On length of the period of the continued fraction of nd

For a given quadratic irrational [Formula: see text], let us denote by [Formula: see text] the length of the periodic part of the continued fraction expansion of [Formula: see text]. We prove that for every positive integer [Formula: see text], which is not a perfect square, there are infinitely many even integers [Formula: see text], for which the equality [Formula: see text] holds for infinitely many integers [Formula: see text].