DefinitionDefinition: 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.

Integers represented, and primitively represented, by a binary quadratic form

Definition

Let $f (x, y) = a x^{2} + b x y + c y^{2}$ be an integral binary quadratic form (Integral binary quadratic forms) and let $n \in Z$ .

We say that $f$ represents $n$ when there exist integers $r, s$ such that

$n = f (r, s) = a r^{2} + b r s + c s^{2} .$

We say that $f$ primitively represents $n$ when there exist integers $r, s$ with

$n = f (r, s) and \gcd (r, s) = 1$

(Common divisor, and the greatest common divisor $\gcd (a, b)$ , with the convention $\gcd (0, 0) : = 0$ ).

Remarks

Primitive representation is a condition on the representing pair $(r, s)$ , not on the coefficients of the form.
A form can represent an integer without primitively representing it.

Depends on

Used by

Dependency tree · two levels

8 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, Chapter 9 (standard reference, not scraped)
Andrew Granville, Binary Quadratic Forms, Chapter 4 (standard reference, not scraped)