Periodic continued fraction of a quadratic irrational

The simple continued fraction of a quadratic irrational is eventually periodic. For a nonsquare positive integer

D

D

is periodic after its integer part.

Purely periodic continued fraction

A continued fraction is purely periodic when its repeating block begins at its first partial quotient:

[\overline{a_{0}, \dots, a_{n}}] = [a_{0}, \dots, a_{n}, a_{0}, \dots, a_{n}, \dots] .

Let

θ = [\overline{a_{0}, \dots, a_{n}}]

, and let

θ^{'}

M (a_{0}) \dots M (a_{n})

. Then

- \frac{1}{θ ^{'}} = [\overline{a_{n}, \dots, a_{0}}] .

The identity follows by transposing the continued-fraction matrix and substituting

X = - 1/ Y

Successive complete quotients repeat after two steps:

3 = [1; \overline{1, 2}] .

Rationalizing successive complete quotients gives

11 = [3; \overline{3, 6}] .

p_{n} / q_{n}

are the convergents of

11

, then

p_{n + 1} + q_{n + 1} 11 = ⎩ ⎨ ⎧ (3 + 11) (p_{n} + q_{n} 11), \frac{3 + 11}{2} (p_{n} + q_{n} 11), n odd, n even .

For the even convergents of

11

p_{2 m} + q_{2 m} 11 = (3 + 11) (10 + 311)^{m} .

Since the second factor has norm one, every such convergent satisfies

p_{2 m}^{2} - 11 q_{2 m}^{2} = - 2

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