ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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.

Squarefree sums of two squares up to $30$

Example

Among the squarefree positive integers at most $30$ , those representable as sums of two squares are

$1, 2, 5, 10, 13, 17, 26, 29.$

Every displayed representation below is primitive.

Facts & Assumptions

Given: Positive integers at most $30$ .

[F1]

A two-square representation is primitive when its coordinate gcd is $1$ (Representations and primitive representations as sums of two squares).

[F2]

A positive integer $n$ is squarefree if no square of a prime divides $n$ (Squarefree positive integers).

[L1]

A squarefree positive integer is a sum of two squares if and only if none of its odd prime factors is congruent to $3$ modulo $4$ ; every such representation is primitive (Squarefree sums of two squares).

Verification

technique · direct

1.1

Trial division gives the squarefree positive integers at most $30$ as $1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30$ .

F2algebra

2.1

Applying [L1] removes exactly those having an odd prime factor congruent to $3$ modulo $4$ , leaving $1, 2, 5, 10, 13, 17, 26, 29$ .

step 1.1L1algebra

3.1

The complete witness list is $1 = 1^{2} + 0^{2}$ , $2 = 1^{2} + 1^{2}$ , $5 = 1^{2} + 2^{2}$ , $10 = 1^{2} + 3^{2}$ , $13 = 2^{2} + 3^{2}$ , $17 = 1^{2} + 4^{2}$ , $26 = 1^{2} + 5^{2}$ , and $29 = 2^{2} + 5^{2}$ . Each coordinate gcd is one, as [F1] and [L1] require.

step 2.1F1L1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

P. Hackman, Elementary Number Theory, Chapter E, Theorems E.II.2 and E.II.4 (standard reference, not scraped)

Squarefree sums of two squares up to 3030