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The negative Pell equation for $D = 5$

Example

For $D = 5$ one has $\sqrt{5} = [2; \overline{4}] .$ So the negative Pell equation has least positive solution $2 + \sqrt{5},$ and squaring it gives the fundamental positive solution $9 + 4 \sqrt{5} .$

Facts & Assumptions

Given: The positive nonsquare integer $D = 5$ .

[F1]

Negative Pell is soluble exactly for odd period length, and in that case the least positive negative-Pell solution is $p_{ℓ - 1} / q_{ℓ - 1}$ while the least positive norm-one solution is $p_{2 ℓ - 1} / q_{2 ℓ - 1}$ (Negative Pell is soluble exactly for odd period length).

Verification

technique · direct

1.1

A direct continued-fraction calculation gives $\sqrt{5} = [2; \overline{4}]$ , so $ℓ = 1$ . Fact [F1] therefore gives the least positive negative-Pell solution $p_{0} / q_{0} = 2 / 1,$ that is, $2 + \sqrt{5}$ , and indeed $2^{2} - 5 \cdot 1^{2} = - 1.$

F1givenalgebra

2.1

Squaring the negative-Pell solution yields $(2 + \sqrt{5})^{2} = 9 + 4 \sqrt{5},$ and $9^{2} - 5 \cdot 4^{2} = 1.$ By [F1], this is the least positive norm-one solution.

F1step 1.1algebra∎

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 negative Pell equation for D=5D=5