DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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
Real quadratic units and Pell's equation Example
Generalized Pell solutions fall into finitely many Pell orbits Theorem
Negative Pell is soluble exactly for odd period length Theorem

Dependency tree · one level

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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)