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V.I. Arnold has experimentally established that the limit of the statisticsof incomplete quotients of partial continued fractions of quadraticirrationalities coincides with the Gauss--Kuz'min statistics. Below we brieflyprove this fact for roots of the equation $r x^2+p x=q$ with fixed $p$ and $r$($r>0$), and with random $q$, $q\le R$, $R\to \infty$. In Section 3 we estimatethe sum of incomplete quotients of the period. According to the obtained bound,prior to the passage to the limit, incomplete quotients in average arelogarithmically small. We also upper estimate the proportion of the "red"numbers among those representable as a sum of two squares.

Statistics of incomplete quotients of continued fractions of quadratic irrationalities