Schnirelmann density
Let an agent explain Schnirelmann density, relay its definition, properties and use in Schnirelmann theorem from mathematical references, and distinguish it from natural and other densities.
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Understand What Schnirelmann density is.
Definition
Definition.
Definition
Definition.
- How is Schnirelmann density defined, and why does missing 1 give density zero? definition
- Is the question about Schnirelmann density, natural density or another density? boundary
Examples
Examples.
Examples
Examples.
- What are the Schnirelmann densities of the odd numbers, the squares and the primes? measurement
- Which entry fits the specific set? action
Theory Theorems.
Sumsets
Sumset inequalities.
Sumsets
Sumsets.
- What do the Schnirelmann and Mann inequalities say about sumsets? provenance
- Which references are standard? provenance
Primes
Sums of primes.
Primes
Primes.
- How does Schnirelmann theorem show every integer is a sum of boundedly many primes, and what is known about the constant? provenance
- Which entry fits Goldbach conjecture? action
Compare Comparison.
Densities
Other densities.
Densities
Densities.
- How does Schnirelmann density compare with natural, lower, upper and Banach densities? provenance
- Which sources are cited? provenance
Uses
Uses.
Uses
Uses.
- Where else is Schnirelmann density used in additive combinatorics? provenance
- Which entry fits additive combinatorics? action
Context History.
History
History.
History
History.
- Who was Lev Schnirelmann, and how did his work lead to later results by Mann, Vinogradov and Helfgott? provenance
- Which entry fits the history of number theory? action
Teaching
Teaching.
Teaching
Teaching.
- How is Schnirelmann density presented in additive number theory courses? provenance
- Which entry fits mathematics education? action
Classifiers Filled
- Family
- Thing Registry
- Category
- Cross-cutting context
- Entry kind
- thing
- Plane
- XCT
- Domain
- XCT.QTY
What it is Filled
A measure in additive number theory of how dense a set of natural numbers is, defined by Schnirelmann in 1930 as the infimum over positive integers n of the number of set members from 1 through n divided by n, so that any set missing 1 has density zero; for sets A and B of nonnegative integers both containing zero, with densities alpha and beta calculated without counting zero, their sumset has density at least alpha + beta - alpha * beta; the density underlies Schnirelmann's theorem that every integer greater than one is a sum of a bounded number of primes and is distinct from natural density.
Why it exists Filled
Let an agent explain Schnirelmann density, relay its definition, properties and use in Schnirelmann theorem from mathematical references, and distinguish it from natural and other densities.
Distinguishing features Filled
- Infimum definition
- Sensitivity to small numbers
- Sumset inequality
- Prime sum application
What robots and AI may and may not do Filled
Must not
- Confuse Schnirelmann density with asymptotic or natural density.
- Present conjectures as proved results.
- Misattribute results in additive number theory.
Only with a human decision
- Announcing a claimed proof of an open problem using the density.
May
- Compute and explain Schnirelmann density for given sets of natural numbers.
- Cite the theorems that use it with their sources.
Moral aspects Filled
- Credit for results belongs to those who proved them.
- False claims about open problems mislead students and waste reviewers' time.
Who is affected
- Mathematicians
- Students
Owners Filled
Steward
Nobody owns the concept; it is held in common.
Links to other meta-models Filled
parent
- Q614112 - registry parent class
related
- number of entities - in registry terms, as a limiting proportion
- natural density - which is a limit
- Schnirelmann theorem - on sums of primes
- Mann theorem - on sumset densities
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.schnirelmann-density
- Wikidata: Q2482753 (https://www.wikidata.org/wiki/Q2482753)
Direct properties not applicable Not applicable
- defined: 1930 year - Lev Schnirelmann
- Schnirelmann constant: at most 7 value - Ramare 1995; 3 with Helfgott ternary result
Plane XCT: no invented physical properties.
Recognition optional Filled
- Infimum density of a set of naturals
- Schnirelmann density and Schnirelmann theorem on sums of primes
- Natural density is a limit; upper and lower densities are limsup and liminf
- Not a visible object; an infimum of ratios.
Capabilities and actions required Filled
- explain the definition
- relay properties and theorems
- describe applications
- distinguish from other densities
Hazards and failure modes required Filled
- Confusing with natural density
- Forgetting that missing 1 gives density zero
- Overstating the Schnirelmann constant results
Standards and interfaces required Filled
- No regulation; standard mathematical definitions
Context of use required Filled
- Not applicable; a mathematical concept.
- Schnirelmann density of sets
- Mann theorem strengthening the sumset inequality
- Schnirelmann theorem and constant
- related densities: natural, lower, upper, Banach
Sources Filled
- Wikidata item Q2482753: Schnirelmann density - identity and sense of the item
- Wikipedia: Schnirelmann density - general description of the item
Open questions
- Should Schnirelmann theorem be a separate entry?
- How should mathematical references be linked?
- How should the constant history be recorded?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q2482753/spec.json