DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)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 fundamental Pell solution

Definition

By Every Pell equation has a positive nontrivial integral solution, the set of positive nontrivial solutions of Pell's equation is nonempty. The fundamental Pell solution is the positive solution

(x_{1}, y_{1}) \in Z_{> 0}^{2}, x_{1}^{2} - D y_{1}^{2} = 1,

whose first coordinate $x_{1}$ is least among all positive solutions.

Write $ε_{D} : = x_{1} + y_{1} \sqrt{D} \in Z [\sqrt{D}] .$ Since $x_{1} > 0$ and $y_{1} > 0$ , one has $ε_{D} > 1$ .

Depends on

Every Pell equation has a positive nontrivial integral solution

Used by

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

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