Brun sieve¶
A combinatorial sieve that estimates integers avoiding specified prime divisibility conditions by truncating inclusion-exclusion with alternating upper and lower bounds.
Core Idea¶
Brun's pure sieve introduced weighted control of sifted sets and enabled results about almost primes, including convergence of the sum of reciprocals of twin primes, while lacking the parity resolution needed for primes themselves. Divisibility subsets are combined by inclusion-exclusion over square-free products; truncation and combinatorial weights bound the uncomputed remainder and produce upper or lower estimates for the sifted set. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Brun sieve belongs to analytic number theory and is useful where the analyst can specify the typed analytic number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the finite sequence or weighted set, sieving primes and level, divisibility counts and remainder terms, square-free products, inclusion-exclusion weights, upper or lower bound, uniformity assumptions, and parity limitation are explicit. The scope is broad within that domain but bounded by the need for the finite sequence or weighted set, sieving primes and level, divisibility counts and remainder terms, square-free products, inclusion-exclusion weights, upper or lower bound, uniformity assumptions, and parity limitation are explicit.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the finite sequence or weighted set, sieving primes and level, divisibility counts and remainder terms, square-free products, inclusion-exclusion weights, upper or lower bound, uniformity assumptions, and parity limitation are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Brun sieve. Brun sieve compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed analytic number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the finite sequence or weighted set, sieving primes and level, divisibility counts and remainder terms, square-free products, inclusion-exclusion weights, upper or lower bound, uniformity assumptions, and parity limitation are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of analytic number theory because they reuse the typed analytic number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Divisibility subsets are combined by inclusion-exclusion over square-free products; truncation and combinatorial weights bound the uncomputed remainder and produce upper or lower estimates for the sifted set., and type the carrier, state every parameter and convention in the definition, test that the finite sequence or weighted set, sieving primes and level, divisibility counts and remainder terms, square-free products, inclusion-exclusion weights, upper or lower bound, uniformity assumptions, and parity limitation are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Brun sieve Domain-specific
Parents (1) — more general patterns this builds on
-
Brun sieve is a kind of Structural Filtering Prime
The proposed strict upward parent is
prime:structural_filtering.
Hierarchy path (1) — routes to 1 parentless root
- Brun sieve → Structural Filtering → Selection
Neighborhood in Abstraction Space¶
Brun sieve sits in a crowded region of the domain-specific corpus (25th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Arithmetic Functions & Number Sequences (16 abstractions)
Nearest neighbors
- Lucky number — 0.92
- Faulhaber's formula — 0.91
- Dirichlet density — 0.91
- Highly totient number — 0.90
- Niven's constant — 0.90
Computed from structural-signature embeddings · 2026-09-08