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The Pell and negative Pell equations for $D = 2$

Example

For $D = 2$ one has $\sqrt{2} = [1; \overline{2}] .$ Hence $1^{2} - 2 \cdot 1^{2} = - 1, 3^{2} - 2 \cdot 2^{2} = 1,$ so the negative Pell equation has least positive solution $1 + \sqrt{2}$ , while the fundamental Pell solution is $ε_{2} = 3 + 2 \sqrt{2} = (1 + \sqrt{2})^{2} .$

The first positive Pell solutions are therefore $3 + 2 \sqrt{2}, 17 + 12 \sqrt{2}, 99 + 70 \sqrt{2}, \dots .$

Facts & Assumptions

Given: The positive nonsquare integer $D = 2$ .

[F1]

Negative Pell is soluble exactly for odd period length, and when the period length is odd 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).

[F2]

Every positive Pell solution is a unique positive power of the fundamental unit $ε_{D}$ (All positive Pell solutions are powers of the fundamental solution).

Verification

technique · direct

1.1

A direct continued-fraction calculation gives $\sqrt{2} = [1; \overline{2}]$ , so the period length is $ℓ = 1$ . Fact [F1] therefore gives the least positive negative-Pell solution $p_{0} / q_{0} = 1 / 1,$ hence $1 + \sqrt{2}$ , and the least positive norm-one solution $p_{1} / q_{1} = 3 / 2,$ hence $ε_{2} = 3 + 2 \sqrt{2} = (1 + \sqrt{2})^{2}$ .

F1givenalgebra

2.1

By [F2], every positive Pell solution for $D = 2$ is a power of $ε_{2}$ . Multiplying out the first powers gives $ε_{2}^{2} = (3 + 2 \sqrt{2})^{2} = 17 + 12 \sqrt{2}, ε_{2}^{3} = (17 + 12 \sqrt{2}) (3 + 2 \sqrt{2}) = 99 + 70 \sqrt{2} .$

F2step 1.1algebra∎

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Sources

Keith Conrad, Pell's Equation, I (standard reference, not scraped)
Peter Hackman, Elementary Number Theory (standard reference, not scraped)

The Pell and negative Pell equations for D=2D=2