Heegner number¶

One of the nine square-free positive integers d for which the imaginary quadratic field Q(√−d) has class number one, equivalently unique factorization in its ring of integers.

Version: v1 · 2026-09-08 · History
Domain-specific #: 4846
Origin domain: algebraic number theory
Subdomain: quadratic fields and class numbers

Core Idea¶

A Heegner number is a square-free positive d such that the imaginary quadratic field of discriminant associated with −d has class number one; the complete list is 1,2,3,7,11,19,43,67,163. Class number one makes every ideal principal, giving unique factorization of ideals and elements up to the ring's unit qualifications. The Stark-Heegner theorem proves only nine imaginary quadratic fields have this property. 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¶

Heegner number belongs to algebraic number theory and is useful where the analyst can specify a square-free positive integer d, the imaginary quadratic field Q(√−d), its ring of integers, ideal class group, and class number, then evaluate d is positive and square-free and the correctly computed ring-of-integers class number of Q(√−d) equals one. The scope is broad within that domain but bounded by the need for d is positive and square-free and the correctly computed ring-of-integers class number of Q(√−d) equals one. 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 d is positive and square-free and the correctly computed ring-of-integers class number of Q(√−d) equals one 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 Heegner number 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 Heegner number. Heegner number 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: a square-free positive integer d, the imaginary quadratic field Q(√−d), its ring of integers, ideal class group, and class number. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express d is positive and square-free and the correctly computed ring-of-integers class number of Q(√−d) equals one independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of algebraic number theory because they reuse a square-free positive integer d, the imaginary quadratic field Q(√−d), its ring of integers, ideal class group, and class number, Class number one makes every ideal principal, giving unique factorization of ideals and elements up to the ring's unit qualifications. The Stark-Heegner theorem proves only nine imaginary quadratic fields have this property., and type the carrier, state every parameter and convention in the definition, test that d is positive and square-free and the correctly computed ring-of-integers class number of Q(√−d) equals one, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Heegner number Domain-specific

Parents (1) — more general patterns this builds on

Heegner number is a kind of Classification Prime

The proposed strict upward parent is prime:classification.

Hierarchy path (1) — routes to 1 parentless root

Heegner number → Classification

Neighborhood in Abstraction Space¶

Heegner number sits in a moderately populated region (47^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Elliptic Arithmetic & Zeta Values (7 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08