DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-31

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 two-square representation function r_2

Definition

For each positive integer $n$ , define

r_{2} (n) : = # {(x, y) \in Z^{2} : x^{2} + y^{2} = n} .

This counts order and signs separately, in the sense of Representations and primitive representations as sums of two squares.

Remarks

The domain is the positive integers, because $r_{2}$ is being used here as an arithmetic function. In particular, $r_{2} (0)$ is outside the present convention.
The ordered-sign convention gives $r_{2} (1) = 4$ .

Depends on

Used by

The normalized two-square count is multiplicative with the expected prime-power values Lemma

Dependency tree · two levels

5 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

Peter Hackman, Elementary Number Theory (standard reference, not scraped)
Karl-Dieter Crisman, Number Theory: In Context and Interactive (standard reference, not scraped)