CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 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.

Proper equivalence of positive-definite integral binary quadratic forms is decidable

Statement

There is an algorithm to decide whether two positive-definite integral binary quadratic forms are properly equivalent: reduce both forms, and compare the two reduced triples.

Facts & Assumptions

Given: Positive-definite integral binary quadratic forms $f$ and $g$ .

[L1]

Every positive-definite integral binary quadratic form is properly equivalent to a reduced form (Every positive-definite integral binary quadratic form is properly equivalent to a reduced form).

[L2]

Each proper-equivalence class contains exactly one reduced form (Each proper-equivalence class of positive-definite integral binary quadratic forms contains exactly one reduced form).

Proof

technique · direct

1.1

By [L1], choose reduced forms $f_{r e d}$ and $g_{r e d}$ properly equivalent to $f$ and $g$ , respectively.

L1choose

2.1

If $f$ and $g$ are properly equivalent, then $f_{r e d}$ and $g_{r e d}$ lie in the same proper-equivalence class, so [L2] gives $f_{r e d} = g_{r e d}$ .

L2step 1.1

2.2

Conversely, if $f_{r e d} = g_{r e d}$ , then $f$ and $g$ are both properly equivalent to that same reduced form, hence are properly equivalent to each other.

step 1.1algebra

3.1

Therefore $f$ and $g$ are properly equivalent exactly when their reduced representatives are equal. Since equality of two explicit coefficient triples is decidable, proper equivalence is decidable.

step 2.1step 2.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 results within two dependency steps 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, Section 9.3.2 (standard reference, not scraped)
Andrew Granville, Binary Quadratic Forms, Exercises 4.1e-4.1g (standard reference, not scraped)