^{1/2}"usually" even? For example, if N runs over the ninety non-squares less than 100, then only 20 times does the continued fraction expansion of N

^{1/2}have an odd period. Of the 992 non-squares less than 1024, 157 have an odd period. Of the 9900 squares less than 10

^{4}, 1322 have an odd period. This is a sign that something is going on under the hood -- naively you'd expect half the periods to be odd.

Arnold has observed this, but only empirically; I first observed it from this problem from Project Euler.

The period of the continued fraction of N

^{1/2}is odd if and only if x

^{2}- Ny

^{2}= -1 has solutions in integers. All such integers, it turns out, have no prime factors congruent to 3 mod 4, which is pretty rare for large numbers. (The number of positive integers less than N with no prime factors congruent to 3 mod 4 is about N(log N)

^{-1/2}.) For integers having no prime factors congruent to 3 mod 4, though, a paper of Etienne Fouvry and Jurgen Kluners shows that asymptotically at least 52% of such numbers have odd period, and at most two-thirds do.

## 3 comments:

The paper(s) of Fouvry and Klueners have links to the original conjecture of P. Stevenhagen -- based on strong heuristics -- that proposed a conjecture and model for this problem:

"The Number of Real Quadratic Fields Having Units of Negative Norm", in Experimental Math. (1993)

Arnold, unfortunately, ignores this completely.

Emmanuel, thanks for the pointer.

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