On the continued fraction expansions of $(1+\protect \sqrt{pq})/2$ and $\protect \sqrt{pq}$

Abstract

The evenness and the values modulo $4$ of the lengths of the periods of the continued fraction expansions of $\sqrt{p}$ and $\sqrt{2p}$ for $p\equiv 3\pmod {4}$ a prime are known. Here we prove similar results for the continued fraction expansion of $\sqrt{pq}$, where $p,q\equiv 3\pmod {4}$ are distinct primes.