Continued Fraction Expansion of Irrational Square Root

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Theorem

Let $n \in \Z$ such that $n$ is not a square.

Then the continued fraction expansion of $\sqrt n$ is of the form:

$\left[{a_1 \left \langle{b_1, b_2, \ldots, b_{m - 1}, b_m, b_{m - 1}, \ldots, b_2, b_1, 2 a_1}\right \rangle}\right]$

or

$\left[{a_1 \left \langle{b_1, b_2, \ldots, b_{m - 1}, b_m, b_m, b_{m - 1}, \ldots, b_2, b_1, 2 a_1}\right \rangle}\right]$

where $m \in \Z: m \ge 0$.


That is, it has the form as follows:

It is periodic
It starts with an integer $a_1$
Its cycle starts with a palindromic section, either:
$b_1, b_2, \ldots, b_{m - 1}, b_m, b_{m - 1}, \ldots, b_2, b_1$
or:
$b_1, b_2, \ldots, b_{m - 1}, b_m, b_m, b_{m - 1}, \ldots, b_2, b_1$
which may be of length zero
Its cycle ends with twice the first partial quotient.


Proof

We use:

Expansion of Associated Reduced Quadratic Irrational to establish a series of reduced quadratic irrationals associated to $n$

and then use

Finitely Many Reduced Associated Quadratic Irrationals to assert this series must repeat (hence be periodic) due to the finite number of such irrationals.


Let:

$a_0 = \left \lfloor {\sqrt n} \right \rfloor$
$\sqrt n = a_0 + \dfrac 1 {\alpha_0}$


For all $i \in \N: i > 0$, let:

$a_{i + 1} = \left \lfloor {\alpha_i} \right \rfloor$
$\alpha_i = a_{i + 1} + \dfrac 1 {\alpha_{i + 1}}$


From here, we will prove:

$\alpha_0$ is a reduced quadratic irrational associated to $n$
If all $\alpha_i: 0 \le i \le k$ are all reduced quadratic irrationals associated to $n$, then so is $\alpha_{k + 1}$.


Since $\dfrac 1 {\alpha_0}$ is the fractional part of the irrational $\sqrt n$, we have:

$0 < \dfrac 1 {\alpha_0} < 1 \implies \alpha_0 > 1$

By simple algebra, we have:

$\alpha_0 = \dfrac {a_0 + \sqrt n} {n - a_0^2}$
$\tilde{\alpha_0} = \dfrac {a_0 - \sqrt n} {n - a_0^2}$

where $\tilde{\alpha_0}$ is the conjugate of $\alpha_0$.


Since $a_0$ is the floor of $\sqrt n$, we know that:

$a_0 - \sqrt n < 0 \implies \tilde{\alpha_0} < 0$

Since $n \in \Z \implies \sqrt n > 1$ and $\sqrt n > a_0$, we have:

\(\displaystyle 1\) \(<\) \(\displaystyle \sqrt n + a_0\)
\(\displaystyle \implies \ \ \) \(\displaystyle \sqrt n - a_0\) \(<\) \(\displaystyle n - a_0^2\)
\(\displaystyle \implies \ \ \) \(\displaystyle a_0 - \sqrt n\) \(>\) \(\displaystyle -\left({n - a_0^2}\right)\)
\(\displaystyle \implies \ \ \) \(\displaystyle \tilde{\alpha_0}\) \(>\) \(\displaystyle -1\)


Thus $\alpha_0$ is a reduced quadratic irrational.


Since $P = a_0$ and $Q = n - a_0^2 = n - P^2$, $Q$ clearly divides $n - P^2$.

Thus $\alpha_0$ is associated to $n$ as well.


We have that each $\alpha_i$ is a reduced quadratic irrational.

So, following the recurrence defined, each $a_i \ge 1$.

Also, by Expansion of Associated Reduced Quadratic Irrational, each $\alpha_{i + 1}$ is reduced and associated to $n$ since $\alpha_0$ is.

By Finitely Many Reduced Associated Quadratic Irrationals, we only have finitely many choices for these.

Hence there must be some smallest $k$ for which $\alpha_k = \alpha_0$.

Since $\alpha_{i + 1}$ is determined completely by $\alpha_i$ we will then have:

$\alpha_{k + j} = \alpha_j$

for all $j > 0$.

Hence the $\alpha_i$ are periodic.

Similarly, as the $a_i$ for $i > 0$ are determined completely by $\alpha_{i - 1}$, the $a_i$ must be periodic as well.

This forces the continued fraction expansion:

$\sqrt n = a_0 + \cfrac 1 {a_1 + \cfrac 1 {a_2 + \ddots} }$

to be periodic.

Examples

\(\displaystyle \sqrt 2\) \(=\) \(\displaystyle \left[{1, \left \langle{2}\right \rangle}\right]\)
\(\displaystyle \sqrt 3\) \(=\) \(\displaystyle \left[{1, \left \langle{1, 2}\right \rangle}\right]\)
\(\displaystyle \sqrt 5\) \(=\) \(\displaystyle \left[{2, \left \langle{4}\right \rangle}\right]\)
\(\displaystyle \sqrt 6\) \(=\) \(\displaystyle \left[{2, \left \langle{2, 4}\right \rangle}\right]\)
\(\displaystyle \sqrt 7\) \(=\) \(\displaystyle \left[{2, \left \langle{1, 1, 1, 4}\right \rangle}\right]\)
\(\displaystyle \sqrt {13}\) \(=\) \(\displaystyle \left[{3, \left \langle{1, 1, 1, 1, 6}\right \rangle}\right]\)
\(\displaystyle \sqrt {19}\) \(=\) \(\displaystyle \left[{4, \left \langle{2, 1, 3, 1, 2, 8}\right \rangle}\right]\)
\(\displaystyle \sqrt {28}\) \(=\) \(\displaystyle \left[{5, \left \langle{3, 2, 3, 10}\right \rangle}\right]\)
\(\displaystyle \sqrt {31}\) \(=\) \(\displaystyle \left[{5, \left \langle{1, 1, 3, 5, 3, 1, 1, 10}\right \rangle}\right]\)