## 04 June 2009

### Odd periods in continued fractions

Here's a question. Why is the period of the quotients in the continued fraction of N1/2 "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 N1/2 have an odd period. Of the 992 non-squares less than 1024, 157 have an odd period. Of the 9900 squares less than 104, 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 N1/2 is odd if and only if x2 - Ny2 = -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.

Emmanuel Kowalski said...

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.

Michael Lugo said...
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Michael Lugo said...

Emmanuel, thanks for the pointer.