Farey sequence¶
The increasing sequence of reduced rational numbers between zero and one whose denominators do not exceed a chosen order.
Core Idea¶
Adjacent terms a/b and c/d in the order-n Farey sequence satisfy bc-ad=1 under the standard convention, and mediants generate new terms as the order increases. All eligible reduced fractions are enumerated, duplicates are removed through coprimality and exact rational comparison sorts them; determinant and mediant relations expose the sequence's local structure. 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¶
Farey sequence belongs to elementary number theory and is useful where the analyst can specify the typed elementary number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the positive order n, interval convention, reduced fractions, denominator bound, endpoint inclusion and increasing rational order are explicit. The scope is broad within that domain but bounded by the need for the positive order n, interval convention, reduced fractions, denominator bound, endpoint inclusion and increasing rational order are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the positive order n, interval convention, reduced fractions, denominator bound, endpoint inclusion and increasing rational order 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. A bare label is insufficient because the name Farey sequence can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.
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 Farey sequence. Farey sequence 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 elementary 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 positive order n, interval convention, reduced fractions, denominator bound, endpoint inclusion and increasing rational order are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of elementary number theory because they reuse the typed elementary number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, All eligible reduced fractions are enumerated, duplicates are removed through coprimality and exact rational comparison sorts them; determinant and mediant relations expose the sequence's local structure., and type the carrier, state every parameter and convention in the definition, test that the positive order n, interval convention, reduced fractions, denominator bound, endpoint inclusion and increasing rational order are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Farey sequence Domain-specific
Parents (1) — more general patterns this builds on
-
Farey sequence is a kind of Local Sequence Legality Prime
The proposed strict upward parent is
prime:local_sequence_legality.
Hierarchy path (1) — routes to 1 parentless root
- Farey sequence → Local Sequence Legality → Local-to-Global Aggregation
Neighborhood in Abstraction Space¶
Farey sequence sits in a crowded region of the domain-specific corpus (22nd percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Number-Theoretic Sequences & Classes (37 abstractions)
Nearest neighbors
- Composite number — 0.92
- Complete sequence — 0.91
- Square number — 0.91
- Geometric progression — 0.91
- Gauss's lemma (number theory) — 0.91
Computed from structural-signature embeddings · 2026-09-08