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.

Generalized and negative Pell equations

Definition

Keep the same positive nonsquare integer $D$ as in Pell's equation, and let $N \in Z$ with $N \neq 0$ . The equation $x^{2} - D y^{2} = N, x, y \in Z,$ is the generalized Pell equation for $(D, N)$ .

The special case $x^{2} - D y^{2} = - 1$ is the negative Pell equation for $D$ .

The restriction $N \neq 0$ is part of the present convention: the page studies the nonzero norm classes whose solutions move by multiplication with norm-one Pell solutions.

Depends on

Pell's equation

Used by

One generalized Pell solution generates infinitely many more Corollary
Pell-equivalence for generalized solutions Definition
Generalized Pell solutions fall into finitely many Pell orbits Theorem
Negative Pell is soluble exactly for odd period length Theorem

Dependency tree · one level

1 result within one dependency step 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

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