Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck 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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The Pell equation for D=3

Example

For D=3 one finds 3=[1;1,2‾], so the period length is even. The least positive solution of x2−3y2=1 is 2+3, and the negative Pell equation x2−3y2=−1 has no integral solution.

Facts & Assumptions

Given: The positive nonsquare integer D=3.

[F1]

Negative Pell is soluble exactly for odd period length, and when the period length is even the convergent pℓ−1/qℓ−1 is the least positive norm-one solution (Negative Pell is soluble exactly for odd period length).

Verification

technique · direct
1.1F1givenalgebra

A direct continued-fraction calculation gives 3=[1;1,2‾], so the period length is ℓ=2. Then [F1] gives the least positive norm-one solution at p1/q1=2/1, hence 2+3, with 22−3⋅12=1.

2.1F1step 1.1∎

Because the period length is even, [F1] also says that x2−3y2=−1 has no integral solution.

Depends on

Used by

Nothing in the library uses this result yet.

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