Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-26
How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
The discriminant of a binary quadratic form
Definition
The discriminant of the integral binary quadratic form
is the integer
When the form is clear from context, we also write its discriminant as .
Depends on
Used by
- An indefinite proper-equivalence class can contain a cycle of reduced forms Counterexample
- Forms of discriminant -24 need not be properly equivalent Counterexample
- The class number of primitive positive-definite binary quadratic forms of discriminant Δ Definition
- The principal binary quadratic form of a discriminant Definition
- A reduced positive-definite form of discriminant Δ satisfies a≤√|Δ|/3 Lemma
- Properly equivalent reduced forms with the same leading coefficient are equal Lemma
- A positive integer n is primitively represented by some discriminant Δ form exactly when Δ is a square modulo 4n Proposition
- An integer is the discriminant of an integral binary quadratic form exactly when it is congruent to 0 or 1 modulo 4 Proposition
- An integral binary quadratic form is positive definite exactly when its leading coefficient is positive and its discriminant is negative Proposition
- Proper equivalence preserves discriminant and primitivity of the form Proposition
Dependency tree · one level
1 result within one dependency step of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- William Stein, Elementary Number Theory and Elliptic Curves, Definition 9.2.6 (standard reference, not scraped)
- Andrew Granville, Binary Quadratic Forms, Chapter 4 (standard reference, not scraped)