture? provenance
- Which entry fits Chen theorem? action
Context Applications and history.
Applications
Applications.
Applications
Applications.
- Where do semiprimes and almost primes appear in cryptography and elsewhere? provenance
- Which entry fits RSA? action
History
History.
History
History.
- How did the study of almost primes develop in twentieth-century number theory? provenance
- Which sources are cited? provenance
Classifiers Filled
- Family
- Thing Registry
- Category
- Cross-cutting context
- Entry kind
- thing
- Plane
- XCT
- Domain
- XCT.QTY
What it is Filled
A positive integer with exactly k prime factors counted with multiplicity, called a k-almost prime, so that primes are 1-almost primes, semiprimes such as 6, 9 and 15 are 2-almost primes, and numbers such as 8, 12 and 18 are 3-almost primes; almost primes are studied in analytic number theory, where sieve methods prove results about them that approach results about primes, as in Chen's theorem that every sufficiently large even integer is the sum of a prime and a number with at most two prime factors counted with multiplicity.
Why it exists Filled
Let an agent define almost primes and their classes, give examples, relay their role in sieve theory and results such as Chen theorem, and help classify a number by its count of prime factors.
Distinguishing features Filled
- Counts factors with multiplicity
- Primes as the k equals 1 case
- Sieve theory tool
- Chen theorem
What robots and AI may and may not do Filled
Must not
- Forget multiplicity when counting prime factors.
- Confuse almost primes with squarefree numbers.
- Overstate results as statements about primes.
- Report unverified computations as confirmed.
Only with a human decision
- Publishing new claimed results.
May
- Define k-almost primes and check whether a number is one, showing factorisation.
- Explain related results with sources.
Moral aspects Filled
- Accurate definitions matter for students.
- Correct credit for results matters to mathematicians.
Who is affected
- Students
- Mathematicians
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
- semiprime - the case k equals 2
- prime number - the case k equals 1
- sieve theory - in analytic number theory
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.almost-prime
- Wikidata: Q936614 (https://www.wikidata.org/wiki/Q936614)
- OEIS: A001358 for semiprimes, A014612 for 3-almost primes
Direct properties not applicable Not applicable
- 2-almost primes: 4, 6, 9, 10, 14, 15, 21, 22 sequence - semiprimes
- 3-almost primes: 8, 12, 18, 20, 27, 28, 30 sequence
Plane XCT: no invented physical properties.
Recognition optional Filled
- Exactly k prime factors with multiplicity
- Primes, semiprimes and higher classes
- Squarefree numbers and composite numbers are different classifications
- Not a visible object; integers classified by factor count.
Capabilities and actions required Filled
- define k-almost primes
- give examples and counts
- relay sieve theory results
- classify numbers
Hazards and failure modes required Filled
- Forgetting multiplicity in counting factors
- Confusing almost primes with squarefree numbers
- Overstating results as statements about primes
Standards and interfaces required Filled
- No regulation; the OEIS is the reference database
Context of use required Filled
- A concept in number theory and sieve methods.
- 1-almost primes, the primes
- 2-almost primes, the semiprimes
- 3-almost primes and higher classes
- almost primes in arithmetic progressions
- almost prime values of polynomials
Sources Filled
- Wikidata item Q936614: almost prime - identity and sense of the item
- Wikipedia: Almost prime - general description of the item
Open questions
- Should semiprime be a separate primary entry?
- How should the OEIS be linked?
- How should sieve theory results be recorded?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q936614/spec.json