CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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Generalized Pell solubility is decidable by bounded search

Statement

For fixed positive nonsquare $D$ and fixed nonzero $N \in Z$ , the generalized Pell equation $x^{2} - D y^{2} = N$ is decidable by a finite search over the explicit representative bounds of Generalized Pell solutions fall into finitely many Pell orbits.

Facts & Assumptions

Given: A fixed pair $(D, N)$ with $D$ a positive nonsquare integer and $N \neq 0$ .

[F1]

Every solution of $x^{2} - D y^{2} = N$ is Pell-equivalent to a solution in an explicit bounded rectangle, and there are only finitely many Pell-equivalence classes (Generalized Pell solutions fall into finitely many Pell orbits).

Proof

technique · direct

1.1

Step through all integer pairs $(x, y)$ in the finite rectangle supplied by [F1], keep the pairs satisfying $x^{2} - D y^{2} = N$ , and retain one pair from each Pell-equivalence class among those finitely many solutions.

F1given

2.1

By [F1], the search succeeds exactly when the generalized Pell equation has an integral solution, and any retained list of representatives generates every solution by multiplying with powers of the fundamental Pell unit. So the finite search decides solubility.

F1step 1.1∎

Depends on

Generalized Pell solutions fall into finitely many Pell orbits

Used by

Bounded representatives for x² - 7y² = 57 Example

Dependency tree · two levels

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Sources

Keith Conrad, Pell's Equation, II (standard reference, not scraped)