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

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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.
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The primitive two-square criterion distinguishes $289$ , $34$ , and $833$

Example

The integers $289$ and $34$ have primitive representations, whereas $833$ is representable but has no primitive two-square representation:

$289 = 8^{2} + 15^{2}, 34 = 3^{2} + 5^{2}, 833 = 28^{2} + 7^{2} .$

Facts & Assumptions

Given: The three displayed integers and coordinate pairs.

[F1]

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

[L1]

A positive integer $n$ has a primitive two-square representation if and only if $v_{2} (n) \leq 1$ and no prime $q \equiv 3 (m o d 4)$ divides $n$ (Characterisation of primitive sums of two squares).

Verification

technique · direct

1.1

One has $8^{2} + 15^{2} = 289$ with $\gcd (8, 15) = 1$ , and $3^{2} + 5^{2} = 34$ with $\gcd (3, 5) = 1$ , so both displayed representations are primitive by [F1].

F1L1algebra

1.2

One has $28^{2} + 7^{2} = 833$ , but $\gcd (28, 7) = 7$ , so the displayed representation is not primitive.

F1algebra

2.1

Since $833 = 7^{2} \cdot 17$ and $7 \equiv 3 (m o d 4)$ , [L1] excludes every primitive representation of $833$ , not just the displayed one.

step 1.2L1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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, Example E.II.7(a)–(c) (standard reference, not scraped)

The primitive two-square criterion distinguishes 289289, 3434, and 833833