CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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.

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The elementary Pell order can miss units from the larger quadratic order

Statement refuted

Every unit of a quadratic order containing $Z [\sqrt{D}]$ already lies in the explicit Pell order $Z [\sqrt{D}]$ .

Witness

Take $D = 5$ . The larger explicit order $Z ⁣ [\frac{1 + \sqrt{5}}{2}]$ contains the unit $u : = \frac{1 + \sqrt{5}}{2},$ but $u \notin Z [\sqrt{5}]$ .

Facts & Assumptions

Given: The explicit Pell order $Z [\sqrt{5}]$ from The norm on the explicit order $Z [\sqrt{D}]$ .

[F1]

The explicit Pell order $Z [\sqrt{D}]$ consists exactly of the elements $x + y \sqrt{D}$ with $x, y \in Z$ (The norm on the explicit order $Z [\sqrt{D}]$ ).

Counterexample

technique · direct

1.1

The element $u = \frac{1 + \sqrt{5}}{2}$ is a unit in $Z ⁣ [\frac{1 + \sqrt{5}}{2}]$ because $u (u - 1) = \frac{1 + \sqrt{5}}{2} \cdot \frac{\sqrt{5} - 1}{2} = 1.$

givenalgebra

2.1

If $u$ lay in $Z [\sqrt{5}]$ , then [F1] would give integers $x, y$ with $\frac{1 + \sqrt{5}}{2} = x + y \sqrt{5} .$ Comparing coefficients of $1$ and $\sqrt{5}$ yields $x = \frac{1}{2}$ and $y = \frac{1}{2}$ , impossible in $Z$ . So $u$ is a unit of the larger quadratic order that is not in $Z [\sqrt{5}]$ .

F1step 1.1algebra∎

Depends on

The norm on the explicit order $\mathbb{Z}[\sqrt{D}]$

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

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Sources

Keith Conrad, Pell's Equation, I (standard reference, not scraped)
Peter Hackman, Elementary Number Theory (standard reference, not scraped)