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The Pell equation for $D = 3$

Example

For $D = 3$ one finds $\sqrt{3} = [1; \overline{1, 2}],$ so the period length is even. The least positive solution of $x^{2} - 3 y^{2} = 1$ is $2 + \sqrt{3},$ and the negative Pell equation $x^{2} - 3 y^{2} = - 1$ has no integral solution.

Facts & Assumptions

Given: The positive nonsquare integer $D = 3$ .

[F1]

Negative Pell is soluble exactly for odd period length, and when the period length is even the convergent $p_{ℓ - 1} / q_{ℓ - 1}$ is the least positive norm-one solution (Negative Pell is soluble exactly for odd period length).

Verification

technique · direct

1.1

A direct continued-fraction calculation gives $\sqrt{3} = [1; \overline{1, 2}]$ , so the period length is $ℓ = 2$ . Then [F1] gives the least positive norm-one solution at $p_{1} / q_{1} = 2 / 1,$ hence $2 + \sqrt{3}$ , with $2^{2} - 3 \cdot 1^{2} = 1.$

F1givenalgebra

2.1

Because the period length is even, [F1] also says that $x^{2} - 3 y^{2} = - 1$ has no integral solution.

F1step 1.1∎

Depends on

Negative Pell is soluble exactly for odd period length

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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)

The Pell equation for D=3D=3