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Each proper-equivalence class of positive-definite integral binary quadratic forms contains exactly one reduced form
Statement
Each proper-equivalence class of positive-definite integral binary quadratic forms contains exactly one reduced form.
Facts & Assumptions
Given: A positive-definite integral binary quadratic form .
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).
In a reduced class, the leading coefficient is minimal among all properly equivalent forms (The leading coefficient of a reduced positive-definite form is minimal in its proper-equivalence class).
Properly equivalent reduced forms with the same leading coefficient are equal (Properly equivalent reduced forms with the same leading coefficient are equal).
Proof
By [L1], the proper-equivalence class of contains at least one reduced form.
Suppose and are reduced forms properly equivalent to . Then and are properly equivalent to each other. Applying [L2] first to against and then to against shows that their leading coefficients are equal.
With equal leading coefficients, [L3] gives . So the reduced representative is unique.
Existence from step 1.1 and uniqueness from step 3.1 prove the theorem.
Depends on
Used by
- For each negative discriminant, there are only finitely many proper-equivalence classes of positive-definite integral binary quadratic forms Corollary
- Proper equivalence of positive-definite integral binary quadratic forms is decidable Corollary
- An indefinite proper-equivalence class can contain a cycle of reduced forms Counterexample
- Distinct reduced forms can represent the same integers Counterexample
- The reduced primitive forms of discriminant -20 Example
- The reduced primitive forms of discriminant -23 Example
- The reduced primitive forms of discriminant -4 Example
- The reduced primitive forms of discriminant -8 Example
Dependency tree · two levels
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Sources
- William Stein, Elementary Number Theory and Elliptic Curves, Theorem 9.3.2 (standard reference, not scraped)
- Andrew Granville, Binary Quadratic Forms, Exercise 4.1f(c) (standard reference, not scraped)