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

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One generalized Pell solution generates infinitely many more

Statement

If the generalized Pell equation $x^{2} - D y^{2} = N, N \neq 0,$ has one integral solution, then it has infinitely many integral solutions.

Facts & Assumptions

Given: A nonzero solution $α = x + y \sqrt{D}$ of $N_{D} (α) = N$ and the fundamental Pell solution $ε_{D} > 1$ .

[F1]

The Pell norm is multiplicative (The Pell norm is multiplicative).

[F2]

The fundamental Pell solution satisfies $ε_{D} > 1$ (The fundamental Pell solution).

Proof

technique · direct

1.1

For every integer $k$ , multiplicativity [F1] gives $N_{D} (α ε_{D}^{k}) = N_{D} (α) N_{D} (ε_{D})^{k} = N,$ so each $α ε_{D}^{k}$ is again a solution of the same generalized Pell equation.

F1givenalgebra

2.1

These solutions are all distinct: if $k < ℓ$ , then [F2] gives $∣ α ε_{D}^{ℓ} ∣ = ∣ α ε_{D}^{k} ∣ ε_{D}^{ℓ - k} > ∣ α ε_{D}^{k} ∣ .$ Hence the first real embeddings have distinct absolute values, so the elements themselves are distinct. Therefore one solution generates infinitely many others.

F2step 1.1algebra∎

Depends on

Used by

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Dependency tree · two levels

6 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, II (standard reference, not scraped)
Peter Hackman, Elementary Number Theory (standard reference, not scraped)