CounterexampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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A generalized Pell solution need not be a convergent

Statement refuted

Every integral solution of a generalized Pell equation $x^{2} - D y^{2} = N$ is a convergent of $\sqrt{D}$ .

Witness

The element $3 + \sqrt{6}$ solves $x^{2} - 6 y^{2} = 3,$ but the rational number $3 / 1$ is not a convergent of $\sqrt{6}$ .

Facts & Assumptions

Given: The generalized Pell solution $3 + \sqrt{6}$ .

[F1]

If a reduced rational number $r / s$ satisfies $∣ \sqrt{D} - \frac{r}{s} ∣ < \frac{1}{2 s^{2}},$ then $r / s$ is a convergent of $\sqrt{D}$ (Legendre's criterion for convergents).

Counterexample

technique · direct

1.1givenalgebra

One checks directly that $3^{2} - 6 \cdot 1^{2} = 3,$ so $3 + \sqrt{6}$ is a generalized Pell solution. A direct continued-fraction calculation gives $\sqrt{6} = [2; \overline{2, 4}],$ whose convergents begin $2, \frac{5}{2}, \frac{22}{9}, \dots .$ Thus $3 / 1$ is not a convergent of $\sqrt{6}$ .

2.1F1step 1.1algebra∎

The missing hypothesis is exactly the small-error bound in [F1]: here $∣ \sqrt{6} - \frac{3}{1} ∣ = 3 - \sqrt{6} > \frac{1}{2} = \frac{1}{2 \cdot 1^{2}},$ so [F1] does not apply. Therefore generalized Pell solutions need not all be convergents.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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, I (standard reference, not scraped)
Peter Hackman, Elementary Number Theory (standard reference, not scraped)