# Revision history [back]

As a continuation of the reply by Benjamin, Let me give some references and give more information.

Both Theorems appear in the book "Continued Fractions" By Andrew Mansfield Rockett and Peter Szüsz. The first is Theorem 3 on page 45, and the second is a remark before Theorem 1 on page 50. The second is not stated correctly (there should not be any coeffiecient before the square root):

Theorem(Lagrange Estimate): Let $t= \frac{P_0+\sqrt{D}}{Q_0}$ with $D$ not a perfect square and $P_0,Q_0$ are integers. Then the length of the period of $t$ is at most $2D$.

A better asymptotics is given by the following (which is Theorem 1 of page 50):

Theorem: Let $t$ be as above with the addtional assumption that $Q_0$ divide $D-P_0^2$. Then if $L(t)$ denoted the length of the period of $t$ then $$L(t)=O(\sqrt{D}log(D))$$

I'm not sure what the constant is (it is related to the divisor function)... I didn't read the proof yet!