The parity of the period of the continued fraction of d

We call a positive square-free integer d special, if d is not divisible by primes congruent to 3 mod 4. We show that the period of the expansion of in continued fractions is asymptotically more often odd than even, when we restrict to special integers. We note that this period is always even for a non-special square-free integer d. It is well known that the above period is odd if and only if the negative Pell equation x²−dy²=−1 is solvable. The latter problem is solvable if and only if the narrow and the ordinary class groups of ℚ(√d) are equal. In a prior work we fully described the asymptotics of the 4-ranks of those class groups. Here we get the first non-trivial results about the asymptotic behavior of the 8-rank of the narrow class group. For example, we show that more than 76% of the quadratic fields ℚ(√d), where d is special, have the property that the 8-rank of the narrow class group is zero.