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Generalized Pell orbits for $x^{2} - 6 y^{2} = 3$

Example

For $x^{2} - 6 y^{2} = 3,$ the bounded representatives are exactly $\pm (3 + \sqrt{6}),$ and every integral solution is $\pm (3 + \sqrt{6}) (5 + 2 \sqrt{6})^{k}, k \in Z .$

Facts & Assumptions

Given: The equation $x^{2} - 6 y^{2} = 3$ .

[F1]

Every positive Pell solution is a power of the fundamental unit (All positive Pell solutions are powers of the fundamental solution).

[F2]

Every generalized Pell solution is Pell-equivalent to one in the explicit bounded rectangle given by the orbit theorem (Generalized Pell solutions fall into finitely many Pell orbits).

Verification

technique · direct

1.1F1F2givenalgebra

The element $5 + 2 \sqrt{6}$ has norm $1$ , and there is no positive norm-one solution with $y = 1$ because $x^{2} = 7$ is impossible; hence $5 + 2 \sqrt{6}$ is the fundamental Pell unit, so [F1] identifies the Pell powers with $(5 + 2 \sqrt{6})^{k}$ . The element $3 + \sqrt{6}$ has norm $3$ . For $N = 3$ and $ε_{6} = 5 + 2 \sqrt{6}$ , the bound of [F2] is $∣ x_{0} ∣ \leq \frac{\sqrt{3}}{2} (\sqrt{5 + 2 \sqrt{6}} + (5 + 2 \sqrt{6})^{- 1 / 2}) = 3,$ $∣ y_{0} ∣ \leq \frac{\sqrt{3}}{2 \sqrt{6}} (\sqrt{5 + 2 \sqrt{6}} + (5 + 2 \sqrt{6})^{- 1 / 2}) < 2.$

2.1F2step 1.1algebra∎

The only integer solutions in that box are $(x, y) = (\pm 3, \pm 1)$ . Moreover, $(3 - \sqrt{6}) = (3 + \sqrt{6}) (5 - 2 \sqrt{6}) = (3 + \sqrt{6}) (5 + 2 \sqrt{6})^{- 1},$ so $(3, 1)$ and $(3, - 1)$ lie in the same Pell orbit, and multiplying by $- 1$ gives the negative orbit. Therefore every solution of $x^{2} - 6 y^{2} = 3$ is $\pm (3 + \sqrt{6}) (5 + 2 \sqrt{6})^{k} (k \in Z) .$

Depends on

Used by

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

Generalized Pell orbits for x2−6y2=3x2−6y2=3