Even and odd periods in continued fractions of square roots

The continued fraction for $\sqrt{N}$, where $N$ is a positive integer, has the periodic form $\sqrt{N}=[a_0,\overline{a_1,a_2,\ldots, a_l}\,],$ where $a_1,a_2,\ldots,a_{l-1}$ is a palindrome and $a_l=2a_0$. The period $l=l(N)$ is assumed to be of minimal length. We give several new results concerning the intriguing question: How can we distinguish between those integers $N$ for which $l(N)$ is even and those for which $l(N)$ is odd