DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge 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.

Pell's equation

Definition

Fix a positive integer $D$ that is not a square. The Diophantine equation $x^{2} - D y^{2} = 1, x, y \in Z,$ is Pell's equation for $D$ .

An integral solution of Pell's equation is a pair $(x, y) \in Z^{2}$ satisfying the displayed equation. A positive nontrivial solution is an integral solution with $x > 0, y > 0.$ The adjective "nontrivial" excludes the obvious solutions $(\pm 1, 0)$ .

Used by

Generalized and negative Pell equations Definition
Integral Pell solutions form an abelian group Proposition
Every Pell equation has a positive nontrivial integral solution Theorem
Negative Pell is soluble exactly for odd period length Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

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