The parity of the period of the continued fraction of d

Abstract

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 �ad 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(�ad) 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(�ad), where d is special, have the property that the 8-rank of the narrow class group is zero.