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I want to know the proof that the continued fraction of $\sqrt{n}$ has a period.

This question and answer prove that when the continued fraction has a period, it can be represented by quadratic form. However, it doesn't prove that the continued fraction of quadratic form have a period. According to wikipedia, Lagrange prove it.

"Amazingly, Lagrange's discovery implies that the canonical continued fraction expansion of the square root of every non-square integer is periodic"

Does anybody know how to prove this?

ueir
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  • use this link https://math.stackexchange.com/questions/3369147/problem-understanding-lagranges-continued-fraction-theorem –  Feb 12 '20 at 10:47

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I would recommend to read the posts of Ben Lynn on continued fractions. In particular, in the second theorem of this post, he proves what you are basically asking.

The sketch of the prove is the following:

  1. Show that tail fractions $x_k=[a_k, a_{k+1},\ldots]$ of the number $\sqrt{n}=x=[a_0;a_1,a_2,\ldots]$ are the roots of a quadratic equation $Az^2+Bz+C=0$ with integer coefficients.
  2. Show that coefficients $A$, $B$, $C$ are bounded, i.e. $|A|,|B|,|C|<f(x)$.
  3. Since $A,B,C$ are integer and bounded, the number of possible equations $Az^2+Bz+C=0$ is finite, and so is the number of possible values for $x_k$.
  4. That means for some $k,m$: $x_k=x_m$, which means the continued fraction has to repeat itself.
Vasily Mitch
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  • Thank you. How he get $|ru-ts|=1$? – ueir Feb 13 '20 at 14:02
  • By induction you can show that $\begin{pmatrix}p_n&p_{n-1}\q_n&q_{n-1}\end{pmatrix}=\begin{pmatrix}a_0&1\1&0\end{pmatrix}\begin{pmatrix}a_1&1\1&0\end{pmatrix}\cdots\begin{pmatrix}a_n&1\1&0\end{pmatrix}$. Taking the determinant, you show that $p_nq_{n-1}-p_{n-1}q_n=(-1)^n$ – Vasily Mitch Feb 14 '20 at 10:47