DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-26

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Positive-definite binary quadratic forms

Definition

An integral binary quadratic form $f (x, y) = a x^{2} + b x y + c y^{2}$ (Integral binary quadratic forms) is positive definite when

$f (x, y) > 0$

for every real pair $(x, y) \neq (0, 0)$ .

Remarks

The condition is about positivity on $R^{2}$ , not only on $Z^{2}$ .
The next proposition turns this intrinsic condition into the coefficient test $a > 0$ and $Δ < 0$ .

Depends on

Integral binary quadratic forms

Used by

Dependency tree · one level

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Sources

Andrew Granville, Binary Quadratic Forms, Chapter 4 (standard reference, not scraped)
William Stein, Elementary Number Theory and Elliptic Curves, Section 9.2.4 (standard reference, not scraped)